#set document(title: "11.5 Understand Slope of a Line", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 11.5#h(0.6em)Understand Slope of a Line #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Before you get started, take this readiness quiz. Simplify: #math.equation(block: false, alt: "the fraction 1 minus 4 over 8 minus 2 .")[$frac(1 − 4, 8 − 2) .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Divide: #math.equation(block: false, alt: "the fraction 0 over 4 , the fraction 4 over 0 .")[$frac(0, 4) , frac(4, 0) .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "0, undefined")[$"0, undefined"$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 15 over −3 , the fraction −15 over 3 , the fraction −15 over −3 .")[$frac(15, −3) , frac(−15, 3) , frac(−15, −3) .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "minus 5 , minus 5 , 5")[$− 5 , − 5 , 5$] ] ] As we’ve been graphing linear equations, we’ve seen that some lines slant up as they go from left to right and some lines slant down. Some lines are very steep and some lines are flatter. What determines whether a line slants up or down, and if its slant is steep or flat? The steepness of the slant of a line is called the #strong[slope of the line]. The concept of slope has many applications in the real world. The pitch of a roof and the grade of a highway or wheelchair ramp are just some examples in which you literally see slopes. And when you ride a bicycle, you feel the slope as you pump uphill or coast downhill. === Use Geoboards to Model Slope In this section, we will explore the concepts of slope. Using rubber bands on a #strong[geoboard] gives a concrete way to model lines on a coordinate grid. By stretching a rubber band between two pegs on a geoboard, we can discover how to find the slope of a line. We’ll start by stretching a rubber band between two pegs to make a line as shown. #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 4 and the point in column 4 row 2.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 4 and the point in column 4 row 2.", caption: none) Does it look like a line? Now we stretch one part of the rubber band straight up from the left peg and around a third peg to make the sides of a right triangle as shown. We carefully make a #math.equation(block: false, alt: "90 °")[$90 °$] angle around the third peg, so that one side is vertical and the other is horizontal. #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 4,and column 4 row 2.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 4,and column 4 row 2.", caption: none) To find the slope of the line, we measure the distance along the vertical and horizontal legs of the triangle. The vertical distance is called the #strong[#emph[rise]] and the horizontal distance is called the #strong[#emph[run]], as shown. #figure(figph[This figure shows two arrows. The first arrow is vertical and is labeled “rise”. The second arrow begins at the end of the first arrow extending to the right and is labeled “run”.], alt: "This figure shows two arrows. The first arrow is vertical and is labeled “rise”. The second arrow begins at the end of the first arrow extending to the right and is labeled “run”.", caption: none) To help remember the terms, it may help to think of the images shown. #figure(figph[The figure shows an image of a hot air balloon signifying rise as the balloon rises straight up, similar to a y-axis. The second image is of a person jogging, signifying run as the person runs as if they are on an x-axis], alt: "The figure shows an image of a hot air balloon signifying rise as the balloon rises straight up, similar to a y-axis. The second image is of a person jogging, signifying run as the person runs as if they are on an x-axis", caption: none) On our geoboard, the rise is #math.equation(block: false, alt: "2")[$2$] units because the rubber band goes up #math.equation(block: false, alt: "2")[$2$] spaces on the vertical leg. What is the run? Be sure to count the spaces between the pegs rather than the pegs themselves! The rubber band goes across #math.equation(block: false, alt: "3")[$3$] spaces on the horizontal leg, so the run is #math.equation(block: false, alt: "3")[$3$] units. #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 4, and column 4 row 2. The triangle has a rise of 2 units and a run of 3 units.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 4, and column 4 row 2. The triangle has a rise of 2 units and a run of 3 units.", caption: none) The slope of a line is the ratio of the rise to the run. So the slope of our line is #math.equation(block: false, alt: "the fraction 2 over 3 .")[$frac(2, 3) .$] In mathematics, the slope is always represented by the letter #math.equation(block: false, alt: "m .")[$m .$] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Slope of a line] The slope of a line is #math.equation(block: false, alt: "m equals the fraction rise over run .")[$m = frac("rise", "run") .$] The #strong[rise] measures the vertical change and the #strong[run] measures the horizontal change. ] What is the slope of the line on the geoboard in ? #math.equation(block: true, alt: "m equals the fraction rise over run")[$#h(1em) m = frac("rise", "run")$]#math.equation(block: true, alt: "m equals the fraction 2 over 3")[$m = frac(2, 3)$]#math.equation(block: true, alt: "The line has slope the fraction 2 over 3 .")[$"The line has slope" #h(0.2em) frac(2, 3) .$]When we work with geoboards, it is a good idea to get in the habit of starting at a peg on the left and connecting to a peg to the right. Then we stretch the rubber band to form a right triangle. If we start by going up the rise is positive, and if we stretch it down the rise is negative. We will count the run from left to right, just like you read this paragraph, so the run will be positive. Since the slope formula has rise over run, it may be easier to always count out the rise first and then the run. #examplebox("Example 1")[][ What is the slope of the line on the geoboard shown? #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 5 and the point in column 5 row 2.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 5 and the point in column 5 row 2.", caption: none) #solutionbox[ Use the definition of slope. #math.equation(block: true, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$] Start at the left peg and make a right triangle by stretching the rubber band up and to the right to reach the second peg. Count the rise and the run as shown. #linebreak() #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 5,and column 5 row 2.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 5,and column 5 row 2.", caption: none) #math.equation(block: true, alt: "The rise is 3 units ., m equals the fraction 3 over run; The run is 4 units ., m equals the fraction 3 over 4; The slope is the fraction 3 over 4 .")[$"The rise is" #h(0.2em) 3 #h(0.2em) "units" . & & & m = frac(3, "run") \ "The run is" #h(0.2em) 4 #h(0.2em) "units" . & & & m = frac(3, 4) \ & & & "The slope is" #h(0.2em) frac(3, 4) .$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ What is the slope of the line on the geoboard shown? #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 5 and the point in column 4 row 1.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 5 and the point in column 4 row 1.", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 4 over 3")[$frac(4, 3)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ What is the slope of the line on the geoboard shown? #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 4 and the point in column 5 row 3.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 4 and the point in column 5 row 3.", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 4")[$frac(1, 4)$] ] ] #examplebox("Example 2")[][ What is the slope of the line on the geoboard shown? #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 3 and the point in column 4 row 4.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 3 and the point in column 4 row 4.", caption: none) #solutionbox[ Use the definition of slope. #math.equation(block: true, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$] Start at the left peg and make a right triangle by stretching the rubber band to the peg on the right. This time we need to stretch the rubber band down to make the vertical leg, so the rise is negative. #linebreak() #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 3, column 1 row 4,and column 4 row 4.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 3, column 1 row 4,and column 4 row 4.", caption: none) #math.equation(block: true, alt: "The rise is −1 ., m equals the fraction −1 over run; The run is 3 ., m equals the fraction −1 over 3; m equals minus the fraction 1 over 3; The slope is minus the fraction 1 over 3 .")[$"The rise is" #h(0.2em) −1 . & & & m = frac(−1, "run") \ "The run is" #h(0.2em) 3 . & & & m = frac(−1, 3) \ & & & m = − frac(1, 3) \ & & & "The slope is" #h(0.2em) − frac(1, 3) .$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ What is the slope of the line on the geoboard? #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 2 and the point in column 4 row 4.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 2 and the point in column 4 row 4.", caption: none) #solutionbox[ #math.equation(block: true, alt: "minus the fraction 2 over 3")[$− frac(2, 3)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ What is the slope of the line on the geoboard? #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 1 and the point in column 4 row 5.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 1 and the point in column 4 row 5.", caption: none) #solutionbox[ #math.equation(block: true, alt: "minus the fraction 4 over 3")[$− frac(4, 3)$] ] ] Notice that in the first example, the slope is positive and in the second example the slope is negative. Do you notice any difference in the two lines shown. #figure(figph[The figure shows two grids of evenly spaced dots. There are 5 rows and 5 columns in each. In the left grid A, a rubberband loops to connect the point in column 1, row 1 and the point in column 5, row 4. In the right grid B, a rubber band loops to connect the point in column 1, row 4 and the point in column 4, row 2.], alt: "The figure shows two grids of evenly spaced dots. There are 5 rows and 5 columns in each. In the left grid A, a rubberband loops to connect the point in column 1, row 1 and the point in column 5, row 4. In the right grid B, a rubber band loops to connect the point in column 1, row 4 and the point in column 4, row 2.", caption: none) As you read from left to right, the line in Figure A, is going up; it has positive slope. The line Figure B is going down; it has negative slope. #figure(figph[This image shows two arrows: the left arrow is labeled positive slope and points upward towards the right. The right arrow is labeled negative slope and points downward towards the right.], alt: "This image shows two arrows: the left arrow is labeled positive slope and points upward towards the right. The right arrow is labeled negative slope and points downward towards the right.", caption: none) #examplebox("Example 3")[][ Use a geoboard to model a line with slope #math.equation(block: false, alt: "the fraction 1 over 2 .")[$frac(1, 2) .$] #solutionbox[ To model a line with a specific slope on a geoboard, we need to know the rise and the run. #figure(table( columns: 2, align: left, inset: 6pt, table.header([Use the slope formula.], [#math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$]]), [Replace #math.equation(block: false, alt: "m")[$m$] with #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$].], [#math.equation(block: false, alt: "the fraction 1 over 2 equals the fraction rise over run")[$frac(1, 2) = frac("rise", "run")$]], )) So, the rise is #math.equation(block: false, alt: "1")[$1$] unit and the run is #math.equation(block: false, alt: "2")[$2$] units. Start at a peg in the lower left of the geoboard. Stretch the rubber band up #math.equation(block: false, alt: "1")[$1$] unit, and then right #math.equation(block: false, alt: "2")[$2$] units. #linebreak() #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 3, column 1 row 4,and column 3 row 3.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 3, column 1 row 4,and column 3 row 3.", caption: none) The hypotenuse of the right triangle formed by the rubber band represents a line with a slope of #math.equation(block: false, alt: "the fraction 1 over 2 .")[$frac(1, 2) .$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use a geoboard to model a line with the given slope: #math.equation(block: false, alt: "m equals the fraction 1 over 3 .")[$m = frac(1, 3) .$] #solutionbox[ #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 2 row 3, column 2 row 4,and column 5 row 3.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 2 row 3, column 2 row 4,and column 5 row 3.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use a geoboard to model a line with the given slope: #math.equation(block: false, alt: "m equals the fraction 3 over 2 .")[$m = frac(3, 2) .$] #solutionbox[ #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 1, column 1 row 4,and column 3 row 1.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 1, column 1 row 4,and column 3 row 1.", caption: none) ] ] #examplebox("Example 4")[][ Use a geoboard to model a line with slope #math.equation(block: false, alt: "the fraction −1 over 4 ,")[$frac(−1, 4) ,$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Use the slope formula.], [#math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$]]), [Replace #math.equation(block: false, alt: "m")[$m$] with #math.equation(block: false, alt: "− the fraction 1 over 4")[$− frac(1, 4)$].], [#math.equation(block: false, alt: "− the fraction 1 over 4 equals the fraction rise over run")[$− frac(1, 4) = frac("rise", "run")$]], )) So, the rise is #math.equation(block: false, alt: "−1")[$−1$] and the run is #math.equation(block: false, alt: "4 .")[$4 .$] Since the rise is negative, we choose a starting peg on the upper left that will give us room to count down. We stretch the rubber band down #math.equation(block: false, alt: "1")[$1$] unit, then to the right #math.equation(block: false, alt: "4")[$4$] units. #linebreak() #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 3,and column 5 row 3.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 3,and column 5 row 3.", caption: none) The hypotenuse of the right triangle formed by the rubber band represents a line whose slope is #math.equation(block: false, alt: "minus the fraction 1 over 4 .")[$− frac(1, 4) .$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use a geoboard to model a line with the given slope: #math.equation(block: false, alt: "m equals the fraction −2 over 1 .")[$m = frac(−2, 1) .$] #solutionbox[ #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 2 row 3, column 2 row 5,and column 3 row 5.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 2 row 3, column 2 row 5,and column 3 row 5.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use a geoboard to model a line with the given slope: #math.equation(block: false, alt: "m equals the fraction −1 over 3 .")[$m = frac(−1, 3) .$] #solutionbox[ #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 1, column 1 row 2,and column 4 row 2.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 1, column 1 row 2,and column 4 row 2.", caption: none) ] ] === Find the Slope of a Line from its Graph Now we’ll look at some graphs on a #strong[coordinate grid] to find their slopes. The method will be very similar to what we just modeled on our geoboards. To find the slope, we must count out the #strong[rise] and the #strong[run]. But where do we start? We locate any two points on the line. We try to choose points with coordinates that are integers to make our calculations easier. We then start with the point on the left and sketch a right triangle, so we can count the rise and run. #examplebox("Example 5")[][ Find the slope of the line shown: #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -1 to 6. The y-axis runs from -4 to 2. A line passes through the points “ordered pair 5, 1” and “ordered pair 0, -3”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -1 to 6. The y-axis runs from -4 to 2. A line passes through the points “ordered pair 5, 1” and “ordered pair 0, -3”.", caption: none) #solutionbox[ Locate two points on the graph, choosing points whose coordinates are integers. We will use #math.equation(block: false, alt: "open parenthesis 0 , −3 close parenthesis")[$( 0 , −3 )$] and #math.equation(block: false, alt: "open parenthesis 5 , 1 close parenthesis .")[$( 5 , 1 ) .$] Starting with the point on the left, #math.equation(block: false, alt: "open parenthesis 0 , −3 close parenthesis ,")[$( 0 , −3 ) ,$] sketch a right triangle, going from the first point to the second point, #math.equation(block: false, alt: "open parenthesis 5 , 1 close parenthesis .")[$( 5 , 1 ) .$] #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -1 to 6. The y-axis runs from -4 to 2. A line passes through the points “ordered pair 5, 1” and “ordered pair 0, -3”. Two line segments form a triangle with the line. A horizontal line connects “ordered pair 0, 1” and “ordered pair 5,1 ”. A vertical line segment connects “ordered pair 0, -3” and “ordered pair 0, 1”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -1 to 6. The y-axis runs from -4 to 2. A line passes through the points “ordered pair 5, 1” and “ordered pair 0, -3”. Two line segments form a triangle with the line. A horizontal line connects “ordered pair 0, 1” and “ordered pair 5,1 ”. A vertical line segment connects “ordered pair 0, -3” and “ordered pair 0, 1”.", caption: none)]), [Count the rise on the vertical leg of the triangle.], [The rise is 4 units.], [Count the run on the horizontal leg.], [The run is 5 units.], [Use the slope formula.], [#math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$]], [Substitute the values of the rise and run.], [#math.equation(block: false, alt: "m equals the fraction 4 over 5")[$m = frac(4, 5)$]], [], [The slope of the line is #math.equation(block: false, alt: "the fraction 4 over 5")[$frac(4, 5)$].], )) Notice that the slope is positive since the line slants upward from left to right. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope of the line: #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -8 to 1. The y-axis runs from -1 to 4. A line passes through the points “ordered pair -8, 1” and “ordered pair 0, 3”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -8 to 1. The y-axis runs from -1 to 4. A line passes through the points “ordered pair -8, 1” and “ordered pair 0, 3”.", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 2 over 5")[$frac(2, 5)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope of the line: #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -2 to 6. The y-axis runs from -2 to 4. A line passes through the points “ordered pair 4, 2” and “ordered pair 0, -1”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -2 to 6. The y-axis runs from -2 to 4. A line passes through the points “ordered pair 4, 2” and “ordered pair 0, -1”.", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over 4")[$frac(3, 4)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Find the slope from a graph.] + Locate two points on the line whose coordinates are integers. + Starting with the point on the left, sketch a right triangle, with the hypotenuse going from the first point to the second point. + Count the rise and the run on the legs of the triangle. + Take the ratio of rise to run to find the slope. #math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$] ] #examplebox("Example 6")[][ Find the slope of the line shown: #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points “ordered pair 4, 2” and “ordered pair 3, 3”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points “ordered pair 4, 2” and “ordered pair 3, 3”.", caption: none) #solutionbox[ Locate two points on the graph. Look for points with coordinates that are integers. We can choose any points, but we will use (0, 5) and (3, 3). Starting with the point on the left, sketch a right triangle, with the hypotenuse going from the first point to the second point. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points “ordered pair 0, 5” and “ordered pair 3, 3”. Two line segments form a triangle with the line. A horizontal line connects “ordered pair 0, 3” and “ordered pair 3, 3 ”. A vertical line segment connects “ordered pair 0, 3” and “ordered pair 0, 5”. It is labeled “rise”. ], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points “ordered pair 0, 5” and “ordered pair 3, 3”. Two line segments form a triangle with the line. A horizontal line connects “ordered pair 0, 3” and “ordered pair 3, 3 ”. A vertical line segment connects “ordered pair 0, 3” and “ordered pair 0, 5”. It is labeled “rise”.", caption: none)]), [Count the rise – it is negative.], [The rise is −2.], [Count the run.], [The run is 3.], [Use the slope formula.], [#math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$]], [Substitute the values of the rise and run.], [#math.equation(block: false, alt: "m equals the fraction −2 over 3")[$m = frac(−2, 3)$]], [Simplify.], [#math.equation(block: false, alt: "m equals minus the fraction 2 over 3")[$m = − frac(2, 3)$]], [], [The slope of the line is #math.equation(block: false, alt: "minus the fraction 2 over 3 .")[$− frac(2, 3) .$]], )) Notice that the slope is negative since the line slants downward from left to right. What if we had chosen different points? Let’s find the slope of the line again, this time using different points. We will use the points #math.equation(block: false, alt: "open parenthesis −3 , 7 close parenthesis")[$( −3 , 7 )$] and #math.equation(block: false, alt: "open parenthesis 6 , 1 close parenthesis .")[$( 6 , 1 ) .$] #linebreak() #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points “ordered pair 0, 5” and “ordered pair 3, 3”. .], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points “ordered pair 0, 5” and “ordered pair 3, 3”. .", caption: none) Starting at #math.equation(block: false, alt: "open parenthesis −3 , 7 close parenthesis ,")[$( −3 , 7 ) ,$] sketch a right triangle to #math.equation(block: false, alt: "open parenthesis 6 , 1 close parenthesis .")[$( 6 , 1 ) .$] #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points “ordered pair 0, 5” and “ordered pair 3, 3”. Two line segments form a triangle with the line. A vertical line connects “ordered pair 0, 3” and “ordered pair 3, 3 ”. A vertical line segment connects “ordered pair 0, 3” and “ordered pair 0, 5”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points “ordered pair 0, 5” and “ordered pair 3, 3”. Two line segments form a triangle with the line. A vertical line connects “ordered pair 0, 3” and “ordered pair 3, 3 ”. A vertical line segment connects “ordered pair 0, 3” and “ordered pair 0, 5”.", caption: none)]), [Count the rise.], [The rise is −6.], [Count the run.], [The run is 9.], [Use the slope formula.], [#math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$]], [Substitute the values of the rise and run.], [#math.equation(block: false, alt: "m equals the fraction −6 over 9")[$m = frac(−6, 9)$]], [Simplify the fraction.], [#math.equation(block: false, alt: "m equals minus the fraction 2 over 3")[$m = − frac(2, 3)$]], [], [The slope of the line is #math.equation(block: false, alt: "minus the fraction 2 over 3 .")[$− frac(2, 3) .$]], )) It does not matter which points you use—the slope of the line is always the same. The slope of a line is constant! ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope of the line: #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -1 to 5. The y-axis runs from -6 to 1. A line passes through the points “ordered pair 3, -6” and “ordered pair 0, -2”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -1 to 5. The y-axis runs from -6 to 1. A line passes through the points “ordered pair 3, -6” and “ordered pair 0, -2”.", caption: none) #solutionbox[ #math.equation(block: true, alt: "minus the fraction 4 over 3")[$− frac(4, 3)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope of the line: #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -3 to 6. The y-axis runs from -3 to 2. A line passes through the points “ordered pair 5, -2” and “ordered pair 0, 1”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -3 to 6. The y-axis runs from -3 to 2. A line passes through the points “ordered pair 5, -2” and “ordered pair 0, 1”.", caption: none) #solutionbox[ #math.equation(block: true, alt: "minus the fraction 3 over 5")[$− frac(3, 5)$] ] ] The lines in the previous examples had #math.equation(block: false, alt: "y")[$y$]-intercepts with integer values, so it was convenient to use the #strong[#emph[y]-intercept] as one of the points we used to find the slope. In the next example, the #math.equation(block: false, alt: "y")[$y$]-intercept is a fraction. The calculations are easier if we use two points with integer coordinates. #examplebox("Example 7")[][ Find the slope of the line shown: #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from 0 to 7. The y-axis runs from 0 to 8. A line passes through the points “ordered pair 2, 3” and “ordered pair 7, 6”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from 0 to 7. The y-axis runs from 0 to 8. A line passes through the points “ordered pair 2, 3” and “ordered pair 7, 6”.", caption: none) #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Locate two points on the graph whose coordinates are integers.], [#math.equation(block: false, alt: "open parenthesis 2 , 3 close parenthesis")[$( 2 , 3 )$] and #math.equation(block: false, alt: "open parenthesis 7 , 6 close parenthesis")[$( 7 , 6 )$]]), [Which point is on the left?], [#math.equation(block: false, alt: "open parenthesis 2 , 3 close parenthesis")[$( 2 , 3 )$]], [Starting at #math.equation(block: false, alt: "open parenthesis 2 , 3 close parenthesis")[$( 2 , 3 )$], sketch a right angle to #math.equation(block: false, alt: "open parenthesis 7 , 6 close parenthesis")[$( 7 , 6 )$] as shown below.], [], )) #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from 0 to 7. The y-axis runs from 0 to 8. Two unlabeled points are drawn at “ordered pair 2, 3” and “ordered pair 7, 6”. A line passes through the points. Two line segments form a triangle with the line. A vertical line connects “ordered pair 2, 3” and “ordered pair 2, 6 ”. It is labeled “rise”. A horizontal line segment connects “ordered pair 2, 6” and “ordered pair 7, 6”. It is labeled “run”. ], alt: "The graph shows the x y-coordinate plane. The x-axis runs from 0 to 7. The y-axis runs from 0 to 8. Two unlabeled points are drawn at “ordered pair 2, 3” and “ordered pair 7, 6”. A line passes through the points. Two line segments form a triangle with the line. A vertical line connects “ordered pair 2, 3” and “ordered pair 2, 6 ”. It is labeled “rise”. A horizontal line segment connects “ordered pair 2, 6” and “ordered pair 7, 6”. It is labeled “run”.", caption: none)]), [Count the rise.], [The rise is 3.], [Count the run.], [The run is 5.], [Use the slope formula.], [#math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$]], [Substitute the values of the rise and run.], [#math.equation(block: false, alt: "m equals the fraction 3 over 5")[$m = frac(3, 5)$]], [], [The slope of the line is #math.equation(block: false, alt: "the fraction 3 over 5 .")[$frac(3, 5) .$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope of the line: #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -4 to 2. The y-axis runs from -5 to 2. A line passes through the points “ordered pair -3, -4” and “ordered pair 1, 1”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -4 to 2. The y-axis runs from -5 to 2. A line passes through the points “ordered pair -3, -4” and “ordered pair 1, 1”.", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 5 over 4")[$frac(5, 4)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope of the line: #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -1 to 4. The y-axis runs from -2 to 3. A line passes through the points “ordered pair 3, 2” and “ordered pair 1, -1”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -1 to 4. The y-axis runs from -2 to 3. A line passes through the points “ordered pair 3, 2” and “ordered pair 1, -1”.", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over 2")[$frac(3, 2)$] ] ] === Find the Slope of Horizontal and Vertical Lines Do you remember what was special about horizontal and vertical lines? Their equations had just one variable. - horizontal line #math.equation(block: false, alt: "y equals b ;")[$y = b ;$] all the #math.equation(block: false, alt: "y")[$y$]-coordinates are the same. - vertical line #math.equation(block: false, alt: "x equals a ;")[$x = a ;$] all the #math.equation(block: false, alt: "x")[$x$]-coordinates are the same. So how do we find the slope of the horizontal line #math.equation(block: false, alt: "y equals 4 ?")[$y = 4 ?$] One approach would be to graph the horizontal line, find two points on it, and count the rise and the run. Let’s see what happens in . We’ll use the two points #math.equation(block: false, alt: "open parenthesis 0 , 4 close parenthesis")[$( 0 , #h(0.2em) 4 )$] and #math.equation(block: false, alt: "open parenthesis 3 , 4 close parenthesis")[$( 3 , #h(0.2em) 4 )$] to count the rise and run. #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -1 to 5. The y-axis runs from -1 to 7. A horizontal line passes through the labeled points “ordered pair 0, 4” and “ordered pair 3, 4”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -1 to 5. The y-axis runs from -1 to 7. A horizontal line passes through the labeled points “ordered pair 0, 4” and “ordered pair 3, 4”.", caption: none) #figure(table( columns: 2, align: left, inset: 6pt, table.header([What is the rise?], [The rise is 0.]), [What is the run?], [The run is 3.], [What is the slope?], [#math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$]], [], [#math.equation(block: false, alt: "m equals the fraction 0 over 3")[$m = frac(0, 3)$]], [], [#math.equation(block: false, alt: "m equals 0")[$m = 0$]], )) The slope of the horizontal line #math.equation(block: false, alt: "y equals 4")[$y = 4$] is #math.equation(block: false, alt: "0 .")[$0 .$] All horizontal lines have slope #math.equation(block: false, alt: "0")[$0$]. When the #math.equation(block: false, alt: "y")[$y$]-coordinates are the same, the rise is #math.equation(block: false, alt: "0")[$0$]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Slope of a Horizontal Line] The #strong[slope] of a #strong[horizontal line], #math.equation(block: false, alt: "y equals b ,")[$y = b ,$] is #math.equation(block: false, alt: "0 .")[$0 .$] ] Now we’ll consider a vertical line, such as the line #math.equation(block: false, alt: "x equals 3")[$x = 3$], shown. We’ll use the two points #math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis")[$( 3 , #h(0.2em) 0 )$] and #math.equation(block: false, alt: "open parenthesis 3 , 2 close parenthesis")[$( 3 , #h(0.2em) 2 )$] to count the rise and run. #figure(figph[The graph shows the x y-coordinate plane. Both axes run from -5 to 5. A vertical line passes through the labeled points “ordered pair 3, 2” and “ordered pair 3, 0”.], alt: "The graph shows the x y-coordinate plane. Both axes run from -5 to 5. A vertical line passes through the labeled points “ordered pair 3, 2” and “ordered pair 3, 0”.", caption: none) #figure(table( columns: 2, align: left, inset: 6pt, table.header([What is the rise?], [The rise is 2.]), [What is the run?], [The run is 0.], [What is the slope?], [#math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$]], [], [#math.equation(block: false, alt: "m equals the fraction 2 over 0")[$m = frac(2, 0)$]], )) But we can’t divide by #math.equation(block: false, alt: "0 .")[$0 .$] Division by #math.equation(block: false, alt: "0")[$0$] is undefined. So we say that the slope of the vertical line #math.equation(block: false, alt: "x equals 3")[$x = 3$] is undefined. The slope of all vertical lines is undefined, because the run is #math.equation(block: false, alt: "0 .")[$0 .$] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Slope of a Vertical Line] The #strong[slope] of a #strong[vertical line], #math.equation(block: false, alt: "x equals a ,")[$x = a ,$] is undefined. ] #examplebox("Example 8")[][ Find the slope of each line: + ⓐ #math.equation(block: false, alt: "x equals 8")[$#h(0.2em) x = 8 #h(0.2em)$] + ⓑ #math.equation(block: false, alt: "y equals −5")[$#h(0.2em) y = −5$] #solutionbox[ ⓐ #math.equation(block: false, alt: "x equals 8")[$#h(0.2em) x = 8$] This is a vertical line, so its slope is undefined. ⓑ #math.equation(block: false, alt: "y equals −5")[$#h(0.2em) y = −5$] This is a horizontal line, so its slope is #math.equation(block: false, alt: "0 .")[$0 .$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope of the line: #math.equation(block: false, alt: "x equals −4 .")[$x = −4 .$] #solutionbox[ undefined ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope of the line: #math.equation(block: false, alt: "y equals 7 .")[$y = 7 .$] #solutionbox[ 0 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Quick Guide to the Slopes of Lines] #figure(figph[The figure shows 4 arrows. The first rises from left to right with the arrow point upwards. It is labeled “positive”. The second goes down from left to right with the arrow pointing downwards. It is labeled “negative”. The third is horizontal with arrow heads on both ends. It is labeled “zero”. The last is vertical with arrow heads on both ends. It is labeled “undefined.”], alt: "The figure shows 4 arrows. The first rises from left to right with the arrow point upwards. It is labeled “positive”. The second goes down from left to right with the arrow pointing downwards. It is labeled “negative”. The third is horizontal with arrow heads on both ends. It is labeled “zero”. The last is vertical with arrow heads on both ends. It is labeled “undefined.”", caption: none) ] === Use the Slope Formula to find the Slope of a Line between Two Points Sometimes we need to find the slope of a line between two points and we might not have a graph to count out the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but there is a way to find the slope without graphing. Before we get to it, we need to introduce some new algebraic notation. We have seen that an #strong[ordered pair] #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$] gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$] be used to represent two different points? Mathematicians use #strong[subscripts] to distinguish between the points. A subscript is a small number written to the right of, and a little lower than, a variable. - #math.equation(block: false, alt: "open parenthesis x sub 1 , y sub 1 close parenthesis read x sub 1 , y sub 1")[$( x_(1) , y_(1) ) #h(0.2em) "read" #h(0.2em) x #h(0.2em) "sub" #h(0.2em) 1 , y #h(0.2em) "sub" #h(0.2em) 1$] - #math.equation(block: false, alt: "open parenthesis x sub 2 , y sub 2 close parenthesis read x sub 2 , y sub 2")[$( x_(2) , y_(2) ) #h(0.2em) "read" #h(0.2em) x #h(0.2em) "sub" #h(0.2em) 2 , y #h(0.2em) "sub" #h(0.2em) 2$] We will use #math.equation(block: false, alt: "open parenthesis x sub 1 , y sub 1 close parenthesis")[$( x_(1) , y_(1) )$] to identify the first point and #math.equation(block: false, alt: "open parenthesis x sub 2 , y sub 2 close parenthesis")[$( x_(2) , y_(2) )$] to identify the second point. If we had more than two points, we could use #math.equation(block: false, alt: "open parenthesis x sub 3 , y sub 3 close parenthesis , open parenthesis x sub 4 , y sub 4 close parenthesis ,")[$( x_(3) , y_(3) ) , ( x_(4) , y_(4) ) ,$] and so on. To see how the rise and run relate to the coordinates of the two points, let’s take another look at the slope of the line between the points #math.equation(block: false, alt: "open parenthesis 2 , 3 close parenthesis")[$( 2 , 3 )$] and #math.equation(block: false, alt: "open parenthesis 7 , 6 close parenthesis")[$( 7 , 6 )$] in . #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from 0 to 7. The y-axis runs from 0 to 7. A line runs through the labeled points 2, 3 and 7, 6. A line segment runs from the point 2, 3 to the unlabeled point 2, 6. It is labeled y sub 2 minus y sub 1, 6 minus 3, 3. A line segment runs from the point 7, 6 to the unlabeled point 2, 6. It os labeled x sub 2 minus x sub 1, 7 minus 2, 5. ], alt: "The graph shows the x y-coordinate plane. The x-axis runs from 0 to 7. The y-axis runs from 0 to 7. A line runs through the labeled points 2, 3 and 7, 6. A line segment runs from the point 2, 3 to the unlabeled point 2, 6. It is labeled y sub 2 minus y sub 1, 6 minus 3, 3. A line segment runs from the point 7, 6 to the unlabeled point 2, 6. It os labeled x sub 2 minus x sub 1, 7 minus 2, 5.", caption: none) Since we have two points, we will use subscript notation. #math.equation(block: true, alt: "open parenthesis 2 , 3 close parenthesis x sub 1 , y sub 1 open parenthesis 7 , 6 close parenthesis x sub 2 , y sub 2")[$limits(( 2 , 3 ))^(x_(1) , y_(1)) #h(1.0em) limits(( 7 , 6 ))^(x_(2) , y_(2))$]On the graph, we counted the rise of #math.equation(block: false, alt: "3 .")[$3 .$] The rise can also be found by subtracting the #math.equation(block: false, alt: "y -coordinates")[$y "-coordinates"$] of the points. #math.equation(block: true, alt: "y sub 2 minus y sub 1; 6 minus 3; 3")[$y_(2) − y_(1) \ 6 − 3 \ 3$]We counted a run of #math.equation(block: false, alt: "5 .")[$5 .$] The run can also be found by subtracting the #math.equation(block: false, alt: "x -coordinates .")[$x "-coordinates" .$] #math.equation(block: true, alt: "x sub 2 minus x sub 1; 7 minus 2; 5")[$x_(2) − x_(1) \ 7 − 2 \ 5$]#figure(table( columns: 2, align: left, inset: 6pt, table.header([We know], [#math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$]]), [So], [#math.equation(block: false, alt: "m equals the fraction 3 over 5")[$m = frac(3, 5)$]], [We rewrite the rise and run by putting in the coordinates.], [#math.equation(block: false, alt: "m equals the fraction 6 minus 3 over 7 minus 2")[$m = frac(6 − 3, 7 − 2)$]], [But 6 is the #math.equation(block: false, alt: "y")[$y$]-coordinate of the second point, #math.equation(block: false, alt: "y sub 2")[$y_(2)$] #linebreak() and 3 is the #math.equation(block: false, alt: "y")[$y$]-coordinate of the first point #math.equation(block: false, alt: "y sub 1")[$y_(1)$]. #linebreak() So we can rewrite the rise using subscript notation.], [#math.equation(block: false, alt: "m equals the fraction y sub 2 minus y sub 1 over 7 minus 2")[$m = frac(y_(2) − y_(1), 7 − 2)$]], [Also 7 is the #math.equation(block: false, alt: "x")[$x$]-coordinate of the second point, #math.equation(block: false, alt: "x sub 2")[$x_(2)$] #linebreak() and 2 is the #math.equation(block: false, alt: "x")[$x$]-coordinate of the first point #math.equation(block: false, alt: "x sub 2")[$x_(2)$]. #linebreak() So we rewrite the run using subscript notation.], [#math.equation(block: false, alt: "m equals the fraction y sub 2 minus y sub 1 over x sub 2 minus x sub 1")[$m = frac(y_(2) − y_(1), x_(2) − x_(1))$]], )) We’ve shown that #math.equation(block: false, alt: "m equals the fraction y sub 2 minus y sub 1 over x sub 2 minus x sub 1")[$m = frac(y_(2) − y_(1), x_(2) − x_(1))$] is really another version of #math.equation(block: false, alt: "m equals the fraction rise over run .")[$m = frac("rise", "run") .$] We can use this formula to find the slope of a line when we have two points on the line. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Slope Formula] The #strong[slope] of the line between two points #math.equation(block: false, alt: "open parenthesis x sub 1 , y sub 1 close parenthesis")[$( x_(1) , y_(1) )$] and #math.equation(block: false, alt: "open parenthesis x sub 2 , y sub 2 close parenthesis")[$( x_(2) , y_(2) )$] is #math.equation(block: true, alt: "m equals the fraction y sub 2 minus y sub 1 over x sub 2 minus x sub 1")[$m = frac(y_(2) − y_(1), x_(2) − x_(1))$] ] Say the formula to yourself to help you remember it: #math.equation(block: true, alt: "Slope is y of the second point minus y of the first point")[$"Slope is" #h(0.2em) y #h(0.2em) "of the second point minus" #h(0.2em) y #h(0.2em) "of the first point"$]#math.equation(block: true, alt: "over")[$"over"$]#math.equation(block: true, alt: "x of the second point minus x of the first point.")[$x #h(0.2em) "of the second point minus" #h(0.2em) x #h(0.2em) "of the first point."$]#examplebox("Example 9")[][ Find the slope of the line between the points #math.equation(block: false, alt: "open parenthesis 1 , 2 close parenthesis")[$( 1 , 2 )$] and #math.equation(block: false, alt: "open parenthesis 4 , 5 close parenthesis .")[$( 4 , 5 ) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([We’ll call #math.equation(block: false, alt: "open parenthesis 1 , 2 close parenthesis")[$( 1 , 2 )$] point \#1 and #math.equation(block: false, alt: "open parenthesis 4 , 5 close parenthesis")[$( 4 , 5 )$]point \#2.], [#math.equation(block: false, alt: "open parenthesis 1 , 2 close parenthesis x sub 1 , y sub 1 and open parenthesis 4 , 5 close parenthesis x sub 2 , y sub 2")[$limits(( 1 , 2 ))^(x_(1) , y_(1)) #h(0.2em) "and" #h(0.2em) limits(( 4 , 5 ))^(x_(2) , y_(2))$]]), [Use the slope formula.], [#math.equation(block: false, alt: "m equals the fraction y sub 2 minus y sub 1 over x sub 2 minus x sub 1")[$m = frac(y_(2) − y_(1), x_(2) − x_(1))$]], [Substitute the values in the slope formula:], [], [#math.equation(block: false, alt: "y")[$y #h(0.2em)$] of the second point minus #math.equation(block: false, alt: "y")[$#h(0.2em) y #h(0.2em)$] of the first point], [#math.equation(block: false, alt: "m equals the fraction 5 minus 2 over x sub 2 minus x sub 1")[$m = frac(5 − 2, x_(2) − x_(1))$]], [#math.equation(block: false, alt: "x")[$x #h(0.2em)$] of the second point minus #math.equation(block: false, alt: "x")[$#h(0.2em) x #h(0.2em)$] of the first point], [#math.equation(block: false, alt: "m equals the fraction 5 minus 2 over 4 minus 1")[$m = frac(5 − 2, 4 − 1)$]], [Simplify the numerator and the denominator.], [#math.equation(block: false, alt: "m equals the fraction 3 over 3")[$m = frac(3, 3)$]], [], [#math.equation(block: false, alt: "m equals 1")[$m = 1$]], )) Let’s confirm this by counting out the slope on the graph. #linebreak() #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -1 to 7. The y-axis runs from -1 to 7. Two labeled points are drawn at “ordered pair 1, 2” and “ordered pair 4, 5”. A line passes through the points. Two line segments form a triangle with the line. A vertical line connects “ordered pair 1, 2” and “ordered pair 1, 5 ”. It is labeled “rise”. A horizontal line segment connects “ordered pair 1, 5” and “ordered pair 4, 5”. It is labeled “run”. ], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -1 to 7. The y-axis runs from -1 to 7. Two labeled points are drawn at “ordered pair 1, 2” and “ordered pair 4, 5”. A line passes through the points. Two line segments form a triangle with the line. A vertical line connects “ordered pair 1, 2” and “ordered pair 1, 5 ”. It is labeled “rise”. A horizontal line segment connects “ordered pair 1, 5” and “ordered pair 4, 5”. It is labeled “run”.", caption: none) The rise is #math.equation(block: false, alt: "3")[$3$] and the run is #math.equation(block: false, alt: "3 ,")[$3 ,$] so #math.equation(block: true, alt: "m equals the fraction rise over run; m equals the fraction 3 over 3; m equals 1")[$m = frac("rise", "run") \ m = frac(3, 3) \ m = 1$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope of the line through the given points: #math.equation(block: false, alt: "open parenthesis 8 , 5 close parenthesis")[$( 8 , 5 )$] and #math.equation(block: false, alt: "open parenthesis 6 , 3 close parenthesis .")[$( 6 , 3 ) .$] #solutionbox[ 1 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope of the line through the given points: #math.equation(block: false, alt: "open parenthesis 1 , 5 close parenthesis")[$( 1 , 5 )$] and #math.equation(block: false, alt: "open parenthesis 5 , 9 close parenthesis .")[$( 5 , 9 ) "."$] #solutionbox[ 1 ] ] How do we know which point to call \#1 and which to call \#2? Let’s find the slope again, this time switching the names of the points to see what happens. Since we will now be counting the run from right to left, it will be negative. #figure(table( columns: 2, align: left, inset: 6pt, table.header([We’ll call #math.equation(block: false, alt: "open parenthesis 4 , 5 close parenthesis")[$( 4 , 5 )$] point \#1 and #math.equation(block: false, alt: "open parenthesis 1 , 2 close parenthesis")[$( 1 , 2 )$] point \#2.], [#math.equation(block: false, alt: "open parenthesis 4 , 5 close parenthesis x sub 1 , y sub 1 and open parenthesis 1 , 2 close parenthesis x sub 2 , y sub 2")[$limits(( 4 , 5 ))^(x_(1) , y_(1)) #h(0.2em) "and" #h(0.2em) limits(( 1 , 2 ))^(x_(2) , y_(2))$]]), [Use the slope formula.], [#math.equation(block: false, alt: "m equals the fraction y sub 2 minus y sub 1 over x sub 2 minus x sub 1")[$m = frac(y_(2) − y_(1), x_(2) − x_(1))$]], [Substitute the values in the slope formula:], [], [#math.equation(block: false, alt: "y")[$y #h(0.2em)$] of the second point minus #math.equation(block: false, alt: "y")[$#h(0.2em) y #h(0.2em)$] of the first point], [#math.equation(block: false, alt: "m equals the fraction 2 minus 5 over x sub 2 minus x sub 1")[$m = frac(2 − 5, x_(2) − x_(1))$]], [#math.equation(block: false, alt: "x")[$x #h(0.2em)$] of the second point minus #math.equation(block: false, alt: "x")[$#h(0.2em) x #h(0.2em)$] of the first point], [#math.equation(block: false, alt: "m equals the fraction 2 minus 5 over 1 minus 4")[$m = frac(2 − 5, 1 − 4)$]], [Simplify the numerator and the denominator.], [#math.equation(block: false, alt: "m equals the fraction −3 over −3")[$m = frac(−3, −3)$]], [], [#math.equation(block: false, alt: "m equals 1")[$m = 1$]], )) The slope is the same no matter which order we use the points. #examplebox("Example 10")[][ Find the slope of the line through the points #math.equation(block: false, alt: "open parenthesis −2 , −3 close parenthesis")[$( −2 , −3 )$] and #math.equation(block: false, alt: "open parenthesis −7 , 4 close parenthesis .")[$( −7 , 4 ) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([We’ll call #math.equation(block: false, alt: "open parenthesis −2 , −3 close parenthesis")[$( −2 , −3 )$] point \#1 and #math.equation(block: false, alt: "open parenthesis −7 , 4 close parenthesis")[$( −7 , 4 )$] point \#2.], [#math.equation(block: false, alt: "open parenthesis −2 , −3 close parenthesis x sub 1 , y sub 1 and open parenthesis −7 , 4 close parenthesis x sub 2 , y sub 2")[$limits(( −2 , −3 ))^(x_(1) , y_(1)) #h(0.2em) "and" #h(0.2em) limits(( −7 , 4 ))^(x_(2) , y_(2))$]]), [Use the slope formula.], [#math.equation(block: false, alt: "m equals the fraction y sub 2 minus y sub 1 over x sub 2 minus x sub 1")[$m = frac(y_(2) − y_(1), x_(2) − x_(1))$]], [Substitute the values], [], [#math.equation(block: false, alt: "y")[$y #h(0.2em)$] of the second point minus #math.equation(block: false, alt: "y")[$#h(0.2em) y #h(0.2em)$] of the first point], [#math.equation(block: false, alt: "m equals the fraction 4 minus open parenthesis −3 close parenthesis over x sub 2 minus x sub 1")[$m = frac(4 − ( −3 ), x_(2) − x_(1))$]], [#math.equation(block: false, alt: "x")[$x #h(0.2em)$] of the second point minus #math.equation(block: false, alt: "x")[$#h(0.2em) x #h(0.2em)$] of the first point], [#math.equation(block: false, alt: "m equals the fraction 4 minus open parenthesis −3 close parenthesis over −7 minus open parenthesis −2 close parenthesis")[$m = frac(4 − ( −3 ), −7 − ( −2 ))$]], [Simplify.], [#math.equation(block: false, alt: "m equals the fraction 7 over −5")[$m = frac(7, −5)$]], [], [#math.equation(block: false, alt: "m equals minus the fraction 7 over 5")[$m = − frac(7, 5)$]], )) Let’s confirm this on the graph shown. #linebreak() #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -8 to 2. The y-axis runs from -6 to 5. Two unlabeled points are drawn at “ordered pair -7, 4” and “ordered pair -2, -3”. A line passes through the points. Two line segments form a triangle with the line. A vertical line connects “ordered pair -7, 4” and “ordered pair -7, -3 ”. It is labeled “rise”. A horizontal line segment connects “ordered pair -7, -3” and “ordered pair -2, -3”. It is labeled “run”. ], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -8 to 2. The y-axis runs from -6 to 5. Two unlabeled points are drawn at “ordered pair -7, 4” and “ordered pair -2, -3”. A line passes through the points. Two line segments form a triangle with the line. A vertical line connects “ordered pair -7, 4” and “ordered pair -7, -3 ”. It is labeled “rise”. A horizontal line segment connects “ordered pair -7, -3” and “ordered pair -2, -3”. It is labeled “run”.", caption: none) #math.equation(block: true, alt: "m equals the fraction rise over run; m equals the fraction −7 over 5; m equals minus the fraction 7 over 5")[$#h(0.1em) m = frac("rise", "run") \ m = frac(−7, 5) \ #h(0.3em) m = − frac(7, 5)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope of the line through the pair of points: #math.equation(block: false, alt: "open parenthesis −3 , 4 close parenthesis")[$( −3 , 4 )$] and #math.equation(block: false, alt: "open parenthesis 2 , −1 close parenthesis .")[$( 2 , −1 ) .$] #solutionbox[ −1 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope of the line through the pair of points: #math.equation(block: false, alt: "open parenthesis −2 , 6 close parenthesis")[$( −2 , 6 )$] and #math.equation(block: false, alt: "open parenthesis −3 , −4 close parenthesis .")[$( −3 , −4 ) .$] #solutionbox[ 10 ] ] === Graph a Line Given a Point and the Slope In this chapter, we graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines. Another method we can use to graph lines is the #strong[point-slope method]. Sometimes, we will be given one point and the slope of the line, instead of its equation. When this happens, we use the definition of slope to draw the graph of the line. #examplebox("Example 11")[][ Graph the line passing through the point #math.equation(block: false, alt: "open parenthesis 1 , −1 close parenthesis")[$( 1 , −1 )$] whose slope is #math.equation(block: false, alt: "m equals the fraction 3 over 4 .")[$m = frac(3, 4) .$] #solutionbox[ Plot the given point, #math.equation(block: false, alt: "open parenthesis 1 , −1 close parenthesis .")[$( 1 , −1 ) .$] #linebreak() #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -1 to 7. The y-axis runs from -3 to 4. A labeled point is drawn at “ordered pair 1, -1”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -1 to 7. The y-axis runs from -3 to 4. A labeled point is drawn at “ordered pair 1, -1”.", caption: none) Use the slope formula #math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$] to identify the rise and the run. #math.equation(block: true, alt: "m equals the fraction 3 over 4; the fraction rise over run equals the fraction 3 over 4; rise equals 3; run equals 4")[$#h(0.8em) m = frac(3, 4) \ frac("rise", "run") = frac(3, 4) \ \ \ #h(0.2em) "rise" = 3 \ #h(0.33em) "run" = 4$]Starting at the point we plotted, count out the rise and run to mark the second point. We count #math.equation(block: false, alt: "3")[$3$] units up and #math.equation(block: false, alt: "4")[$4$] units right. #linebreak() #figure(figph[The graph shows the x y-coordinate plane. Both axes run from -5 to 5. Two line segments are drawn. A vertical line segment connects the points “ordered pair 1, -1” and “order pair “1, 2”. It is labeled “3”. A horizontal line segment starts at the top of the vertical line segment and goes to the right, connecting the points “ordered pair 1, 2” and “ordered pair 5, 2”. It is labeled “4”.], alt: "The graph shows the x y-coordinate plane. Both axes run from -5 to 5. Two line segments are drawn. A vertical line segment connects the points “ordered pair 1, -1” and “order pair “1, 2”. It is labeled “3”. A horizontal line segment starts at the top of the vertical line segment and goes to the right, connecting the points “ordered pair 1, 2” and “ordered pair 5, 2”. It is labeled “4”.", caption: none) Then we connect the points with a line and draw arrows at the ends to show it continues. #linebreak() #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -3 to 5. The y-axis runs from -1 to 7. Two unlabeled points are drawn at “ordered pair 1, -1” and “ordered pair 5, 2”. A line passes through the points. Two line segments form a triangle with the line. A vertical line connects “ordered pair 1, -1” and “ordered pair 1, 2 ”. A horizontal line segment connects “ordered pair 1, 2” and “ordered pair 5, 2”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -3 to 5. The y-axis runs from -1 to 7. Two unlabeled points are drawn at “ordered pair 1, -1” and “ordered pair 5, 2”. A line passes through the points. Two line segments form a triangle with the line. A vertical line connects “ordered pair 1, -1” and “ordered pair 1, 2 ”. A horizontal line segment connects “ordered pair 1, 2” and “ordered pair 5, 2”.", caption: none) We can check our line by starting at any point and counting up #math.equation(block: false, alt: "3")[$3$] and to the right #math.equation(block: false, alt: "4 .")[$4 .$] We should get to another point on the line. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the line passing through the point with the given slope: #math.equation(block: true, alt: "open parenthesis 2 , −2 close parenthesis , m equals the fraction 4 over 3")[$( 2 , −2 ) , m = frac(4, 3)$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from -12 to 12. A line passes through the points “ordered pair -2, 3” and “ordered pair 8, 6”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from -12 to 12. A line passes through the points “ordered pair -2, 3” and “ordered pair 8, 6”.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the line passing through the point with the given slope: #math.equation(block: true, alt: "open parenthesis −2 , 3 close parenthesis , m equals the fraction 1 over 4")[$( −2 , 3 ) , m = frac(1, 4)$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from -12 to 12. A line passes through the points “ordered pair -2, 3” and “ordered pair 2, 4”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from -12 to 12. A line passes through the points “ordered pair -2, 3” and “ordered pair 2, 4”.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Graph a line given a point and a slope.] + Plot the given point. + Use the slope formula to identify the rise and the run. + Starting at the given point, count out the rise and run to mark the second point. + Connect the points with a line. ] #examplebox("Example 12")[][ Graph the line with #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$] and slope #math.equation(block: false, alt: "m equals minus the fraction 2 over 3 .")[$m = − frac(2, 3) .$] #solutionbox[ Plot the given point, the #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis .")[$( 0 , 2 ) .$] #linebreak() #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -1 to 4. The y-axis runs from -1 to 3. The point “ordered pair 0, 2” is labeled.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -1 to 4. The y-axis runs from -1 to 3. The point “ordered pair 0, 2” is labeled.", caption: none) Use the slope formula #math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$] to identify the rise and the run. #math.equation(block: true, alt: "m equals minus the fraction 2 over 3; the fraction rise over run equals the fraction −2 over 3; rise equals –2; run equals 3")[$#h(0.8em) m = − frac(2, 3) \ frac("rise", "run") = frac(−2, 3) \ \ \ #h(0.2em) "rise" = –2 \ #h(0.3em) "run" = 3$]Starting at #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis ,")[$( 0 , 2 ) ,$] count the rise and the run and mark the second point. #linebreak() #figure(figph[The graph shows the x y-coordinate plane. Both axes run from -5 to 5. A vertical line segment connects points at “ordered pair 0, 2” and “ordered pair 0, 0” and is labeled “down 2”. A horizontal line segment connects “ordered pair 0, 0” and “ordered pair 0, 3” and is labeled “right 3”.], alt: "The graph shows the x y-coordinate plane. Both axes run from -5 to 5. A vertical line segment connects points at “ordered pair 0, 2” and “ordered pair 0, 0” and is labeled “down 2”. A horizontal line segment connects “ordered pair 0, 0” and “ordered pair 0, 3” and is labeled “right 3”.", caption: none) Connect the points with a line. #linebreak() #figure(figph[The graph shows the x y-coordinate plane. Both axes run from -5 to 5. Two labeled points are drawn at “ordered pair 0, 2” and “ordered pair 3, 0”. A line passes through the points. Two line segments form a triangle with the line. A vertical line connects “ordered pair 0, 2” and “ordered pair 0, 0 ”. A horizontal line segment connects “ordered pair 0, 0” and “ordered pair 3, 0”. ], alt: "The graph shows the x y-coordinate plane. Both axes run from -5 to 5. Two labeled points are drawn at “ordered pair 0, 2” and “ordered pair 3, 0”. A line passes through the points. Two line segments form a triangle with the line. A vertical line connects “ordered pair 0, 2” and “ordered pair 0, 0 ”. A horizontal line segment connects “ordered pair 0, 0” and “ordered pair 3, 0”.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the line with the given intercept and slope: #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "4 , m equals minus the fraction 5 over 2")[$4 , m = − frac(5, 2)$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from -12 to 12. A line passes through the points “ordered pair 0, 4” and “ordered pair 4, -6”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from -12 to 12. A line passes through the points “ordered pair 0, 4” and “ordered pair 4, -6”.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the line with the given intercept and slope: #math.equation(block: false, alt: "x")[$x$]-intercept #math.equation(block: false, alt: "−3 , m equals minus the fraction 3 over 4")[$−3 , m = − frac(3, 4)$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from -12 to 12. A line passes through the points “ordered pair -3, 0” and “ordered pair 8, -8”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from -12 to 12. A line passes through the points “ordered pair -3, 0” and “ordered pair 8, -8”.", caption: none) ] ] #examplebox("Example 13")[][ Graph the line passing through the point #math.equation(block: false, alt: "open parenthesis −1 , −3 close parenthesis")[$( −1 , −3 )$] whose slope is #math.equation(block: false, alt: "m equals 4 .")[$m = 4 .$] #solutionbox[ Plot the given point. #linebreak() #figure(figph[The graph shows the x y-coordinate plane. Both axes run from -5 to 5. The point “ordered pair -1, -3” is labeled.], alt: "The graph shows the x y-coordinate plane. Both axes run from -5 to 5. The point “ordered pair -1, -3” is labeled.", caption: none) #figure(table( columns: 2, align: left, inset: 6pt, table.header([Identify the rise and the run.], [#math.equation(block: false, alt: "m equals 4")[$m = 4$]]), [Write 4 as a fraction.], [#math.equation(block: false, alt: "the fraction rise over run equals the fraction 4 over 1")[$frac("rise", "run") = frac(4, 1)$]], [], [#math.equation(block: false, alt: "rise equals 4 run equals 1")[$"rise" = 4 #h(0.2em) "run" = 1$]], )) Count the rise and run. #linebreak() #figure(figph[The graph shows the x y-coordinate plane. Both axes run from -5 to 5. The y-axis runs from -4 to 2. A vertical line segment connects points at “ordered pair -1, -3” and “ordered pair -1, 1” and is labeled “up 4”. A horizontal line segment connects “ordered pair -1, 1” and “ordered pair 0, 1” and is labeled “over 1”.], alt: "The graph shows the x y-coordinate plane. Both axes run from -5 to 5. The y-axis runs from -4 to 2. A vertical line segment connects points at “ordered pair -1, -3” and “ordered pair -1, 1” and is labeled “up 4”. A horizontal line segment connects “ordered pair -1, 1” and “ordered pair 0, 1” and is labeled “over 1”.", caption: none) Mark the second point. Connect the two points with a line. #linebreak() #figure(figph[The graph shows the x y-coordinate plane. Both axes run from -5 to 5. Two labeled points are drawn at “ordered pair -1, -3” and “ordered pair -1, 1”. A line passes through the points. Two line segments form a triangle with the line. A vertical line connects “ordered pair -1, -3” and “ordered pair -1, 1 ”. It is labeled “up 4” A horizontal line segment connects “ordered pair -1, 1” and “ordered pair 0, 1”. It is labeled “over 1”], alt: "The graph shows the x y-coordinate plane. Both axes run from -5 to 5. Two labeled points are drawn at “ordered pair -1, -3” and “ordered pair -1, 1”. A line passes through the points. Two line segments form a triangle with the line. A vertical line connects “ordered pair -1, -3” and “ordered pair -1, 1 ”. It is labeled “up 4” A horizontal line segment connects “ordered pair -1, 1” and “ordered pair 0, 1”. It is labeled “over 1”", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the line passing through the point #math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis")[$( −2 , 1 )$] and with slope #math.equation(block: false, alt: "m equals 3 .")[$m = 3 .$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -7 to 7. The y-axis runs from -7 to 7. A line passes through the points “ordered pair -2, 1” and “ordered pair 0, 7”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -7 to 7. The y-axis runs from -7 to 7. A line passes through the points “ordered pair -2, 1” and “ordered pair 0, 7”.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the line passing through the point #math.equation(block: false, alt: "open parenthesis 4 , −2 close parenthesis")[$( 4 , −2 )$] and with slope #math.equation(block: false, alt: "m equals −2 .")[$m = −2 .$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -7 to 7. The y-axis runs from -7 to 7. A line passes through the points “ordered pair 0, 6” and “ordered pair 0, 3”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -7 to 7. The y-axis runs from -7 to 7. A line passes through the points “ordered pair 0, 6” and “ordered pair 0, 3”.", caption: none) ] ] === Solve Slope Applications At the beginning of this section, we said there are many applications of slope in the real world. Let’s look at a few now. #examplebox("Example 14")[][ The pitch of a building’s roof is the slope of the roof. Knowing the pitch is important in climates where there is heavy snowfall. If the roof is too flat, the weight of the snow may cause it to collapse. What is the slope of the roof shown? #figure(figph[This figure shows a house with a sloped roof. The roof on one half of the building is labeled “pitch of the roof”. There is a line segment with arrows at each end measuring the vertical length of the roof and is labeled “rise = 9 feet”. There is a line segment with arrows at each end measuring the horizontal length of the root and is labeled “run = 18 feet”.], alt: "This figure shows a house with a sloped roof. The roof on one half of the building is labeled “pitch of the roof”. There is a line segment with arrows at each end measuring the vertical length of the roof and is labeled “rise = 9 feet”. There is a line segment with arrows at each end measuring the horizontal length of the root and is labeled “run = 18 feet”.", caption: none) #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Use the slope formula.], [#math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$]]), [Substitute the values for rise and run.], [#math.equation(block: false, alt: "m equals the fraction 9 ft over 18 ft")[$m = frac("9 ft", "18 ft")$]], [Simplify.], [#math.equation(block: false, alt: "m equals the fraction 1 over 2")[$m = frac(1, 2)$]], [], [The slope of the roof is #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$].], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope given rise and run: A roof with a rise #math.equation(block: false, alt: "equals 14")[$= 14$] and run #math.equation(block: false, alt: "equals 24 .")[$= 24 .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 7 over 12")[$frac(7, 12)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope given rise and run: A roof with a rise #math.equation(block: false, alt: "equals 15")[$= 15$] and run #math.equation(block: false, alt: "equals 36 .")[$= 36 .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 5 over 12")[$frac(5, 12)$] ] ] Have you ever thought about the sewage pipes going from your house to the street? Their slope is an important factor in how they take waste away from your house. #examplebox("Example 15")[][ Sewage pipes must slope down #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$] inch per foot in order to drain properly. What is the required slope? #figure(figph[This figure shows a right triangle. The short leg is vertical and is labeled “1 over 4 inch”. The long leg labeled “1 foot”.], alt: "This figure shows a right triangle. The short leg is vertical and is labeled “1 over 4 inch”. The long leg labeled “1 foot”.", caption: none) #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Use the slope formula.], [#math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$]]), [], [#math.equation(block: false, alt: "m equals the fraction minus the fraction 1 over 4 in . over 1 ft")[$m = frac(− frac(1, 4) #h(0.2em) "in" ".", 1 #h(0.2em) "ft")$]], [], [#math.equation(block: false, alt: "m equals the fraction minus the fraction 1 over 4 in . over 1 ft")[$m = frac(− frac(1, 4) #h(0.2em) "in" ".", 1 #h(0.2em) "ft")$]], [Convert 1 foot to 12 inches.], [#math.equation(block: false, alt: "m equals the fraction minus the fraction 1 over 4 in . over 12 in.")[$m = frac(− frac(1, 4) #h(0.2em) "in" ".", 12 #h(0.2em) "in.")$]], [Simplify.], [#math.equation(block: false, alt: "m equals minus the fraction 1 over 48")[$m = − frac(1, 48)$]], [], [The slope of the pipe is #math.equation(block: false, alt: "minus the fraction 1 over 48 .")[$− frac(1, 48) .$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope of the pipe: The pipe slopes down #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] inch per foot. #solutionbox[ #math.equation(block: true, alt: "minus the fraction 1 over 36")[$− frac(1, 36)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope of the pipe: The pipe slopes down #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$] inch per yard. #solutionbox[ #math.equation(block: true, alt: "minus the fraction 1 over 48")[$− frac(1, 48)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[ACCESS ADDITIONAL ONLINE RESOURCES] - #link("https://www.openstax.org/l/24posslope")[Determine Positive slope from a Graph] - #link("https://www.openstax.org/l/24negslope")[Determine Negative slope from a Graph] - #link("https://www.openstax.org/l/24slopetwopts")[Determine Slope from Two Points] ] === Key Concepts - #emph[Find the slope from a graph]+ Locate two points on the line whose coordinates are integers. + Starting with the point on the left, sketch a right triangle, with the hypotenuse going from the first point to the second point. + Count the rise and the run on the legs of the triangle. + Take the ratio of rise to run to find the slope, #math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$] - #emph[Slope of a Horizontal Line]- The slope of a horizontal line, #math.equation(block: false, alt: "y equals b")[$y = b$], is 0. - #emph[Slope of a Vertical Line]- The slope of a vertical line, #math.equation(block: false, alt: "x equals a")[$x = a$], is undefined. - #emph[Slope Formula]- The slope of the line between two points #math.equation(block: false, alt: "open parenthesis x sub 1 , y sub 1 close parenthesis")[$( x_(1) , y_(1) )$] and #math.equation(block: false, alt: "open parenthesis x sub 2 , y sub 2 close parenthesis")[$( x_(2) , y_(2) )$] is #math.equation(block: false, alt: "m equals the fraction y sub 2 minus y sub 1 over x sub 2 minus x sub 1")[$m = frac(y_(2) − y_(1), x_(2) − x_(1))$] - #emph[Graph a line given a point and a slope.]+ Plot the given point. + Use the slope formula to identify the rise and the run. + Starting at the given point, count out the rise and run to mark the second point. + Connect the points with a line. === Section Exercises ==== Practice Makes Perfect #strong[Use Geoboards to Model Slope] In the following exercises, find the slope modeled on each geoboard. #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 3 and the point in column 5 row 2.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 3 and the point in column 5 row 2.", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 4")[$frac(1, 4)$] ] #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 2 row 4 and the point in column 5 row 2.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 2 row 4 and the point in column 5 row 2.", caption: none) #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 2 row 1 and the point in column 4 row 4.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 2 row 1 and the point in column 4 row 4.", caption: none) #solutionbox[ #math.equation(block: true, alt: "minus the fraction 3 over 2")[$− frac(3, 2)$] ] #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 2 row 1 and the point in column 4 row 4.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 2 row 1 and the point in column 4 row 4.", caption: none) In the following exercises, model each slope. Draw a picture to show your results. #math.equation(block: true, alt: "the fraction 2 over 3")[$frac(2, 3)$] #solutionbox[ #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 2 row 5 and the point in column 5 row 3.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 2 row 5 and the point in column 5 row 3.", caption: none) ] #math.equation(block: true, alt: "the fraction 3 over 4")[$frac(3, 4)$] #math.equation(block: true, alt: "the fraction 1 over 4")[$frac(1, 4)$] #solutionbox[ #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 4 and the point in column 5 row 3.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 4 and the point in column 5 row 3.", caption: none) ] #math.equation(block: true, alt: "the fraction 4 over 3")[$frac(4, 3)$] #math.equation(block: true, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$] #solutionbox[ #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 4 and the point in column 3 row 5.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 4 and the point in column 3 row 5.", caption: none) ] #math.equation(block: true, alt: "minus the fraction 3 over 4")[$− frac(3, 4)$] #math.equation(block: true, alt: "minus the fraction 2 over 3")[$− frac(2, 3)$] #solutionbox[ #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 2 and the point in column 4 row 4.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 2 and the point in column 4 row 4.", caption: none) ] #math.equation(block: true, alt: "minus the fraction 3 over 2")[$− frac(3, 2)$] #strong[Find the Slope of a Line from its Graph] In the following exercises, find the slope of each line shown. #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair 0, -4” and “ordered pair 10, 0”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair 0, -4” and “ordered pair 10, 0”.", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 2 over 5")[$frac(2, 5)$] ] #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair 0, -5” and “ordered pair 3, 0”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair 0, -5” and “ordered pair 3, 0”.", caption: none) #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from -12 to 12. A line passes through the points “ordered pair 0, -1” and “ordered pair 1, 0”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from -12 to 12. A line passes through the points “ordered pair 0, -1” and “ordered pair 1, 0”.", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 5 over 4")[$frac(5, 4)$] ] #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair 0, 3” and “ordered pair 6, 0”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair 0, 3” and “ordered pair 6, 0”.", caption: none) #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair 0, 2” and “ordered pair 6, 0”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair 0, 2” and “ordered pair 6, 0”.", caption: none) #solutionbox[ #math.equation(block: true, alt: "minus the fraction 1 over 3")[$− frac(1, 3)$] ] #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair 0, 2” and “ordered pair 6, 0”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair 0, 2” and “ordered pair 6, 0”.", caption: none) #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair 0, 6” and “ordered pair 8, 0”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair 0, 6” and “ordered pair 8, 0”.", caption: none) #solutionbox[ #math.equation(block: true, alt: "minus the fraction 3 over 4")[$− frac(3, 4)$] ] #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair -1, 0” and “ordered pair 0, -1”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair -1, 0” and “ordered pair 0, -1”.", caption: none) #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair -4, 0” and “ordered pair -4, 6”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair -4, 0” and “ordered pair -4, 6”.", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over 4")[$frac(3, 4)$] ] #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair -2, 0” and “ordered pair 4, 4”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair -2, 0” and “ordered pair 4, 4”.", caption: none) #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. A line passes through the points “ordered pair 0, 4” and “ordered pair 4, -6”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. A line passes through the points “ordered pair 0, 4” and “ordered pair 4, -6”.", caption: none) #solutionbox[ #math.equation(block: true, alt: "minus the fraction 5 over 2")[$− frac(5, 2)$] ] #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. A line passes through the points “ordered pair -8, 8” and “ordered pair 8, -4”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. A line passes through the points “ordered pair -8, 8” and “ordered pair 8, -4”.", caption: none) #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. A line passes through the points “ordered pair 1, 4” and “ordered pair 7, 0”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. A line passes through the points “ordered pair 1, 4” and “ordered pair 7, 0”.", caption: none) #solutionbox[ #math.equation(block: true, alt: "minus the fraction 2 over 3")[$− frac(2, 3)$] ] #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. A line passes through the points “ordered pair 0, 3” and “ordered pair 7, 0”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. A line passes through the points “ordered pair 0, 3” and “ordered pair 7, 0”.", caption: none) #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. A line passes through the points “ordered pair 2, 0” and “ordered pair 10, 4”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. A line passes through the points “ordered pair 2, 0” and “ordered pair 10, 4”.", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 4")[$frac(1, 4)$] ] #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. A line passes through the points “ordered pair 6, 2” and “ordered pair 0, -3”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. A line passes through the points “ordered pair 6, 2” and “ordered pair 0, -3”.", caption: none) #strong[Find the Slope of Horizontal and Vertical Lines] In the following exercises, find the slope of each line. #math.equation(block: true, alt: "y equals 3")[$y = 3$] #solutionbox[ 0 ] #math.equation(block: true, alt: "y equals 1")[$y = 1$] #math.equation(block: true, alt: "x equals 4")[$x = 4$] #solutionbox[ undefined ] #math.equation(block: true, alt: "x equals 2")[$x = 2$] #math.equation(block: true, alt: "y equals −2")[$y = −2$] #solutionbox[ 0 ] #math.equation(block: true, alt: "y equals −3")[$y = −3$] #math.equation(block: true, alt: "x equals −5")[$x = −5$] #solutionbox[ undefined ] #math.equation(block: true, alt: "x equals −4")[$x = −4$] #strong[Use the Slope Formula to find the Slope of a Line between Two Points] In the following exercises, use the slope formula to find the slope of the line between each pair of points. #math.equation(block: true, alt: "open parenthesis 1 , 4 close parenthesis , open parenthesis 3 , 9 close parenthesis")[$( 1 , 4 ) , ( 3 , 9 )$] #solutionbox[ #math.equation(block: true, alt: "the fraction 5 over 2")[$frac(5, 2)$] ] #math.equation(block: true, alt: "open parenthesis 2 , 3 close parenthesis , open parenthesis 5 , 7 close parenthesis")[$( 2 , 3 ) , ( 5 , 7 )$] #math.equation(block: true, alt: "open parenthesis 0 , 3 close parenthesis , open parenthesis 4 , 6 close parenthesis")[$( 0 , 3 ) , ( 4 , 6 )$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over 4")[$frac(3, 4)$] ] #math.equation(block: true, alt: "open parenthesis 0 , 1 close parenthesis , open parenthesis 5 , 4 close parenthesis")[$( 0 , 1 ) , ( 5 , 4 )$] #math.equation(block: true, alt: "open parenthesis 2 , 5 close parenthesis , open parenthesis 4 , 0 close parenthesis")[$( 2 , 5 ) , ( 4 , 0 )$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 5 over 2")[$− frac(5, 2)$] ] #math.equation(block: true, alt: "open parenthesis 3 , 6 close parenthesis , open parenthesis 8 , 0 close parenthesis")[$( 3 , 6 ) , ( 8 , 0 )$] #math.equation(block: true, alt: "open parenthesis −3 , 3 close parenthesis , open parenthesis 2 , −5 close parenthesis")[$( −3 , 3 ) , ( 2 , −5 )$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 8 over 5")[$− frac(8, 5)$] ] #math.equation(block: true, alt: "open parenthesis −2 , 4 close parenthesis , open parenthesis 3 , −1 close parenthesis")[$( −2 , 4 ) , ( 3 , −1 )$] #math.equation(block: true, alt: "open parenthesis −1 , −2 close parenthesis , open parenthesis 2 , 5 close parenthesis")[$( −1 , −2 ) , ( 2 , 5 )$] #solutionbox[ #math.equation(block: true, alt: "the fraction 7 over 3")[$frac(7, 3)$] ] #math.equation(block: true, alt: "open parenthesis −2 , −1 close parenthesis , open parenthesis 6 , 5 close parenthesis")[$( −2 , −1 ) , ( 6 , 5 )$] #math.equation(block: true, alt: "open parenthesis 4 , −5 close parenthesis , open parenthesis 1 , −2 close parenthesis")[$( 4 , −5 ) , ( 1 , −2 )$] #solutionbox[ −1 ] #math.equation(block: true, alt: "open parenthesis 3 , −6 close parenthesis , open parenthesis 2 , −2 close parenthesis")[$( 3 , −6 ) , ( 2 , −2 )$] #strong[Graph a Line Given a Point and the Slope] In the following exercises, graph the line given a point and the slope. #math.equation(block: true, alt: "open parenthesis 1 , −2 close parenthesis ; m equals the fraction 3 over 4")[$( 1 , −2 ) ; m = frac(3, 4)$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from 12 to -12. A line passes through the points “ordered pair 5, 1” and “ordered pair 1, -2”], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from 12 to -12. A line passes through the points “ordered pair 5, 1” and “ordered pair 1, -2”", caption: none) ] #math.equation(block: true, alt: "open parenthesis 1 , −1 close parenthesis ; m equals the fraction 1 over 2")[$( 1 , −1 ) ; m = frac(1, 2)$] #math.equation(block: true, alt: "open parenthesis 2 , 5 close parenthesis ; m equals minus the fraction 1 over 3")[$( 2 , 5 ) ; m = − frac(1, 3)$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from 12 to -12. A line passes through the points “ordered pair 2, 5” and “ordered pair 5, 4”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from 12 to -12. A line passes through the points “ordered pair 2, 5” and “ordered pair 5, 4”.", caption: none) ] #math.equation(block: true, alt: "open parenthesis 1 , 4 close parenthesis ; m equals minus the fraction 1 over 2")[$( 1 , 4 ) ; m = − frac(1, 2)$] #math.equation(block: true, alt: "open parenthesis −3 , 4 close parenthesis ; m equals minus the fraction 3 over 2")[$( −3 , 4 ) ; m = − frac(3, 2)$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from 12 to -12. A line passes through the points “ordered pair -3, 4” and “ordered pair -1, 1”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from 12 to -12. A line passes through the points “ordered pair -3, 4” and “ordered pair -1, 1”.", caption: none) ] #math.equation(block: true, alt: "open parenthesis −2 , 5 close parenthesis ; m equals minus the fraction 5 over 4")[$( −2 , 5 ) ; m = − frac(5, 4)$] #math.equation(block: true, alt: "open parenthesis −1 , −4 close parenthesis ; m equals the fraction 4 over 3")[$( −1 , −4 ) ; m = frac(4, 3)$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from 12 to -12. A line passes through the points “ordered pair 2,0” and “ordered pair -1, -4”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from 12 to -12. A line passes through the points “ordered pair 2,0” and “ordered pair -1, -4”.", caption: none) ] #math.equation(block: true, alt: "open parenthesis −3 , −5 close parenthesis ; m equals the fraction 3 over 2")[$( −3 , −5 ) ; m = frac(3, 2)$] #math.equation(block: true, alt: "open parenthesis 0 , 3 close parenthesis ; m equals minus the fraction 2 over 5")[$( 0 , 3 ) ; m = − frac(2, 5)$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from 12 to -12. A line passes through the points “ordered pair 0, 3” and “ordered pair 5, 1”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from 12 to -12. A line passes through the points “ordered pair 0, 3” and “ordered pair 5, 1”.", caption: none) ] #math.equation(block: true, alt: "open parenthesis 0 , 5 close parenthesis ; m equals minus the fraction 4 over 3")[$( 0 , 5 ) ; m = − frac(4, 3)$] #math.equation(block: true, alt: "open parenthesis −2 , 0 close parenthesis ; m equals the fraction 3 over 4")[$( −2 , 0 ) ; m = frac(3, 4)$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from 12 to -12. A line passes through the points “ordered pair -2,0” and “ordered pair 2, 3”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from 12 to -12. A line passes through the points “ordered pair -2,0” and “ordered pair 2, 3”.", caption: none) ] #math.equation(block: true, alt: "open parenthesis −1 , 0 close parenthesis ; m equals the fraction 1 over 5")[$( −1 , 0 ) ; m = frac(1, 5)$] #math.equation(block: true, alt: "open parenthesis −3 , 3 close parenthesis ; m equals 2")[$( −3 , 3 ) ; m = 2$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from 12 to -12. A line passes through the points “ordered pair -3, 3” and “ordered pair -2, 5”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from 12 to -12. A line passes through the points “ordered pair -3, 3” and “ordered pair -2, 5”.", caption: none) ] #math.equation(block: true, alt: "open parenthesis −4 , 2 close parenthesis ; m equals 4")[$( −4 , 2 ) ; m = 4$] #math.equation(block: true, alt: "open parenthesis 1 , 5 close parenthesis ; m equals −3")[$( 1 , 5 ) ; m = −3$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from 12 to -12. A line passes through the points “ordered pair 1, 5” and “ordered pair 2, 2”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from 12 to -12. A line passes through the points “ordered pair 1, 5” and “ordered pair 2, 2”.", caption: none) ] #math.equation(block: true, alt: "open parenthesis 2 , 3 close parenthesis ; m equals −1")[$( 2 , 3 ) ; m = −1$] #strong[Solve Slope Applications] In the following exercises, solve these slope applications. #strong[Slope of a roof] A fairly easy way to determine the slope is to take a #math.equation(block: false, alt: "12-inch")[$"12-inch"$] level and set it on one end on the roof surface. Then take a tape measure or ruler, and measure from the other end of the level down to the roof surface. You can use these measurements to calculate the slope of the roof. What is the slope of the roof in this picture? #figure(figph[The figure shows a wood board at a diagonal representing a side-view slice of a pitched roof. A vertical line segment with arrows on both ends measures the vertical change in height of the roof and is labeled “4 inches”. A level tool is in a horizontal position above the board and above it is a line segment with arrows on both ends labeled “12 inches”.], alt: "The figure shows a wood board at a diagonal representing a side-view slice of a pitched roof. A vertical line segment with arrows on both ends measures the vertical change in height of the roof and is labeled “4 inches”. A level tool is in a horizontal position above the board and above it is a line segment with arrows on both ends labeled “12 inches”.", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 3")[$frac(1, 3)$] ] What is the slope of the roof shown? #figure(figph[The figure shows a diagonal side-view slice of a pitched roof. A ruler in vertical position is at the bottom of the roof segment and shows unit labels 1 through 8 and extends one further unit. A second ruler starts at the “7” label of the vertical ruler and extends horizontally until it hits the rising roof. The horizontal ruler has unit labels 1 through 11 and extends one further unit.], alt: "The figure shows a diagonal side-view slice of a pitched roof. A ruler in vertical position is at the bottom of the roof segment and shows unit labels 1 through 8 and extends one further unit. A second ruler starts at the “7” label of the vertical ruler and extends horizontally until it hits the rising roof. The horizontal ruler has unit labels 1 through 11 and extends one further unit.", caption: none) #strong[Road grade] A local road has a grade of #math.equation(block: false, alt: "6% .")[$"6%" .$] The grade of a road is its slope expressed as a percent. + ⓐ Find the slope of the road as a fraction and then simplify the fraction. + ⓑ What rise and run would reflect this slope or grade? #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 3 over 50")[$#h(0.2em) frac(3, 50) #h(0.2em)$]ⓑ #math.equation(block: false, alt: "rise equals 3 ; run equals 50")[$#h(0.2em) "rise" = 3 ; #h(0.2em) "run" = 50$] ] #strong[Highway grade] A local road rises #math.equation(block: false, alt: "2")[$2$] feet for every #math.equation(block: false, alt: "50")[$50$] feet of highway. + ⓐ What is the slope of the highway? + ⓑ The grade of a highway is its slope expressed as a percent. What is the grade of this highway? ==== Everyday Math #strong[Wheelchair ramp] The rules for wheelchair ramps require a maximum #math.equation(block: false, alt: "1")[$1$] inch rise for a #math.equation(block: false, alt: "12")[$12$] inch run. + ⓐ What run must the ramp have to accommodate a #math.equation(block: false, alt: "24-inch")[$"24-inch"$] rise to the door? + ⓑ Draw a model of this ramp. #solutionbox[ + ⓐ 288 inches (24 feet) + ⓑ Models will vary. ] #strong[Wheelchair ramp] A #math.equation(block: false, alt: "1-inch")[$"1-inch"$] rise for a #math.equation(block: false, alt: "16-inch")[$"16-inch"$] run makes it easier for the wheelchair rider to ascend the ramp. + ⓐ What run must the ramp have to easily accommodate a #math.equation(block: false, alt: "24-inch")[$"24-inch"$] rise to the door? + ⓑ Draw a model of this ramp. ==== Writing Exercises What does the sign of the slope tell you about a line? #solutionbox[ Answers will vary. ] How does the graph of a line with slope #math.equation(block: false, alt: "m equals the fraction 1 over 2")[$m = frac(1, 2)$] differ from the graph of a line with slope #math.equation(block: false, alt: "m equals 2 ?")[$m = 2 ?$] Why is the slope of a vertical line undefined? #solutionbox[ Answers will vary. ] Explain how you can graph a line given a point and its slope. ==== Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. #figure(figph[A self-assessment checklist for students to rate their understanding and confidence in various skills related to finding, graphing, and applying slope, with options: Confidently, With some help, or No-I don't get it!], alt: "A self-assessment checklist for students to rate their understanding and confidence in various skills related to finding, graphing, and applying slope, with options: Confidently, With some help, or No-I don't get it!", caption: none) ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this? === Chapter Review Exercises ==== Use the Rectangular Coordinate System #strong[Plot Points in a Rectangular Coordinate System] In the following exercises, plot each point in a rectangular coordinate system. #math.equation(block: true, alt: "open parenthesis 1 , 3 close parenthesis , open parenthesis 3 , 1 close parenthesis")[$( 1 , 3 ) , ( 3 , 1 )$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -6 to 6. The y-axis runs from 6 to -6. The points “ordered pair 1,3” and “ordered pair 3,1” are plotted.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -6 to 6. The y-axis runs from 6 to -6. The points “ordered pair 1,3” and “ordered pair 3,1” are plotted.", caption: none) ] #math.equation(block: true, alt: "open parenthesis 2 , 5 close parenthesis , open parenthesis 5 , 2 close parenthesis")[$( 2 , 5 ) , ( 5 , 2 )$] In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located. + ⓐ #math.equation(block: false, alt: "open parenthesis −1 , −5 close parenthesis")[$#h(0.2em) ( −1 , −5 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis −3 , 4 close parenthesis")[$#h(0.2em) ( −3 , 4 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 2 , −3 close parenthesis")[$#h(0.2em) ( 2 , −3 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis 1 , the fraction 5 over 2 close parenthesis")[$#h(0.2em) ( 1 , frac(5, 2) )$] #solutionbox[ + ⓐ III + ⓑ II + ⓒ IV + ⓐ I #figure(figph[A Cartesian coordinate plane, also known as an xy-grid, displays four quadrants (I, II, III, IV) and four distinct points labeled a, b, c, and d, plotted within these quadrants.], alt: "A Cartesian coordinate plane, also known as an xy-grid, displays four quadrants (I, II, III, IV) and four distinct points labeled a, b, c, and d, plotted within these quadrants.", caption: none) ] + ⓐ #math.equation(block: false, alt: "open parenthesis 3 , −2 close parenthesis")[$#h(0.2em) ( 3 , −2 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis −4 , −1 close parenthesis")[$#h(0.2em) ( −4 , −1 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis −5 , 4 close parenthesis")[$#h(0.2em) ( −5 , 4 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis 2 , the fraction 10 over 3 close parenthesis")[$#h(0.2em) ( 2 , frac(10, 3) )$] #strong[Identify Points on a Graph] In the following exercises, name the ordered pair of each point shown in the rectangular coordinate system. #figure(figph[The graph shows the x y-coordinate plane. The axes run from -7 to 7. “a” is plotted at 5, 3, “b” at 2, -1, “c” at -3,-2, and “d” at -1,4. ], alt: "The graph shows the x y-coordinate plane. The axes run from -7 to 7. “a” is plotted at 5, 3, “b” at 2, -1, “c” at -3,-2, and “d” at -1,4.", caption: none) #solutionbox[ + ⓐ (5,3) + ⓑ (2,−1) + ⓒ (−3,−2) + ⓓ (−1,4) ] #figure(figph[The graph shows the x y-coordinate plane. The axes run from -7 to 7. “a” is plotted at -2, 2, “b” at 3, 5, “c” at 4,-1, and “d” at -1,3.], alt: "The graph shows the x y-coordinate plane. The axes run from -7 to 7. “a” is plotted at -2, 2, “b” at 3, 5, “c” at 4,-1, and “d” at -1,3.", caption: none) #figure(figph[The graph shows the x y-coordinate plane. The axes run from -7 to 7. “a” is plotted at 2, 0, “b” at 0, -5, “c” at -4,0, and “d” at 0,3.], alt: "The graph shows the x y-coordinate plane. The axes run from -7 to 7. “a” is plotted at 2, 0, “b” at 0, -5, “c” at -4,0, and “d” at 0,3.", caption: none) #solutionbox[ + ⓐ (2,0) + ⓑ (0,−5) + ⓒ (−4,0) + ⓓ (0,3) ] #figure(figph[The graph shows the x y-coordinate plane. The axes run from -7 to 7. “a” is plotted at 0, 4, “b” at 5, 0, “c” at 0,-1, and “d” at -3,0.], alt: "The graph shows the x y-coordinate plane. The axes run from -7 to 7. “a” is plotted at 0, 4, “b” at 5, 0, “c” at 0,-1, and “d” at -3,0.", caption: none) #strong[Verify Solutions to an Equation in Two Variables] In the following exercises, find the ordered pairs that are solutions to the given equation. #math.equation(block: true, alt: "5 x plus y equals 10")[$5 x + y = 10$] + ⓐ #math.equation(block: false, alt: "open parenthesis 5 , 1 close parenthesis")[$#h(0.2em) ( 5 , 1 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$#h(0.2em) ( 2 , 0 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 4 , −10 close parenthesis")[$#h(0.2em) ( 4 , −10 )$] #solutionbox[ ⓑ (2, 0),ⓒ (4, –10) ] #math.equation(block: true, alt: "y equals 6 x minus 2")[$y = 6 x − 2$] + ⓐ #math.equation(block: false, alt: "open parenthesis 1 , 4 close parenthesis")[$#h(0.2em) ( 1 , 4 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis the fraction 1 over 3 , 0 close parenthesis")[$#h(0.2em) ( frac(1, 3) , 0 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 6 , −2 close parenthesis")[$#h(0.2em) ( 6 , −2 )$] #strong[Complete a Table of Solutions to a Linear Equation in Two Variables] In the following exercises, complete the table to find solutions to each linear equation. #math.equation(block: true, alt: "y equals 4 x minus 1")[$y = 4 x − 1$] #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]), [#math.equation(block: false, alt: "0")[$0$]], [], [], [#math.equation(block: false, alt: "1")[$1$]], [], [], [#math.equation(block: false, alt: "−2")[$−2$]], [], [], )) #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]), [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "−1")[$−1$]], [#math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$( 0 , −1 )$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "open parenthesis 1 , 3 close parenthesis")[$( 1 , 3 )$]], [#math.equation(block: false, alt: "−2")[$−2$]], [#math.equation(block: false, alt: "−9")[$−9$]], [#math.equation(block: false, alt: "open parenthesis −2 , −9 close parenthesis")[$( −2 , −9 )$]], )) ] #math.equation(block: true, alt: "y equals minus the fraction 1 over 2 x plus 3")[$y = − frac(1, 2) x + 3$] #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]), [#math.equation(block: false, alt: "0")[$0$]], [], [], [#math.equation(block: false, alt: "1")[$1$]], [], [], [#math.equation(block: false, alt: "−2")[$−2$]], [], [], )) #math.equation(block: true, alt: "x plus 2 y equals 5")[$x + 2 y = 5$] #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]), [], [#math.equation(block: false, alt: "0")[$0$]], [], [#math.equation(block: false, alt: "1")[$1$]], [], [], [#math.equation(block: false, alt: "−1")[$−1$]], [], [], )) #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]), [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "open parenthesis 5 , 0 close parenthesis")[$( 5 , 0 )$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "open parenthesis 1 , 2 close parenthesis")[$( 1 , 2 )$]], [#math.equation(block: false, alt: "−1")[$−1$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "open parenthesis −1 , 3 close parenthesis")[$( −1 , 3 )$]], )) ] #math.equation(block: true, alt: "3 x minus 2 y equals 6")[$3 x − 2 y = 6$] #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]), [#math.equation(block: false, alt: "0")[$0$]], [], [], [], [#math.equation(block: false, alt: "0")[$0$]], [], [#math.equation(block: false, alt: "−2")[$−2$]], [], [], )) #strong[Find Solutions to a Linear Equation in Two Variables] In the following exercises, find three solutions to each linear equation. #math.equation(block: true, alt: "x plus y equals 3")[$x + y = 3$] #solutionbox[ Answers will vary. ] #math.equation(block: true, alt: "x plus y equals −4")[$x + y = −4$] #math.equation(block: true, alt: "y equals 3 x plus 1")[$y = 3 x + 1$] #solutionbox[ Answers will vary. ] #math.equation(block: true, alt: "y equals minus x minus 1")[$y = − x − 1$] ==== Graphing Linear Equations #strong[Recognize the Relation Between the Solutions of an Equation and its Graph] In each of the following exercises, an equation and its graph is shown. For each ordered pair, decide + ⓐ if the ordered pair is a solution to the equation. + ⓑ if the point is on the line. #math.equation(block: true, alt: "y equals minus x plus 4")[$y = − x + 4$] #linebreak() #figure(figph[The graph shows the x y-coordinate plane. The axes run from -7 to 7. A line passes through the points “ordered pair 0, 4” and “ordered pair 4, 0”.], alt: "The graph shows the x y-coordinate plane. The axes run from -7 to 7. A line passes through the points “ordered pair 0, 4” and “ordered pair 4, 0”.", caption: none) + #math.equation(block: false, alt: "open parenthesis 0 , 4 close parenthesis")[$#h(0.2em) ( 0 , 4 )$] + #math.equation(block: false, alt: "open parenthesis −1 , 3 close parenthesis")[$#h(0.2em) ( −1 , 3 )$] + #math.equation(block: false, alt: "open parenthesis 2 , 2 close parenthesis")[$#h(0.2em) ( 2 , 2 )$] + #math.equation(block: false, alt: "open parenthesis −2 , 6 close parenthesis")[$#h(0.2em) ( −2 , 6 )$] #solutionbox[ + ⓐ yes ⓑ yes + ⓐ no ⓑ no + ⓐ yes ⓑ yes + ⓐ yes ⓑ yes ] #math.equation(block: true, alt: "y equals the fraction 2 over 3 x minus 1")[$y = frac(2, 3) x − 1$] #linebreak() #figure(figph[The graph shows the x y-coordinate plane. The axes run from -7 to 7. A line passes through the points “ordered pair 0, -1” and “ordered pair 3, 1”.], alt: "The graph shows the x y-coordinate plane. The axes run from -7 to 7. A line passes through the points “ordered pair 0, -1” and “ordered pair 3, 1”.", caption: none) + #math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$#h(0.2em) ( 0 , −1 )$] + #math.equation(block: false, alt: "open parenthesis 3 , 1 close parenthesis")[$#h(0.2em) ( 3 , 1 )$] + #math.equation(block: false, alt: "open parenthesis −3 , −3 close parenthesis")[$#h(0.2em) ( −3 , −3 )$] + #math.equation(block: false, alt: "open parenthesis 6 , 4 close parenthesis")[$#h(0.2em) ( 6 , 4 )$] #strong[Graph a Linear Equation by Plotting Points] In the following exercises, graph by plotting points. #math.equation(block: true, alt: "y equals 4 x minus 3")[$y = 4 x − 3$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. Each axis runs from -6 to 6. A line passes through the points “ordered pair 1, 1” and “ordered pair 0, -3”.], alt: "The graph shows the x y-coordinate plane. Each axis runs from -6 to 6. A line passes through the points “ordered pair 1, 1” and “ordered pair 0, -3”.", caption: none) ] #math.equation(block: true, alt: "y equals −3 x")[$y = −3 x$] #math.equation(block: true, alt: "2 x plus y equals 7")[$2 x + y = 7$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. Each axis runs from -6 to 6. A line passes through the points “ordered pair 1, 5” and “ordered pair 0, 7”.], alt: "The graph shows the x y-coordinate plane. Each axis runs from -6 to 6. A line passes through the points “ordered pair 1, 5” and “ordered pair 0, 7”.", caption: none) ] #strong[Graph Vertical and Horizontal lines] In the following exercises, graph the vertical or horizontal lines. #math.equation(block: true, alt: "y equals −2")[$y = −2$] #math.equation(block: true, alt: "x equals 3")[$x = 3$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. Each axis runs from -6 to 6. A vertical line passes through the point “ordered pair 0, 3”.], alt: "The graph shows the x y-coordinate plane. Each axis runs from -6 to 6. A vertical line passes through the point “ordered pair 0, 3”.", caption: none) ] ==== Graphing with Intercepts #strong[Identify the Intercepts on a Graph] In the following exercises, find the #math.equation(block: false, alt: "x minus")[$x −$] and #math.equation(block: false, alt: "y -intercepts .")[$y "-intercepts" .$] #figure(figph[The graph shows the x y-coordinate plane. The axes run from -7 to 7. A line passes through the points “ordered pair 0, 4” and “ordered pair -4, 0”.], alt: "The graph shows the x y-coordinate plane. The axes run from -7 to 7. A line passes through the points “ordered pair 0, 4” and “ordered pair -4, 0”.", caption: none) #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -1 to 6. The y-axis runs from -4 to 2. A line passes through the points “ordered pair 5, 1” and “ordered pair 0, -3”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -1 to 6. The y-axis runs from -4 to 2. A line passes through the points “ordered pair 5, 1” and “ordered pair 0, -3”.", caption: none) #solutionbox[ (0,3) (3,0) ] #strong[Find the Intercepts from an Equation of a Line] In the following exercises, find the intercepts. #math.equation(block: true, alt: "x plus y equals 5")[$x + y = 5$] #math.equation(block: true, alt: "x minus y equals −1")[$x − y = −1$] #solutionbox[ (−1,0) (0,1) ] #math.equation(block: true, alt: "y equals the fraction 3 over 4 x minus 12")[$y = frac(3, 4) x − 12$] #math.equation(block: true, alt: "y equals 3 x")[$y = 3 x$] #solutionbox[ (0,0) ] #strong[Graph a Line Using the Intercepts] In the following exercises, graph using the intercepts. #math.equation(block: true, alt: "minus x plus 3 y equals 3")[$− x + 3 y = 3$] #math.equation(block: true, alt: "x plus y equals −2")[$x + y = −2$] #solutionbox[ #figure(figph[This answer graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The equation x plus y equals -2 is shown. A line passes through the intercepts with coordinates 0, –2 and –2, 0.], alt: "This answer graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The equation x plus y equals -2 is shown. A line passes through the intercepts with coordinates 0, –2 and –2, 0.", caption: none) ] #strong[Choose the Most Convenient Method to Graph a Line] In the following exercises, identify the most convenient method to graph each line. #math.equation(block: true, alt: "x equals 5")[$x = 5$] #math.equation(block: true, alt: "y equals −3")[$y = −3$] #solutionbox[ horizontal line ] #math.equation(block: true, alt: "2 x plus y equals 5")[$2 x + y = 5$] #math.equation(block: true, alt: "x minus y equals 2")[$x − y = 2$] #solutionbox[ intercepts ] #math.equation(block: true, alt: "y equals the fraction 1 over 2 x plus 2")[$y = frac(1, 2) x + 2$] #math.equation(block: true, alt: "y equals the fraction 3 over 4 x minus 1")[$y = frac(3, 4) x − 1$] #solutionbox[ plotting points ] ==== Understand Slope of a Line #strong[Use Geoboards to Model Slope] In the following exercises, find the slope modeled on each geoboard. #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 4 and the point in column 4 row 2.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 4 and the point in column 4 row 2.", caption: none) #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 5 and the point in column 4 row 1.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 5 and the point in column 4 row 1.", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 4 over 3")[$frac(4, 3)$] ] #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 3 and the point in column 4 row 4.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 3 and the point in column 4 row 4.", caption: none) #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 2 and the point in column 4 row 4.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 2 and the point in column 4 row 4.", caption: none) #solutionbox[ #math.equation(block: true, alt: "minus the fraction 2 over 3")[$− frac(2, 3)$] ] In the following exercises, model each slope. Draw a picture to show your results. #math.equation(block: true, alt: "the fraction 1 over 3")[$frac(1, 3)$] #math.equation(block: true, alt: "the fraction 3 over 2")[$frac(3, 2)$] #solutionbox[ #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 5 and the point in column 3 row 2.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 5 and the point in column 3 row 2.", caption: none) ] #math.equation(block: true, alt: "minus the fraction 2 over 3")[$− frac(2, 3)$] #math.equation(block: true, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$] #solutionbox[ #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 2 and the point in column 3 row 3.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 2 and the point in column 3 row 3.", caption: none) ] #strong[Find the Slope of a Line from its Graph] In the following exercises, find the slope of each line shown. #figure(figph[The graph shows the x y-coordinate plane. The axes run from -7 to 7. A line passes through the points “ordered pair 0, 0” and “ordered pair 2, -6”.], alt: "The graph shows the x y-coordinate plane. The axes run from -7 to 7. A line passes through the points “ordered pair 0, 0” and “ordered pair 2, -6”.", caption: none) #figure(figph[The graph shows the x y-coordinate plane. The axes run from -7 to 7. A line passes through the points “ordered pair 0, 4” and “ordered pair -4, 0”.], alt: "The graph shows the x y-coordinate plane. The axes run from -7 to 7. A line passes through the points “ordered pair 0, 4” and “ordered pair -4, 0”.", caption: none) #solutionbox[ 1 ] #figure(figph[The graph shows the x y-coordinate plane. The axes run from -7 to 7. A line passes through the points “ordered pair -4, -4” and “ordered pair 5, -1”.], alt: "The graph shows the x y-coordinate plane. The axes run from -7 to 7. A line passes through the points “ordered pair -4, -4” and “ordered pair 5, -1”.", caption: none) #figure(figph[The graph shows the x y-coordinate plane. The axes run from -7 to 7. A line passes through the points “ordered pair -3, 6” and “ordered pair 5, 2”.], alt: "The graph shows the x y-coordinate plane. The axes run from -7 to 7. A line passes through the points “ordered pair -3, 6” and “ordered pair 5, 2”.", caption: none) #solutionbox[ #math.equation(block: true, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$] ] #strong[Find the Slope of Horizontal and Vertical Lines] In the following exercises, find the slope of each line. #math.equation(block: true, alt: "y equals 2")[$y = 2$] #math.equation(block: true, alt: "x equals 5")[$x = 5$] #solutionbox[ undefined ] #math.equation(block: true, alt: "x equals −3")[$x = −3$] #math.equation(block: true, alt: "y equals −1")[$y = −1$] #solutionbox[ 0 ] #strong[Use the Slope Formula to find the Slope of a Line between Two Points] In the following exercises, use the slope formula to find the slope of the line between each pair of points. #math.equation(block: true, alt: "open parenthesis 2 , 1 close parenthesis , open parenthesis 4 , 5 close parenthesis")[$( 2 , 1 ) , ( 4 , 5 )$] #math.equation(block: true, alt: "open parenthesis −1 , −1 close parenthesis , open parenthesis 0 , −5 close parenthesis")[$( −1 , −1 ) , ( 0 , −5 )$] #solutionbox[ −4 ] #math.equation(block: true, alt: "open parenthesis 3 , 5 close parenthesis , open parenthesis 4 , −1 close parenthesis")[$( 3 , 5 ) , ( 4 , −1 )$] #math.equation(block: true, alt: "open parenthesis −5 , −2 close parenthesis , open parenthesis 3 , 2 close parenthesis")[$( −5 , −2 ) , ( 3 , 2 )$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 2")[$frac(1, 2)$] ] #strong[Graph a Line Given a Point and the Slope] In the following exercises, graph the line given a point and the slope. #math.equation(block: true, alt: "open parenthesis 2 , −2 close parenthesis ; m equals the fraction 5 over 2")[$( 2 , −2 ) ; m = frac(5, 2)$] #math.equation(block: true, alt: "open parenthesis −3 , 4 close parenthesis ; m equals minus the fraction 1 over 3")[$( −3 , 4 ) ; m = − frac(1, 3)$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from -6 to 6. The y-axis runs from -4 to 2. A line passes through the points “ordered pair -3, 4” and “ordered pair 1, 3”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -6 to 6. The y-axis runs from -4 to 2. A line passes through the points “ordered pair -3, 4” and “ordered pair 1, 3”.", caption: none) ] #strong[Solve Slope Applications] In the following exercise, solve the slope application. A roof has rise #math.equation(block: false, alt: "10")[$10$] feet and run #math.equation(block: false, alt: "15")[$15$] feet. What is its slope? === Chapter Practice Test Plot and label these points: + ⓐ #math.equation(block: false, alt: "open parenthesis 2 , 5 close parenthesis")[$#h(0.2em) ( 2 , 5 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis −1 , −3 close parenthesis")[$#h(0.2em) ( −1 , −3 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis −4 , 0 close parenthesis")[$#h(0.2em) ( −4 , 0 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis 3 , −5 close parenthesis")[$#h(0.2em) ( 3 , −5 )$] + ⓔ #math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis")[$#h(0.2em) ( −2 , 1 )$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The axes extend from -6 to 6. a is plotted at 2, 5, b at -1, -3, c at -4, 0, d at 3, -5, and e at -2,1.], alt: "The graph shows the x y-coordinate plane. The axes extend from -6 to 6. a is plotted at 2, 5, b at -1, -3, c at -4, 0, d at 3, -5, and e at -2,1.", caption: none) ] Name the ordered pair for each point shown. #figure(figph[The graph shows the x y-coordinate plane. The axes extend from -7 to 7. A is plotted at -4, 1, B at 3, 2, C at 0, -2, D at -1, -4, and E at 4,-3.], alt: "The graph shows the x y-coordinate plane. The axes extend from -7 to 7. A is plotted at -4, 1, B at 3, 2, C at 0, -2, D at -1, -4, and E at 4,-3.", caption: none) Find the #math.equation(block: false, alt: "x -intercept")[$x "-intercept"$] and #math.equation(block: false, alt: "y -intercept")[$y "-intercept"$] on the line shown. #figure(figph[ The graph shows the x y-coordinate plane. The x-axis runs from -7 to 7. The y-axis runs from -7 to 7. A line passes through the points “ordered pair 4, 0” and “ordered pair 0, -2”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -7 to 7. The y-axis runs from -7 to 7. A line passes through the points “ordered pair 4, 0” and “ordered pair 0, -2”.", caption: none) #solutionbox[ (4,0), (0,−2) ] Find the #math.equation(block: false, alt: "x -intercept")[$x "-intercept"$] and #math.equation(block: false, alt: "y -intercept")[$y "-intercept"$] of the equation #math.equation(block: false, alt: "3 x minus y equals 6 .")[$3 x − y = 6 .$] Is #math.equation(block: false, alt: "open parenthesis 1 , 3 close parenthesis")[$( 1 , 3 )$] a solution to the equation #math.equation(block: false, alt: "x plus 4 y equals 12 ?")[$x + 4 y = 12 ?$] How do you know? #solutionbox[ no; 1 + 4 · 3 ≠ 12 ] Complete the table to find four solutions to the equation #math.equation(block: false, alt: "y equals minus x plus 1 .")[$y = − x + 1 .$] #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]), [#math.equation(block: false, alt: "0")[$0$]], [], [], [#math.equation(block: false, alt: "1")[$1$]], [], [], [#math.equation(block: false, alt: "3")[$3$]], [], [], [#math.equation(block: false, alt: "−2")[$−2$]], [], [], )) Complete the table to find three solutions to the equation #math.equation(block: false, alt: "4 x plus y equals 8")[$4 x + y = 8$] #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]), [#math.equation(block: false, alt: "0")[$0$]], [], [], [], [#math.equation(block: false, alt: "0")[$0$]], [], [#math.equation(block: false, alt: "3")[$3$]], [], [], )) #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]), [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "open parenthesis 0 , 8 close parenthesis")[$( 0 , 8 )$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "−4")[$−4$]], [#math.equation(block: false, alt: "open parenthesis 3 , −4 close parenthesis")[$( 3 , −4 )$]], )) ] In the following exercises, find three solutions to each equation and then graph each line. #math.equation(block: true, alt: "y equals −3 x")[$y = −3 x$] #math.equation(block: true, alt: "2 x plus 3 y equals −6")[$2 x + 3 y = −6$] #solutionbox[ #figure(figph[ The graph shows the x y-coordinate plane. The x-axis runs from -6 to 6. The y-axis runs from -6 to 6. A line passes through the points “ordered pair 0, -2” and “ordered pair -3, 0”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from -6 to 6. The y-axis runs from -6 to 6. A line passes through the points “ordered pair 0, -2” and “ordered pair -3, 0”.", caption: none) ] In the following exercises, find the slope of each line. #figure(figph[The graph shows the x y-coordinate plane. The axes run from -7 to 7. The y-axis runs from -5 to -4. A line passes through the points “ordered pair 6, 4” and “ordered pair 0, -3”.], alt: "The graph shows the x y-coordinate plane. The axes run from -7 to 7. The y-axis runs from -5 to -4. A line passes through the points “ordered pair 6, 4” and “ordered pair 0, -3”.", caption: none) #figure(figph[The graph shows the x y-coordinate plane. The axes run from -7 to 7. A line passes through the points “ordered pair 3, 0” and “ordered pair 1, 5”.], alt: "The graph shows the x y-coordinate plane. The axes run from -7 to 7. A line passes through the points “ordered pair 3, 0” and “ordered pair 1, 5”.", caption: none) #solutionbox[ #math.equation(block: true, alt: "minus the fraction 5 over 2")[$− frac(5, 2)$] ] Use the slope formula to find the slope of the line between #math.equation(block: false, alt: "open parenthesis 0 , −4 close parenthesis")[$( 0 , −4 )$] and #math.equation(block: false, alt: "open parenthesis 5 , 2 close parenthesis .")[$( 5 , 2 ) .$] Find the slope of the line #math.equation(block: false, alt: "y equals 2 .")[$y = 2 .$] #solutionbox[ 0 ] Graph the line passing through #math.equation(block: false, alt: "open parenthesis 1 , 1 close parenthesis")[$( 1 , 1 )$] with slope #math.equation(block: false, alt: "m equals the fraction 3 over 2 .")[$m = frac(3, 2) .$] A bicycle route climbs #math.equation(block: false, alt: "20")[$20$] feet for #math.equation(block: false, alt: "1,000")[$1,000$] feet of horizontal distance. What is the slope of the route? #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 50")[$frac(1, 50)$] ]