#set document(title: "3.2 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")) == 3.2#h(0.6em)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 open parenthesis 1 minus 4 close parenthesis over open parenthesis 8 minus 2 close parenthesis .")[$frac(( 1 − 4 ), ( 8 − 2 )) .$] #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: false, alt: "0")[$0$]; 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")[$− 5 ; #h(0.2em) − 5 ; #h(0.2em) 5$] ] ] === Find the Slope of a Line When you graph linear equations, you may notice that some lines tilt up as they go from left to right and some lines tilt down. Some lines are very steep and some lines are flatter. In mathematics, the measure of the steepness of a line is called the #emph[slope] of the line. The concept of slope has many applications in the real world. In construction the pitch of a roof, the slant of the plumbing pipes, and the steepness of the stairs are all applications of slope. and as you ski or jog down a hill, you definitely experience slope. We can assign a numerical value to the slope of a line by finding the ratio of the rise and run. The #emph[rise] is the amount the vertical distance changes while the #emph[run] measures the horizontal change, as shown in this illustration. Slope is a rate of change. #figure(figph[This figure has a diagram of two arrows. The first arrow is vertical and pointed up and labeled “rise”. The second arrow starts at the end of the first. The second arrow is horizontal and pointed right and labeled “run”.], alt: "This figure has a diagram of two arrows. The first arrow is vertical and pointed up and labeled “rise”. The second arrow starts at the end of the first. The second arrow is horizontal and pointed right and labeled “run”.", caption: none) #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 rise measures the vertical change and the run measures the horizontal change. ] To find the slope of a line, we locate two points on the line whose coordinates are integers. Then we sketch a right triangle where the two points are vertices and one side is horizontal and one side is vertical. To find the slope of the line, we measure the distance along the vertical and horizontal sides of the triangle. The vertical distance is called the #emph[rise] and the horizontal distance is called the #emph[run], #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Find the slope of a line from its graph using #math.equation(block: false, alt: "m equals the fraction rise over run .")[$m = frac("rise", "run") .$]] + Locate two points on the line whose coordinates are integers. + Starting with one point, sketch a right triangle, 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 1")[][ Find the slope of the line shown. #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 1 to 9. The y-axis runs from negative 1 to 7. The line goes through the points (0, 5), (3, 3), and (6, 1).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 1 to 9. The y-axis runs from negative 1 to 7. The line goes through the points (0, 5), (3, 3), and (6, 1).", caption: none) #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Locate two points on the graph whose #linebreak() coordinates are integers.], [#math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis")[$( 0 , 5 )$] and #math.equation(block: false, alt: "open parenthesis 3 , 3 close parenthesis")[$( 3 , 3 )$]]), [Starting at #math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis ,")[$( 0 , 5 ) ,$] sketch a right triangle to #linebreak() #math.equation(block: false, alt: "open parenthesis 3 , 3 close parenthesis")[$( 3 , 3 )$] as shown in this graph.], [#figure(figph[A graph illustrating the concept of slope with a downward-sloping line on a coordinate plane, showing labeled 'rise' and 'run' segments forming a right triangle.], alt: "A graph illustrating the concept of slope with a downward-sloping line on a coordinate plane, showing labeled 'rise' and 'run' segments forming a right triangle.", caption: none)], [Count the rise— since it goes down, it is negative.], [The rise is #math.equation(block: false, alt: "−2 .")[$−2 .$]], [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 .")[$− frac(2, 3) .$]], [], [So #emph[y] decreases by 2 units as #emph[x] increases by 3 units.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope of the line shown. #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 1 to 5. The y-axis runs from negative 6 to 1. The line goes through the points (0, negative 2) and (3, negative 6).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 1 to 5. The y-axis runs from negative 6 to 1. The line goes through the points (0, negative 2) and (3, negative 6).", 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 shown. #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 3 to 6. The y-axis runs from negative 3 to 2. The line goes through the points (0, 1) and (5, negative 2).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 3 to 6. The y-axis runs from negative 3 to 2. The line goes through the points (0, 1) and (5, negative 2).", caption: none) #solutionbox[ #math.equation(block: true, alt: "minus the fraction 3 over 5")[$− frac(3, 5)$] ] ] How do we find the slope of horizontal and vertical lines? To find the slope of the horizontal line, #math.equation(block: false, alt: "y equals 4 ,")[$y = 4 ,$] we could graph the line, find two points on it, and count the rise and the run. Let’s see what happens when we do this, as shown in the graph below. #figure(figph[The figure then shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 1 to 6. The y-axis runs from negative 1 to 8. The line goes through the points (0, 4) and (3, 4). What is the rise? The rise is 0. What is the run? The run is 3. What is the slope? m equals rise divided by run. m equals 0 divided by 3. m equals 0. The slope of the horizontal line y equals 4 is 0.], alt: "The figure then shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 1 to 6. The y-axis runs from negative 1 to 8. The line goes through the points (0, 4) and (3, 4). What is the rise? The rise is 0. What is the run? The run is 3. What is the slope? m equals rise divided by run. m equals 0 divided by 3. m equals 0. The slope of the horizontal line y equals 4 is 0.", 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 is 0.")[$y = 4 #h(0.2em) "is 0."$]], )) Let’s also consider a vertical line, the line #math.equation(block: false, alt: "x equals 3 ,")[$x = 3 ,$] as shown in the graph. #figure(figph[The figure then shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 2 to 6. The y-axis runs from negative 3 to 3. The line goes through the points (3, 0) and (3, 2). What is the rise? The rise is 2. What is the run? The run is 0. What is the slope? m equals rise divided by run. m equals 2 divided by 0.], alt: "The figure then shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 2 to 6. The y-axis runs from negative 3 to 3. The line goes through the points (3, 0) and (3, 2). What is the rise? The rise is 2. What is the run? The run is 0. What is the slope? m equals rise divided by run. m equals 2 divided by 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")[$m = frac(2, 0)$]], )) The slope is undefined since division by zero is undefined. So we say that the slope of the vertical line #math.equation(block: false, alt: "x equals 3")[$x = 3$] is undefined. All horizontal lines have slope 0. When the #emph[y]-coordinates are the same, the rise is 0. The slope of any vertical line is undefined. When the #emph[x]-coordinates of a line are all the same, the run is 0. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Slope of a Horizontal and Vertical Line ] The slope of a horizontal line, #math.equation(block: false, alt: "y equals b ,")[$y = b ,$] is 0. The slope of a vertical line, #math.equation(block: false, alt: "x equals a ,")[$x = a ,$] is undefined. ] #examplebox("Example 2")[][ Find the slope of each line: ⓐ #math.equation(block: false, alt: "x equals 8")[$x = 8$] ⓑ #math.equation(block: false, alt: "y equals −5 .")[$y = −5 .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "x equals 8")[$x = 8$] #linebreak() This is a vertical line. Its slope is undefined. #linebreak() ⓑ #math.equation(block: false, alt: "y equals −5")[$y = −5$] #linebreak() This is a horizontal line. It has slope 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 image shows four arrows. The first arrow is slanted and pointing up and to the right and is labeled “positive”. The second arrow is slanted and pointing down and to the right and labeled “negative”. The third arrow is horizontal and labeled “zero”. The fourth arrow is vertical and labeled “undefined”.], alt: "The image shows four arrows. The first arrow is slanted and pointing up and to the right and is labeled “positive”. The second arrow is slanted and pointing down and to the right and labeled “negative”. The third arrow is horizontal and labeled “zero”. The fourth arrow is vertical and labeled “undefined”.", caption: none) ] Sometimes we’ll need to find the slope of a line between two points when we don’t 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 as we’ll see, there is a way to find the slope without graphing. Before we get to it, we need to introduce some algebraic notation. We have seen that an 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 subscripts to distinguish the points. #math.equation(block: true, alt: "open parenthesis x sub 1 , y sub 1 close parenthesis, read “ x sub 1, y sub 1”; open parenthesis x sub 2 , y sub 2 close parenthesis, read “ x sub 2, y sub 2”")[$( x_(1) , y_(1) ) & "read “" x #h(0.2em) "sub 1," #h(0.2em) y #h(0.2em) "sub 1”" \ ( x_(2) , y_(2) ) & "read “" x #h(0.2em) "sub 2," #h(0.2em) y #h(0.2em) "sub 2”"$]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 ,")[$( x_(3) , y_(3) ) ,$]#math.equation(block: false, alt: "open parenthesis x sub 4 , y sub 4 close parenthesis ,")[$( x_(4) , y_(4) ) ,$] and so on. Let’s see how the rise and run relate to the coordinates of the two points by taking 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 ,")[$( 7 , 6 ) ,$] as shown in this graph. #figure(figph[The figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 1 to 7. The y-axis runs from negative 1 to 7. The line goes through the points (2, 3) and (7, 6). A right triangle is drawn by connecting the three points (2, 3), (2, 6), and (7, 6). The point (2, 3) is labeled (x 1, y 1). The point (7, 6) is labeled (x 2, y 2). The vertical side of the triangle has labels y 2 minus y 1, 6 minus 3, and 3. The horizontal side of the triangle has labels x 2 minus x 1, 7 minus 2, and 5.], alt: "The figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 1 to 7. The y-axis runs from negative 1 to 7. The line goes through the points (2, 3) and (7, 6). A right triangle is drawn by connecting the three points (2, 3), (2, 6), and (7, 6). The point (2, 3) is labeled (x 1, y 1). The point (7, 6) is labeled (x 2, y 2). The vertical side of the triangle has labels y 2 minus y 1, 6 minus 3, and 3. The horizontal side of the triangle has labels x 2 minus x 1, 7 minus 2, and 5.", caption: none) #figure(table( columns: 2, align: left, inset: 6pt, table.header([Since we have two points, we will use subscript notation.], [#math.equation(block: false, alt: "open parenthesis 2 , x sub 1 , 3 y sub 1 close parenthesis open parenthesis 7 , 6 x sub 2 , y sub 2 close parenthesis")[$( limits(2 ",")^(x_(1) ,) limits(3)^(y_(1)) ) #h(0.2em) ( limits(7 "," 6)^(x_(2) , y_(2)) )$]]), [], [#math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$]], [On the graph, we counted the rise of 3 and the run of 5.], [#math.equation(block: false, alt: "m equals the fraction 3 over 5")[$m = frac(3, 5)$]], [Notice that the rise of 3 can be found by subtracting the #linebreak() #emph[y]-coordinates, 6 and 3, and the run of 5 can be found by #linebreak() subtracting the #emph[x]-coordinates 7 and 2.], [], [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 #math.equation(block: false, alt: "y sub 2 ,")[$y_(2) ,$] the #emph[y]-coordinate of the second point and 3 is #math.equation(block: false, alt: "y sub 1 ,")[$y_(1) ,$] the #emph[y]-coordinate #linebreak() of the first point. So we can rewrite the slope 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 #emph[x]-coordinate of the second point and 2 is the #emph[x]-coordinate #linebreak() of the first point. So again we rewrite the slope 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 of a line between two points] 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: true, alt: "m equals the fraction y sub 2 minus y sub 1 over x sub 2 minus x sub 1 .")[$m = frac(y_(2) − y_(1), x_(2) − x_(1)) .$] ] The slope is: #math.equation(block: true, alt: "y of the second point minus y of the first point; over; x of the second point minus x of the first point.")[$y #h(0.2em) "of the second point minus" #h(0.2em) y #h(0.2em) "of the first point" \ "over" \ x #h(0.2em) "of the second point minus" #h(0.2em) x #h(0.2em) "of the first point."$]#examplebox("Example 3")[][ Use the slope formula to 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 x sub 1 ,, y sub 1; minus 2 ,, minus 3 close parenthesis open parenthesis x sub 2 ,, y sub 2; minus 7 ,, 4 close parenthesis")[$( x_(1) , & y_(1) \ − 2 , & − 3 ) #h(0.2em) ( x_(2) , & y_(2) \ − 7 , & 4 )$]]), [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. #linebreak() #math.equation(block: false, alt: "y")[$y$] of the second point minus #math.equation(block: false, alt: "y")[$y$] of the first point], [], [#math.equation(block: false, alt: "x")[$x$] of the second point minus #math.equation(block: false, alt: "x")[$x$] 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 equals minus the fraction 7 over 5")[$m = frac(7, −5) \ m = − frac(7, 5)$]], [Let’s verify this slope on the graph shown.], [], )) #figure(figph[The figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 2. The y-axis runs from negative 6 to 6. The line goes through the points (negative 7, 4) and (negative 2, negative 3). A right triangle is drawn by connecting the three points (negative 7, 4), (negative 7, negative 3), and (negative 2, negative 3). The vertical side of the triangle is labeled “rise”. The horizontal side of the triangle is labeled “run”.], alt: "The figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 2. The y-axis runs from negative 6 to 6. The line goes through the points (negative 7, 4) and (negative 2, negative 3). A right triangle is drawn by connecting the three points (negative 7, 4), (negative 7, negative 3), and (negative 2, negative 3). The vertical side of the triangle is labeled “rise”. The horizontal side of the triangle 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")[$m = frac("rise", "run") \ m = frac(7, −5) \ m = − frac(7, 5)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use the slope formula to 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[ #math.equation(block: true, alt: "−1")[$−1$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use the slope formula to 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 Up to now, in this chapter, we have graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines. We can also graph a line when we know one point and the slope of the line. We will start by plotting the point and then use the definition of slope to draw the graph of the line. #examplebox("Example 4")[How to graph a Line Given a Point and the Slope][ 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[ #figure(figph[Step 1 is to plot the given point. Plot (1, negative 1). The figure then shows the graph of a point on the x y-coordinate plane. The x-axis runs from negative 2 to 6. The y-axis runs from negative 2 to 4. The point (1, negative 1) is plotted and labeled with its coordinates.], alt: "Step 1 is to plot the given point. Plot (1, negative 1). The figure then shows the graph of a point on the x y-coordinate plane. The x-axis runs from negative 2 to 6. The y-axis runs from negative 2 to 4. The point (1, negative 1) is plotted and labeled with its coordinates.", caption: none) #figure(figph[Step 2 is to use the slope formula m equals rise divided by run to identify the rise and the run. Identify the rise and the run. m equals 3 divided by 4. Rise divided by run equals 3 divided by 4. Rise equals 3. Run equals 4.], alt: "Step 2 is to use the slope formula m equals rise divided by run to identify the rise and the run. Identify the rise and the run. m equals 3 divided by 4. Rise divided by run equals 3 divided by 4. Rise equals 3. Run equals 4.", caption: none) #figure(figph[Step 3 is to start at the given point and count out the rise and run to mark the second point. Start at (1, negative 1) and count the rise and the run. Up 3 units, right 4 units. The figure then shows the graph of three points connected by two straight line segments on the x y-coordinate plane. The x-axis runs from negative 2 to 6. The y-axis runs from negative 2 to 4. The points (1, negative 1), (1, 2), and (5, 2) are plotted. A vertical line segment connects (1, negative 1) to (1, 2) and is labeled 3. A horizontal line segment connects (1, 2) to (5, 2) and is labeled 4.], alt: "Step 3 is to start at the given point and count out the rise and run to mark the second point. Start at (1, negative 1) and count the rise and the run. Up 3 units, right 4 units. The figure then shows the graph of three points connected by two straight line segments on the x y-coordinate plane. The x-axis runs from negative 2 to 6. The y-axis runs from negative 2 to 4. The points (1, negative 1), (1, 2), and (5, 2) are plotted. A vertical line segment connects (1, negative 1) to (1, 2) and is labeled 3. A horizontal line segment connects (1, 2) to (5, 2) and is labeled 4.", caption: none) #figure(figph[Step 4 is to connect the points with a line. Connect the two points with a line. . The figure then shows the graph of a straight line, three points, and two line segments on the x y-coordinate plane. The x-axis runs from negative 2 to 6. The y-axis runs from negative 2 to 4. The points (1, negative 1), (1, 2), and (5, 2) are plotted. A vertical line segment connects (1, negative 1) to (1, 2). A horizontal line segment connects (1, 2) to (5, 2) and is labeled 4. A straight line is drawn through the points (1, negative 1) and (5, 2) with arrows on both ends.], alt: "Step 4 is to connect the points with a line. Connect the two points with a line. . The figure then shows the graph of a straight line, three points, and two line segments on the x y-coordinate plane. The x-axis runs from negative 2 to 6. The y-axis runs from negative 2 to 4. The points (1, negative 1), (1, 2), and (5, 2) are plotted. A vertical line segment connects (1, negative 1) to (1, 2). A horizontal line segment connects (1, 2) to (5, 2) and is labeled 4. A straight line is drawn through the points (1, negative 1) and (5, 2) with arrows on both ends.", caption: none) You can check your work by finding a third point. Since the slope is #math.equation(block: false, alt: "m equals the fraction 3 over 4 ,")[$m = frac(3, 4) ,$] it can also be written as #math.equation(block: false, alt: "m equals the fraction −3 over −4")[$m = frac(−3, −4)$] (negative divided by negative is positive!). Go back to #math.equation(block: false, alt: "open parenthesis 1 , −1 close parenthesis")[$( 1 , −1 )$] and count out the rise, #math.equation(block: false, alt: "−3 ,")[$−3 ,$] and the run, #math.equation(block: false, alt: "−4 .")[$−4 .$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the line passing through the point #math.equation(block: false, alt: "open parenthesis 2 , −2 close parenthesis")[$( 2 , −2 )$] with the slope#math.equation(block: false, alt: "m equals the fraction 4 over 3 .")[$m = frac(4, 3) .$] #solutionbox[ #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (2, negative 2) and (5, 2).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (2, negative 2) and (5, 2).", 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 , 3 close parenthesis")[$( −2 , 3 )$] with the slope #math.equation(block: false, alt: "m equals the fraction 1 over 4 .")[$m = frac(1, 4) .$] #solutionbox[ #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (negative 2, 3) and (2, 4).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (negative 2, 3) and (2, 4).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Graph a line given a point and the slope.] + Plot the given point. + 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. + Starting at the given point, count out the rise and run to mark the second point. + Connect the points with a line. ] === Graph a Line Using its Slope and Intercept We have graphed linear equations by plotting points, using intercepts, recognizing horizontal and vertical lines, and using one point and the slope of the line. Once we see how an equation in slope–intercept form and its graph are related, we’ll have one more method we can use to graph lines. Let’s look at the graph of the equation #math.equation(block: false, alt: "y equals the fraction 1 over 2 x plus 3")[$y = frac(1, 2) x + 3$] and find its slope and #emph[y]-intercept. #figure(figph[The figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (0, 3), (2, 4), and (4, 5). A right triangle is drawn by connecting the three points (2, 4), (2, 5), and (4, 5). The vertical side of the triangle is labeled “Rise equals 1”. The horizontal side of the triangle is labeled “Run equals 2”. The line is labeled y equals 1 divided by 2 x plus 3.], alt: "The figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (0, 3), (2, 4), and (4, 5). A right triangle is drawn by connecting the three points (2, 4), (2, 5), and (4, 5). The vertical side of the triangle is labeled “Rise equals 1”. The horizontal side of the triangle is labeled “Run equals 2”. The line is labeled y equals 1 divided by 2 x plus 3.", caption: none) The red lines in the graph show us the rise is 1 and the run is 2. Substituting into the slope formula: #math.equation(block: true, alt: "m equals the fraction rise over run; m equals the fraction 1 over 2")[$m = frac("rise", "run") \ m = frac("1", "2")$]The #emph[y]-intercept is #math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis .")[$( 0 , 3 ) .$] Look at the equation of this line. #figure(figph[The figure shows the equation y equals 1 divided by 2 x plus 3. The 1 divided by 2 is emphasized in red. The 3 is emphasized in blue.], alt: "The figure shows the equation y equals 1 divided by 2 x plus 3. The 1 divided by 2 is emphasized in red. The 3 is emphasized in blue.", caption: none) Look at the slope and #emph[y]-intercept. #figure(figph[Slope m equals 1 divided by 2 and y-intercept (0, 3). The 1 divided by 2 is emphasized in red. The 3 is emphasized in blue.], alt: "Slope m equals 1 divided by 2 and y-intercept (0, 3). The 1 divided by 2 is emphasized in red. The 3 is emphasized in blue.", caption: none) When a linear equation is solved for #emph[y], the coefficient of the #emph[x] term is the slope and the constant term is the #emph[y]-coordinate of the #emph[y]-intercept. We say that the equation #math.equation(block: false, alt: "y equals the fraction 1 over 2 x plus 3")[$y = frac(1, 2) x + 3$] is in slope–intercept form. Sometimes the slope–intercept form is called the “#emph[y]-form.” #figure(figph[m equals 1 divided by 2; y-intercept is (0, 3). y equals 1 divided by 2 x plus 3. y equals m x plus b. The m and 1 divided by 2 are emphasized in red. The b and 3 are emphasized in blue.], alt: "m equals 1 divided by 2; y-intercept is (0, 3). y equals 1 divided by 2 x plus 3. y equals m x plus b. The m and 1 divided by 2 are emphasized in red. The b and 3 are emphasized in blue.", caption: none) #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Slope Intercept Form of an Equation of a Line] The slope–intercept form of an equation of a line with slope #emph[m] and #emph[y]-intercept, #math.equation(block: false, alt: "open parenthesis 0 , b close parenthesis")[$( 0 , b )$] is #math.equation(block: false, alt: "y equals m x plus b .")[$y = m x + b .$] ] Let’s practice finding the values of the slope and #emph[y]-intercept from the equation of a line. #examplebox("Example 5")[][ Identify the slope and #emph[y]-intercept of the line from the equation: ⓐ #math.equation(block: false, alt: "y equals minus the fraction 4 over 7 x minus 2")[$y = − frac(4, 7) x − 2$] ⓑ #math.equation(block: false, alt: "x plus 3 y equals 9")[$x + 3 y = 9$] #solutionbox[ ⓐ We compare our equation to the slope–intercept form of the equation. #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([Write the slope–intercept form of the equation of the line.], [#figure(figph[The equation for a straight line, y = mx + b, is displayed on a white background, with 'm' in red and 'b' in blue to highlight the slope and y-intercept.], alt: "The equation for a straight line, y = mx + b, is displayed on a white background, with 'm' in red and 'b' in blue to highlight the slope and y-intercept.", caption: none)]), [Write the equation of the line.], [#figure(figph[A mathematical equation is displayed against a white background, reading y = -4/7x - 2. The fraction 4/7 is in red, and the number 2 is in light blue.], alt: "A mathematical equation is displayed against a white background, reading y = -4/7x - 2. The fraction 4/7 is in red, and the number 2 is in light blue.", caption: none)], [Identify the slope.], [#figure(figph[The image shows the mathematical equation m = -4/7, where m represents a variable equal to a negative fraction.], alt: "The image shows the mathematical equation m = -4/7, where m represents a variable equal to a negative fraction.", caption: none)], [Identify the #emph[y]-intercept.], [#figure(figph[The image displays the text 'y-intercept is (0, -2)', which represents a coordinate point on a graph where a line or curve crosses the y-axis, indicating that when x is 0, y is -2.], alt: "The image displays the text 'y-intercept is (0, -2)', which represents a coordinate point on a graph where a line or curve crosses the y-axis, indicating that when x is 0, y is -2.", caption: none)], )) #linebreak() ⓑ When an equation of a line is not given in slope–intercept form, our first step will be to solve the equation for #emph[y]. #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([Solve for #emph[y].], [#math.equation(block: false, alt: "x plus 3 y equals 9")[$x + 3 y = 9$]]), [Subtract #emph[x] from each side.], [#figure(figph[A mathematical equation is displayed, reading '3y = -x + 9' on a white background.], alt: "A mathematical equation is displayed, reading '3y = -x + 9' on a white background.", caption: none)], [Divide both sides by 3.], [#figure(figph[A mathematical equation shows '3y divided by 3 equals the quantity of negative x plus 9, all divided by 3'. The equation simplifies to y = (-x+9)/3, demonstrating a step in solving for y.], alt: "A mathematical equation shows '3y divided by 3 equals the quantity of negative x plus 9, all divided by 3'. The equation simplifies to y = (-x+9)/3, demonstrating a step in solving for y.", caption: none)], [Simplify.], [#figure(figph[A mathematical equation is displayed against a white background: y = -1/3x + 3. This is the slope-intercept form of a linear equation, showing a negative slope and a positive y-intercept.], alt: "A mathematical equation is displayed against a white background: y = -1/3x + 3. This is the slope-intercept form of a linear equation, showing a negative slope and a positive y-intercept.", caption: none)], [Write the slope–intercept form of the equation of the line.], [#figure(figph[The equation y = mx + b, representing the slope-intercept form of a linear equation, is displayed. The 'm' is highlighted in red, indicating the slope, and 'b' is highlighted in blue, indicating the y-intercept.], alt: "The equation y = mx + b, representing the slope-intercept form of a linear equation, is displayed. The 'm' is highlighted in red, indicating the slope, and 'b' is highlighted in blue, indicating the y-intercept.", caption: none)], [Write the equation of the line.], [#figure(figph[A mathematical equation, 'y = -1/3x + 3', is displayed in black text with the fraction and the number 3 highlighted in red and light blue, respectively.], alt: "A mathematical equation, 'y = -1/3x + 3', is displayed in black text with the fraction and the number 3 highlighted in red and light blue, respectively.", caption: none)], [Identify the slope.], [#figure(figph[A mathematical equation shows 'm = -1/3', with the '1' and '3' in red font.], alt: "A mathematical equation shows 'm = -1/3', with the '1' and '3' in red font.", caption: none)], [Identify the #emph[y]-intercept.], [#figure(figph[The image displays the mathematical expression 'y-intercept is (0, 3)', indicating the point where a graph crosses the y-axis.], alt: "The image displays the mathematical expression 'y-intercept is (0, 3)', indicating the point where a graph crosses the y-axis.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Identify the slope and #emph[y]-intercept from the equation of the line. ⓐ #math.equation(block: false, alt: "y equals the fraction 2 over 5 x minus 1")[$y = frac(2, 5) x − 1$] ⓑ #math.equation(block: false, alt: "x plus 4 y equals 8")[$x + 4 y = 8$] #solutionbox[ ⓐ #math.equation(block: false, alt: "m equals the fraction 2 over 5 ; open parenthesis 0 , −1 close parenthesis")[$m = frac(2, 5) ; ( 0 , −1 )$] #linebreak() ⓑ #math.equation(block: false, alt: "m equals minus the fraction 1 over 4 ; open parenthesis 0 , 2 close parenthesis")[$m = − frac(1, 4) ; ( 0 , 2 )$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Identify the slope and #emph[y]-intercept from the equation of the line. ⓐ #math.equation(block: false, alt: "y equals minus the fraction 4 over 3 x plus 1")[$y = − frac(4, 3) x + 1$] ⓑ #math.equation(block: false, alt: "3 x plus 2 y equals 12")[$3 x + 2 y = 12$] #solutionbox[ ⓐ #math.equation(block: false, alt: "m equals minus the fraction 4 over 3 ; open parenthesis 0 , 1 close parenthesis")[$m = − frac(4, 3) ; ( 0 , 1 )$] #linebreak() ⓑ #math.equation(block: false, alt: "m equals minus the fraction 3 over 2 ; open parenthesis 0 , 6 close parenthesis")[$m = − frac(3, 2) ; ( 0 , 6 )$] ] ] We have graphed a line using the slope and a point. Now that we know how to find the slope and #emph[y]-intercept of a line from its equation, we can use the #emph[y]-intercept as the point, and then count out the slope from there. #examplebox("Example 6")[][ Graph the line of the equation #math.equation(block: false, alt: "y equals − x plus 4")[$y = "−" x + 4$] using its slope and #emph[y]-intercept. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "y equals m x plus b")[$y = m x + b$]]), [The equation is in slope–intercept form.], [#math.equation(block: false, alt: "y equals − x plus 4")[$y = "−" x + 4$]], [Identify the slope and #emph[y]-intercept.], [#math.equation(block: false, alt: "m equals −1")[$m = −1$] #linebreak() #emph[y]-intercept is #math.equation(block: false, alt: "open parenthesis 0 , 4 close parenthesis")[$( 0 , 4 )$]], [Plot the #emph[y]-intercept.], [See the graph.], [Identify the rise over the run.], [#math.equation(block: false, alt: "m equals the fraction −1 over 1")[$m = frac(−1, 1)$]], [Count out the rise and run to mark the second point.], [rise #math.equation(block: false, alt: "−1")[$−1$], run 1 #linebreak() #figure(figph[A coordinate plane illustrating a downward-sloping line that intersects the y-axis at (0,4) and passes through (1,3).], alt: "A coordinate plane illustrating a downward-sloping line that intersects the y-axis at (0,4) and passes through (1,3).", caption: none)], )) Draw the line as shown in the graph. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the line of the equation #math.equation(block: false, alt: "y equals − x minus 3")[$y = "−" x − 3$] using its slope and #emph[y]-intercept. #solutionbox[ #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (0, negative 3) and (1, negative 4).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (0, negative 3) and (1, negative 4).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the line of the equation#math.equation(block: false, alt: "y equals − x minus 1")[$y = "−" x − 1$] using its slope and #emph[y]-intercept. #solutionbox[ #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (0, negative 1) and (1, negative 2).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (0, negative 1) and (1, negative 2).", caption: none) ] ] Now that we have graphed lines by using the slope and #emph[y]-intercept, let’s summarize all the methods we have used to graph lines. #figure(figph[The table has a title row that reads “Methods to Graph Lines”. Below this are four columns. The first column contains the following: Point Plotting. A blank table with two columns and four rows. The first row is a header row with the headers “x” and “y”. Find three points. Plot the points, make sure they line up, them draw the line. The second column contains: Slope-Intercept. Y equals m x plus b. Find the slope and y-intercept, then count the slope to get a second point. The third column: Intercepts. A table with two columns and four rows. The first row is a header row with the headers “x” and “y”. In the first row there is a 0 in the x column. In the second row there is a 0 in the y column. The remaining spaces are blank. Fourth column. Recognize vertical and horizontal lines. The equation has only one variable. X equals a vertical. Y equals b horizontal.], alt: "The table has a title row that reads “Methods to Graph Lines”. Below this are four columns. The first column contains the following: Point Plotting. A blank table with two columns and four rows. The first row is a header row with the headers “x” and “y”. Find three points. Plot the points, make sure they line up, them draw the line. The second column contains: Slope-Intercept. Y equals m x plus b. Find the slope and y-intercept, then count the slope to get a second point. The third column: Intercepts. A table with two columns and four rows. The first row is a header row with the headers “x” and “y”. In the first row there is a 0 in the x column. In the second row there is a 0 in the y column. The remaining spaces are blank. Fourth column. Recognize vertical and horizontal lines. The equation has only one variable. X equals a vertical. Y equals b horizontal.", caption: none) === Choose the Most Convenient Method to Graph a Line Now that we have seen several methods we can use to graph lines, how do we know which method to use for a given equation? While we could plot points, use the slope–intercept form, or find the intercepts for #emph[any] equation, if we recognize the most convenient way to graph a certain type of equation, our work will be easier. Generally, plotting points is not the most efficient way to graph a line. Let’s look for some patterns to help determine the most convenient method to graph a line. Here are five equations we graphed in this chapter, and the method we used to graph each of them. #math.equation(block: true, alt: "Equation, Method; #1, x equals 2, Vertical line; #2, y equals −1, Horizontal line; #3, − x plus 2 y equals 6, Intercepts; #4, 4 x minus 3 y equals 12, Intercepts; #5, y equals − x plus 4, Slope–intercept")[$& "Equation" & & & #h(4em) "Method" \ "#1" & x = 2 & & & #h(4em) "Vertical line" \ "#2" & y = −1 & & & #h(4em) "Horizontal line" \ "#3" & "−" x + 2 y = 6 & & & #h(4em) "Intercepts" \ "#4" & 4 x − 3 y = 12 & & & #h(4em) "Intercepts" \ "#5" & y = "−" x + 4 & & & #h(4em) "Slope–intercept"$]Equations \#1 and \#2 each have just one variable. Remember, in equations of this form the value of that one variable is constant; it does not depend on the value of the other variable. Equations of this form have graphs that are vertical or horizontal lines. In equations \#3 and \#4, both #emph[x] and #emph[y] are on the same side of the equation. These two equations are of the form #math.equation(block: false, alt: "A x plus B y equals C .")[$A x + B y = C .$] We substituted #math.equation(block: false, alt: "y equals 0")[$y = 0$] to find the #emph[x]- intercept and #math.equation(block: false, alt: "x equals 0")[$x = 0$] to find the #emph[y]-intercept, and then found a third point by choosing another value for #emph[x] or #emph[y]. Equation \#5 is written in slope–intercept form. After identifying the slope and #emph[y-]intercept from the equation we used them to graph the line. This leads to the following strategy. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Strategy for Choosing the Most Convenient Method to Graph a Line] Consider the form of the equation. - If it only has one variable, it is a vertical or horizontal line. - #math.equation(block: false, alt: "x equals a")[$x = a$] is a vertical line passing through the #emph[x]-axis at #emph[a]. - #math.equation(block: false, alt: "y equals b")[$y = b$] is a horizontal line passing through the #emph[y]-axis at #emph[b]. - If #emph[y] is isolated on one side of the equation, in the form #math.equation(block: false, alt: "y equals m x plus b ,")[$y = m x + b ,$] graph by using the slope and #emph[y]-intercept. - Identify the slope and #emph[y]-intercept and then graph. - If the equation is of the form #math.equation(block: false, alt: "A x plus B y equals C ,")[$A x + B y = C ,$] find the intercepts. - Find the #emph[x]- and #emph[y]-intercepts, a third point, and then graph. ] #examplebox("Example 7")[][ Determine the most convenient method to graph each line: ⓐ #math.equation(block: false, alt: "y equals 5")[$y = 5$] ⓑ #math.equation(block: false, alt: "4 x minus 5 y equals 20")[$4 x − 5 y = 20$] ⓒ #math.equation(block: false, alt: "x equals −3")[$x = −3$] ⓓ #math.equation(block: false, alt: "y equals minus the fraction 5 over 9 x plus 8")[$y = − frac(5, 9) x + 8$] #solutionbox[ ⓐ #math.equation(block: false, alt: "y equals 5")[$y = 5$] #linebreak() This equation has only one variable, #emph[y]. Its graph is a horizontal line crossing the #emph[y]-axis at #math.equation(block: false, alt: "5")[$5$]. #linebreak() ⓑ #math.equation(block: false, alt: "4 x minus 5 y equals 20")[$4 x − 5 y = 20$] #linebreak() This equation is of the form #math.equation(block: false, alt: "A x plus B y equals C .")[$A x + B y = C .$] The easiest way to graph it will be to find the intercepts and one more point. #linebreak() ⓒ #math.equation(block: false, alt: "x equals −3")[$x = −3$] #linebreak() There is only one variable, #emph[x]. The graph is a vertical line crossing the #emph[x]-axis at#math.equation(block: false, alt: "−3 .")[$−3 .$] #linebreak() ⓓ #math.equation(block: false, alt: "y equals minus the fraction 5 over 9 x plus 8")[$y = − frac(5, 9) x + 8$] #linebreak() Since this equation is in #math.equation(block: false, alt: "y equals m x plus b")[$y = m x + b$] form, it will be easiest to graph this line by using the slope and #emph[y]-intercepts. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Determine the most convenient method to graph each line: ⓐ #math.equation(block: false, alt: "3 x plus 2 y equals 12")[$3 x + 2 y = 12$] ⓑ #math.equation(block: false, alt: "y equals 4")[$y = 4$] ⓒ #math.equation(block: false, alt: "y equals the fraction 1 over 5 x minus 4")[$y = frac(1, 5) x − 4$] ⓓ #math.equation(block: false, alt: "x equals −7 .")[$x = −7 .$] #solutionbox[ ⓐ intercepts ⓑ horizontal line ⓒ slope-intercept ⓓ vertical line ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Determine the most convenient method to graph each line: ⓐ #math.equation(block: false, alt: "x equals 6")[$x = 6$] ⓑ #math.equation(block: false, alt: "y equals minus the fraction 3 over 4 x plus 1")[$y = − frac(3, 4) x + 1$] ⓒ #math.equation(block: false, alt: "y equals −8")[$y = −8$] ⓓ #math.equation(block: false, alt: "4 x minus 3 y equals −1 .")[$4 x − 3 y = −1 .$] #solutionbox[ ⓐ vertical line ⓑ slope-intercept ⓒ horizontal line #linebreak() ⓓ intercepts ] ] === Graph and Interpret Applications of Slope–Intercept Many real-world applications are modeled by linear equations. We will take a look at a few applications here so you can see how equations written in slope–intercept form relate to real world situations. Usually, when a linear equation models uses real-world data, different letters are used for the variables, instead of using only #emph[x] and #emph[y]. The variable names remind us of what quantities are being measured. Also, we often will need to extend the axes in our rectangular coordinate system to bigger positive and negative numbers to accommodate the data in the application. #examplebox("Example 8")[][ The equation #math.equation(block: false, alt: "F equals the fraction 9 over 5 C plus 32")[$F = frac(9, 5) C + 32$] is used to convert temperatures, #emph[C,] on the Celsius scale to temperatures, #emph[F], on the Fahrenheit scale. ⓐ Find the Fahrenheit temperature for a Celsius temperature of 0. ⓑ Find the Fahrenheit temperature for a Celsius temperature of 20. ⓒ Interpret the slope and #emph[F]-intercept of the equation. ⓓ Graph the equation. #solutionbox[ ⓐ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Find the Fahrenheit temperature for a Celsius temperature of 0.], [#math.equation(block: false, alt: "F equals the fraction 9 over 5 C plus 32")[$F = frac(9, 5) C + 32$]]), [Find #math.equation(block: false, alt: "F")[$F$] when #math.equation(block: false, alt: "C equals 0 .")[$C = 0 .$]], [#math.equation(block: false, alt: "F equals the fraction 9 over 5 open parenthesis 0 close parenthesis plus 32")[$F = frac(9, 5) ( 0 ) + 32$]], [Simplify.], [#math.equation(block: false, alt: "F equals 32")[$F = 32$]], )) ⓑ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Find the Fahrenheit temperature for a Celsius temperature of 20.], [#math.equation(block: false, alt: "F equals the fraction 9 over 5 C plus 32")[$F = frac(9, 5) C + 32$]]), [Find #math.equation(block: false, alt: "F")[$F$] when #math.equation(block: false, alt: "C equals 20 .")[$C = 20 .$]], [#math.equation(block: false, alt: "F equals the fraction 9 over 5 open parenthesis 20 close parenthesis plus 32")[$F = frac(9, 5) ( 20 ) + 32$]], [Simplify.], [#math.equation(block: false, alt: "F equals 36 plus 32")[$F = 36 + 32$]], [Simplify.], [#math.equation(block: false, alt: "F equals 68")[$F = 68$]], )) ⓒ #linebreak() Interpret the slope and #emph[F]-intercept of the equation. #linebreak() Even though this equation uses #emph[F] and #emph[C], it is still in slope–intercept form. #linebreak() #figure(figph[y equals m x plus b. F equals m C plus b. The y and F are emphasized in red. The x and C are emphasized in blue. F equals 9 divided by 5 C plus 32.], alt: "y equals m x plus b. F equals m C plus b. The y and F are emphasized in red. The x and C are emphasized in blue. F equals 9 divided by 5 C plus 32.", caption: none) The slope, #math.equation(block: false, alt: "the fraction 9 over 5 ,")[$frac(9, 5) ,$] means that the temperature Fahrenheit (#emph[F]) increases 9 degrees when the temperature Celsius (#emph[C]) increases 5 degrees. #linebreak() The #emph[F]-intercept means that when the temperature is #math.equation(block: false, alt: "0 °")[$0 "°"$] on the Celsius scale, it is #math.equation(block: false, alt: "32 °")[$32 "°"$] on the Fahrenheit scale. #linebreak() ⓓ Graph the equation. #linebreak() We’ll need to use a larger scale than our usual. Start at the #emph[F]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 32 close parenthesis")[$( 0 , 32 )$], and then count out the rise of 9 and the run of 5 to get a second point as shown in the graph. #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 40 to 80. The y-axis runs from negative 40 to 80. The line goes through the points (0, 32) and (5, 41).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 40 to 80. The y-axis runs from negative 40 to 80. The line goes through the points (0, 32) and (5, 41).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The equation #math.equation(block: false, alt: "h equals 2 s plus 50")[$h = 2 s + 50$] is used to estimate a woman’s height in inches, #emph[h], based on her shoe size, #emph[s]. ⓐ Estimate the height of a child who wears women’s shoe size 0. ⓑ Estimate the height of a woman with shoe size 8. ⓒ Interpret the slope and #emph[h]-intercept of the equation. ⓓ Graph the equation. #solutionbox[ ⓐ 50 inches #linebreak() ⓑ 66 inches #linebreak() ⓒ The slope, 2, means that the height, #emph[h], increases by 2 inches when the shoe size, #emph[s], increases by 1. The #emph[h]-intercept means that when the shoe size is 0, the height is 50 inches. #linebreak() ⓓ #linebreak() #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 1 to 14. The y-axis runs from negative 1 to 80. The line goes through the points (0, 50) and (10, 70).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 1 to 14. The y-axis runs from negative 1 to 80. The line goes through the points (0, 50) and (10, 70).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The equation #math.equation(block: false, alt: "T equals the fraction 1 over 4 n plus 40")[$T = frac(1, 4) n + 40$] is used to estimate the temperature in degrees Fahrenheit, #emph[T], based on the number of cricket chirps, #emph[n], in one minute. ⓐ Estimate the temperature when there are no chirps. ⓑ Estimate the temperature when the number of chirps in one minute is 100. ⓒ Interpret the slope and #emph[T]-intercept of the equation. ⓓ Graph the equation. #solutionbox[ ⓐ 40 degrees #linebreak() ⓑ 65 degrees #linebreak() ⓒ The slope, #math.equation(block: false, alt: "the fraction 1 over 4 ,")[$frac(1, 4) ,$] means that the temperature Fahrenheit (#emph[F]) increases 1 degree when the number of chirps, #emph[n], increases by 4. The #emph[T]-intercept means that when the number of chirps is 0, the temperature is 40°. #linebreak() ⓓ #linebreak() #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 1 to 140. The y-axis runs from negative 1 to 80. The line goes through the points (0, 40) and (40, 50).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 1 to 140. The y-axis runs from negative 1 to 80. The line goes through the points (0, 40) and (40, 50).", caption: none) ] ] The cost of running some types of business has two components—a #emph[fixed cost] and a #emph[variable cost]. The fixed cost is always the same regardless of how many units are produced. This is the cost of rent, insurance, equipment, advertising, and other items that must be paid regularly. The variable cost depends on the number of units produced. It is for the material and labor needed to produce each item. #examplebox("Example 9")[][ Sam drives a delivery van. The equation #math.equation(block: false, alt: "C equals 0.5 m plus 60")[$C = 0.5 m + 60$] models the relation between his weekly cost, #emph[C], in dollars and the number of miles, #emph[m], that he drives. ⓐ Find Sam’s cost for a week when he drives 0 miles. ⓑ Find the cost for a week when he drives 250 miles. ⓒ Interpret the slope and #emph[C]-intercept of the equation. ⓓ Graph the equation. #solutionbox[ ⓐ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Find Sam’s cost for a week when he drives 0 miles.], [#math.equation(block: false, alt: "C equals 0.5 m plus 60")[$C = 0.5 m + 60$]]), [Find #math.equation(block: false, alt: "C")[$C$] when #math.equation(block: false, alt: "m equals 0 .")[$m = 0 .$]], [#math.equation(block: false, alt: "C equals 0.5 open parenthesis 0 close parenthesis plus 60")[$C = 0.5 ( 0 ) + 60$]], [Simplify.], [#math.equation(block: false, alt: "C equals 60")[$C = 60$]], [], [Sam’s costs are \$60 when he drives 0 miles.], )) ⓑ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Find the cost for a week when he drives 250 miles.], [#math.equation(block: false, alt: "C equals 0.5 m plus 60")[$C = 0.5 m + 60$]]), [Find #math.equation(block: false, alt: "C")[$C$] when #math.equation(block: false, alt: "m equals 250 .")[$m = 250 .$]], [#math.equation(block: false, alt: "C equals 0.5 open parenthesis 250 close parenthesis plus 60")[$C = 0.5 ( 250 ) + 60$]], [Simplify.], [#math.equation(block: false, alt: "C equals 185")[$C = 185$]], [], [Sam’s costs are \$185 when he drives 250 miles.], )) ⓒ Interpret the slope and #emph[C]-intercept of the equation. #linebreak() #figure(figph[y equals m x plus b. C equals 0.5 p plus 60. The y and C are emphasized in red. The x and p are emphasized in blue.], alt: "y equals m x plus b. C equals 0.5 p plus 60. The y and C are emphasized in red. The x and p are emphasized in blue.", caption: none) #linebreak() The slope, 0.5, means that the weekly cost, #emph[C], increases by \$0.50 when the number of miles driven, #emph[n,] increases by 1. #linebreak() The #emph[C]-intercept means that when the number of miles driven is 0, the weekly cost is \$60. #linebreak() ⓓ Graph the equation. #linebreak() We’ll need to use a larger scale than our usual. Start at the #emph[C]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 60 close parenthesis")[$( 0 , 60 )$]. To count out the slope #emph[m] = 0.5, we rewrite it as an equivalent fraction that will make our graphing easier. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "m equals 0.5")[$m = 0.5$]]), [Rewrite as a fraction.], [#math.equation(block: false, alt: "m equals the fraction 0.5 over 1")[$m = frac(0.5, 1)$]], [Multiply numerator and denominator by 100.], [#math.equation(block: false, alt: "m equals the fraction 0.5 open parenthesis 100 close parenthesis over 1 open parenthesis 100 close parenthesis")[$m = frac(0.5 ( 100 ), 1 ( 100 ))$]], [Simplify.], [#math.equation(block: false, alt: "m equals the fraction 50 over 100")[$m = frac(50, 100)$]], )) So to graph the next point go up 50 from the intercept of 60 and then to the right 100. The second point will be (100, 110). #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 1 to 350. The y-axis runs from negative 1 to 350. The line goes through the points (0, 60) and (200, 160).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 1 to 350. The y-axis runs from negative 1 to 350. The line goes through the points (0, 60) and (200, 160).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Stella has a home business selling gourmet pizzas. The equation #math.equation(block: false, alt: "C equals 4 p plus 25")[$C = 4 p + 25$] models the relation between her weekly cost, #emph[C], in dollars and the number of pizzas, #emph[p], that she sells. ⓐ Find Stella’s cost for a week when she sells no pizzas. ⓑ Find the cost for a week when she sells 15 pizzas. ⓒ Interpret the slope and #emph[C]-intercept of the equation. ⓓ Graph the equation. #solutionbox[ ⓐ \$25 #linebreak() ⓑ \$85 #linebreak() ⓒ The slope, 4, means that the weekly cost, #emph[C], increases by \$4 when the number of pizzas sold, #emph[p,] increases by 1. The #emph[C]-intercept means that when the number of pizzas sold is 0, the weekly cost is \$25. #linebreak() ⓓ #linebreak() #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 2 to 20. The y-axis runs from negative 10 to \`00. The line goes through the points (0, 25) and (1, 29).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 2 to 20. The y-axis runs from negative 10 to `00. The line goes through the points (0, 25) and (1, 29).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Loreen has a calligraphy business. The equation #math.equation(block: false, alt: "C equals 1.8 n plus 35")[$C = 1.8 n + 35$] models the relation between her weekly cost, #emph[C], in dollars and the number of wedding invitations, #emph[n], that she writes. ⓐ Find Loreen’s cost for a week when she writes no invitations. ⓑ Find the cost for a week when she writes 75 invitations. ⓒ Interpret the slope and #emph[C]-intercept of the equation. ⓓ Graph the equation. #solutionbox[ ⓐ \$35 #linebreak() ⓑ \$170 #linebreak() ⓒ The slope, #math.equation(block: false, alt: "1.8 ,")[$1.8 ,$] means that the weekly cost, #emph[C], increases by #math.equation(block: false, alt: "$ 1.80")[$"$" 1.80$] when the number of invitations, #emph[n], increases by 1. #linebreak() The #emph[C]-intercept means that when the number of invitations is 0, the weekly cost is \$35. #linebreak() ⓓ #linebreak() #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 1 to 350. The y-axis runs from negative 1 to 350. The line goes through the points (0, 35) and (75, 170).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 1 to 350. The y-axis runs from negative 1 to 350. The line goes through the points (0, 35) and (75, 170).", caption: none) ] ] === Use Slopes to Identify Parallel and Perpendicular Lines Two lines that have the same slope are called #strong[parallel lines]. Parallel lines have the same steepness and never intersect. We say this more formally in terms of the rectangular coordinate system. Two lines that have the same slope and different #emph[y]-intercepts are called parallel lines. #figure(figph[This figure shows the graph of a two straight lines on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The first line goes through the points (0, 3) and (5, 5). The second line goes through the points (0, negative 2) and (5, 0). The lines are parallel meaning they will always be the same distance apart and never intersect. They are slanted by the same angle.], alt: "This figure shows the graph of a two straight lines on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The first line goes through the points (0, 3) and (5, 5). The second line goes through the points (0, negative 2) and (5, 0). The lines are parallel meaning they will always be the same distance apart and never intersect. They are slanted by the same angle.", caption: none) Verify that both lines have the same slope, #math.equation(block: false, alt: "m equals the fraction 2 over 5 ,")[$m = frac(2, 5) ,$] and different #emph[y]-intercepts. What about vertical lines? The slope of a vertical line is undefined, so vertical lines don’t fit in the definition above. We say that vertical lines that have different #emph[x]-intercepts are parallel, like the lines shown in this graph. #figure(figph[This figure shows the graph of a two straight vertical line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The first line goes through the points (2, 0) and (2, 1). The second line goes through the points (5, 0) and (5, 1). The lines are parallel meaning they will always be the same distance apart and never intersect.], alt: "This figure shows the graph of a two straight vertical line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The first line goes through the points (2, 0) and (2, 1). The second line goes through the points (5, 0) and (5, 1). The lines are parallel meaning they will always be the same distance apart and never intersect.", caption: none) #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Parallel Lines] #strong[Parallel lines] are lines in the same plane that do not intersect. - Parallel lines have the same slope and different #emph[y]-intercepts. - If #math.equation(block: false, alt: "m sub 1")[$m_(1)$] and #math.equation(block: false, alt: "m sub 2")[$m_(2)$] are the slopes of two parallel lines then #math.equation(block: false, alt: "m sub 1 equals m sub 2 .")[$m_(1) = m_(2) .$] - Parallel vertical lines have different #emph[x]-intercepts. ] Since parallel lines have the same slope and different #emph[y]-intercepts, we can now just look at the slope–intercept form of the equations of lines and decide if the lines are parallel. #examplebox("Example 10")[][ Use slopes and #emph[y]-intercepts to determine if the lines are parallel: ⓐ #math.equation(block: false, alt: "3 x minus 2 y equals 6")[$3 x − 2 y = 6$] and #math.equation(block: false, alt: "y equals the fraction 3 over 2 x plus 1")[$y = frac(3, 2) x + 1$] ⓑ #math.equation(block: false, alt: "y equals 2 x minus 3")[$y = 2 x − 3$] and #math.equation(block: false, alt: "−6 x plus 3 y equals −9 .")[$−6 x + 3 y = −9 .$] #solutionbox[ ⓐ #figure(table( columns: 4, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "3 x minus 2 y, equals 6")[$#h(0.7em) 3 x − 2 y & = 6$]], [and], [#math.equation(block: false, alt: "y, equals the fraction 3 over 2 x plus 1")[$#h(3em) y & = frac(3, 2) x + 1$]]), [Solve the first equation for #math.equation(block: false, alt: "y .")[$y .$]], [#math.equation(block: false, alt: "minus 2 y, equals minus 3 x plus 6; the fraction minus 2 y over minus 2, equals the fraction minus 3 x plus 6 over minus 2")[$#h(1.85em) − 2 y & = − 3 x + 6 \ frac(− 2 y, − 2) & = frac(− 3 x + 6, − 2)$]], [], [], [The equation is now in slope–intercept form.], [#math.equation(block: false, alt: "y, equals the fraction 3 over 2 x minus 3")[$#h(3.3em) y & = frac(3, 2) x − 3$]], [], [], [The equation of the second line is already in slope–intercept form.], [], [], [#math.equation(block: false, alt: "y, equals the fraction 3 over 2 x plus 1")[$#h(3em) y & = frac(3, 2) x + 1$]], [Identify the slope and #math.equation(block: false, alt: "y")[$y$]-intercept of both lines.], [#math.equation(block: false, alt: "y, equals the fraction 3 over 2 x minus 3; y, equals m x plus b; m, equals the fraction 3 over 2")[$#h(3em) y & = frac(3, 2) x − 3 \ y & = m x + b \ m & = frac(3, 2)$]], [], [#math.equation(block: false, alt: "y, equals the fraction 3 over 2 x plus 1; y, equals m x plus b; y, equals the fraction 3 over 2")[$#h(3em) y & = frac(3, 2) x + 1 \ y & = m x + b \ y & = frac(3, 2)$]], [], [#math.equation(block: false, alt: "y -intercept is open parenthesis 0 , −3 close parenthesis")[$y "-intercept is" #h(0.2em) ( 0 , −3 )$]], [], [#math.equation(block: false, alt: "y -intercept is open parenthesis 0 , 1 close parenthesis")[$#h(2em) y "-intercept is" #h(0.2em) ( 0 , 1 )$]], )) The lines have the same slope and different #emph[y]-intercepts and so they are parallel. You may want to graph the lines to confirm whether they are parallel. ⓑ #figure(table( columns: 4, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "y, equals 2 x minus 3")[$#h(2em) y & = 2 x − 3$]], [and], [#math.equation(block: false, alt: "minus 6 x plus 3 y, equals minus 9")[$#h(1.5em) − 6 x + 3 y & = − 9$]]), [The first equation is already in slope–intercept form.], [#math.equation(block: false, alt: "y, equals 2 x minus 3")[$#h(2em) y & = 2 x − 3$]], [], [], [Solve the second equation for #math.equation(block: false, alt: "y .")[$y .$]], [], [], [#math.equation(block: false, alt: "minus 6 x plus 3 y, equals minus 9; 3 y, equals 6 x minus 9; the fraction 3 y over 3, equals the fraction 6 x minus 9 over 3; y, equals 2 x minus 3")[$#h(1.5em) − 6 x + 3 y & = − 9 \ 3 y & = 6 x − 9 \ frac(3 y, 3) & = frac(6 x − 9, 3) \ y & = 2 x − 3$]], [The second equation is now in slope–intercept form.], [], [], [#math.equation(block: false, alt: "y, equals 2 x minus 3")[$#h(4.85em) y & = 2 x − 3$]], [Identify the slope and #math.equation(block: false, alt: "y")[$y$]-intercept of both lines.], [#math.equation(block: false, alt: "y, equals 2 x minus 3; y, equals m x plus b; m, equals 2")[$#h(2em) y & = 2 x − 3 \ y & = m x + b \ m & = 2$]], [], [#math.equation(block: false, alt: "y, equals 2 x minus 3; y, equals m x plus b; m, equals 2")[$#h(4.5em) y & = 2 x − 3 \ y & = m x + b \ m & = 2$]], [], [#math.equation(block: false, alt: "y -intercept is open parenthesis 0 , −3 close parenthesis")[$y "-intercept is" #h(0.2em) ( 0 , −3 )$]], [], [#math.equation(block: false, alt: "y -intercept is open parenthesis 0 , −3 close parenthesis")[$#h(3em) y "-intercept is" #h(0.2em) ( 0 , −3 )$]], )) The lines have the same slope, but they also have the same #emph[y]-intercepts. Their equations represent the same line and we say the lines are coincident. They are not parallel; they are the same line. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use slopes and #emph[y-]intercepts to determine if the lines are parallel: ⓐ #math.equation(block: false, alt: "2 x plus 5 y equals 5")[$2 x + 5 y = 5$] and #math.equation(block: false, alt: "y equals minus the fraction 2 over 5 x minus 4")[$y = − frac(2, 5) x − 4$] ⓑ #math.equation(block: false, alt: "y equals minus the fraction 1 over 2 x minus 1")[$y = − frac(1, 2) x − 1$] and #math.equation(block: false, alt: "x plus 2 y equals −2 .")[$x + 2 y = −2 .$] #solutionbox[ ⓐ parallel ⓑ not parallel; same line ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use slopes and #emph[y-]intercepts to determine if the lines are parallel: ⓐ #math.equation(block: false, alt: "4 x minus 3 y equals 6")[$4 x − 3 y = 6$] and #math.equation(block: false, alt: "y equals the fraction 4 over 3 x minus 1")[$y = frac(4, 3) x − 1$] ⓑ #math.equation(block: false, alt: "y equals the fraction 3 over 4 x minus 3")[$y = frac(3, 4) x − 3$] and #math.equation(block: false, alt: "3 x minus 4 y equals 12 .")[$3 x − 4 y = 12 .$] #solutionbox[ ⓐ parallel ⓑ not parallel; same line ] ] #examplebox("Example 11")[][ Use slopes and #emph[y-]intercepts to determine if the lines are parallel: ⓐ #math.equation(block: false, alt: "y equals −4")[$y = −4$] and #math.equation(block: false, alt: "y equals 3")[$y = 3$] ⓑ #math.equation(block: false, alt: "x equals −2")[$x = −2$] and #math.equation(block: false, alt: "x equals −5 .")[$x = −5 .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "y equals −4")[$y = −4$] and #math.equation(block: false, alt: "y equals 3")[$y = 3$] #linebreak() We recognize right away from the equations that these are horizontal lines, and so we know their slopes are both 0. #linebreak() Since the horizontal lines cross the #emph[y]-axis at #math.equation(block: false, alt: "y equals −4")[$y = −4$] and at #math.equation(block: false, alt: "y equals 3 ,")[$y = 3 ,$] we know the #emph[y]-intercepts are #math.equation(block: false, alt: "open parenthesis 0 , −4 close parenthesis")[$( 0 , −4 )$] and #math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis .")[$( 0 , 3 ) .$] #linebreak() The lines have the same slope and different #emph[y]-intercepts and so they are parallel. #linebreak() ⓑ #math.equation(block: false, alt: "x equals −2")[$x = −2$] and #math.equation(block: false, alt: "x equals −5")[$x = −5$] #linebreak() We recognize right away from the equations that these are vertical lines, and so we know their slopes are undefined. #linebreak() Since the vertical lines cross the #emph[x]-axis at #math.equation(block: false, alt: "x equals −2")[$x = −2$] and #math.equation(block: false, alt: "x equals −5 ,")[$x = −5 ,$] we know the #emph[y]-intercepts are #math.equation(block: false, alt: "open parenthesis −2 , 0 close parenthesis")[$( −2 , 0 )$] and #math.equation(block: false, alt: "open parenthesis −5 , 0 close parenthesis .")[$( −5 , 0 ) .$] #linebreak() The lines are vertical and have different #emph[x]-intercepts and so they are parallel. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use slopes and #emph[y-]intercepts to determine if the lines are parallel: ⓐ #math.equation(block: false, alt: "y equals 8")[$y = 8$] and #math.equation(block: false, alt: "y equals −6")[$y = −6$] ⓑ #math.equation(block: false, alt: "x equals 1")[$x = 1$] and #math.equation(block: false, alt: "x equals −5 .")[$x = −5 .$] #solutionbox[ ⓐ parallel ⓑ parallel ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use slopes and #emph[y-]intercepts to determine if the lines are parallel: ⓐ #math.equation(block: false, alt: "y equals 1")[$y = 1$] and #math.equation(block: false, alt: "y equals −5")[$y = −5$] ⓑ #math.equation(block: false, alt: "x equals 8")[$x = 8$] and #math.equation(block: false, alt: "x equals −6 .")[$x = −6 .$] #solutionbox[ ⓐ parallel ⓑ parallel ] ] Let’s look at the lines whose equations are #math.equation(block: false, alt: "y equals the fraction 1 over 4 x minus 1")[$y = frac(1, 4) x − 1$] and #math.equation(block: false, alt: "y equals −4 x plus 2 ,")[$y = −4 x + 2 ,$] shown. #figure(figph[This figure shows the graph of a two perpendicular straight lines on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The first line goes through the points (0, negative 1) and (4, 0). The first line is labeled y equals 1 divided by 4 x minus 1. The second line goes through the points (0, 2) and (1, negative 2). The second line is labeled y equals negative 4 x plus 2. The lines are perpendicular meaning they form a right angle where they intersect.], alt: "This figure shows the graph of a two perpendicular straight lines on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The first line goes through the points (0, negative 1) and (4, 0). The first line is labeled y equals 1 divided by 4 x minus 1. The second line goes through the points (0, 2) and (1, negative 2). The second line is labeled y equals negative 4 x plus 2. The lines are perpendicular meaning they form a right angle where they intersect.", caption: none) These lines lie in the same plane and intersect in right angles. We call these lines perpendicular. If we look at the slope of the first line, #math.equation(block: false, alt: "m sub 1 equals the fraction 1 over 4 ,")[$m_(1) = frac(1, 4) ,$] and the slope of the second line, #math.equation(block: false, alt: "m sub 2 equals −4 ,")[$m_(2) = −4 ,$] we can see that they are #emph[negative reciprocals] of each other. If we multiply them, their product is #math.equation(block: false, alt: "−1 .")[$−1 .$] #math.equation(block: true, alt: "m sub 1 times m sub 2; the fraction 1 over 4 open parenthesis −4 close parenthesis; −1")[$m_(1) · m_(2) \ frac(1, 4) ( −4 ) \ −1$]This is always true for #strong[perpendicular lines] and leads us to this definition. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Perpendicular Lines] #strong[Perpendicular lines] are lines in the same plane that form a right angle. - If #math.equation(block: false, alt: "m sub 1")[$m_(1)$] and #math.equation(block: false, alt: "m sub 2")[$m_(2)$] are the slopes of two perpendicular lines, then: - their slopes are negative reciprocals of each other, #math.equation(block: false, alt: "m sub 1 equals minus the fraction 1 over m sub 2 .")[$m_(1) = − frac(1, m_(2)) .$] - the product of their slopes is #math.equation(block: false, alt: "−1")[$−1$], #math.equation(block: false, alt: "m sub 1 times m sub 2 equals −1 .")[$m_(1) · m_(2) = −1 .$] - A vertical line and a horizontal line are always perpendicular to each other. ] We were able to look at the slope–intercept form of linear equations and determine whether or not the lines were parallel. We can do the same thing for perpendicular lines. We find the slope–intercept form of the equation, and then see if the slopes are opposite reciprocals. If the product of the slopes is #math.equation(block: false, alt: "−1 ,")[$−1 ,$] the lines are perpendicular. #examplebox("Example 12")[][ Use slopes to determine if the lines are perpendicular: ⓐ #math.equation(block: false, alt: "y equals −5 x minus 4")[$y = −5 x − 4$] and #math.equation(block: false, alt: "x minus 5 y equals 5")[$x − 5 y = 5$] ⓑ #math.equation(block: false, alt: "7 x plus 2 y equals 3")[$7 x + 2 y = 3$] and #math.equation(block: false, alt: "2 x plus 7 y equals 5")[$2 x + 7 y = 5$] #solutionbox[ ⓐ #figure(table( columns: 3, align: left, inset: 6pt, table.header([The first equation is in slope–intercept form.], [#math.equation(block: false, alt: "y equals minus 5 x minus 4")[$#h(1.2em) y = − 5 x − 4$]], []), [Solve the second equation for #math.equation(block: false, alt: "y .")[$y .$]], [], [#math.equation(block: false, alt: "x minus 5 y, equals 5; minus 5 y, equals minus x plus 5; the fraction minus 5 y over minus 5, equals the fraction minus x plus 5 over minus 5; y, equals the fraction 1 over 5 x minus 1")[$#h(4em) x − 5 y & = 5 \ − 5 y & = − x + 5 \ frac(− 5 y, − 5) & = frac(− x + 5, − 5) \ y & = frac(1, 5) x − 1$]], [Identify the slope of each line.], [#math.equation(block: false, alt: "y, equals minus 5 x minus 4; y, equals m x plus b; m sub 1, equals minus 5")[$y & = − 5 x − 4 \ y & = m x + b \ m_(1) & = − 5$]], [#math.equation(block: false, alt: "y, equals the fraction 1 over 5 x minus 1; y, equals m x plus b; m sub 2, equals the fraction 1 over 5")[$#h(5.3em) y & = frac(1, 5) x − 1 \ y & = m x + b \ m_(2) & = frac(1, 5)$]], )) The slopes are negative reciprocals of each other, so the lines are perpendicular. We check by multiplying the slopes, Since #math.equation(block: false, alt: "−5 open parenthesis the fraction 1 over 5 close parenthesis equals −1 ,")[$−5 ( frac(1, 5) ) = −1 ,$] it checks. ⓑ #figure(table( columns: 3, align: left, inset: 6pt, table.header([Solve the equations for #math.equation(block: false, alt: "y .")[$y "."$]], [#math.equation(block: false, alt: "7 x plus 2 y, equals 3; 2 y, equals minus 7 x plus 3; the fraction 2 y over 2, equals the fraction minus 7 x plus 3 over 2; y, equals minus the fraction 7 over 2 x plus the fraction 3 over 2")[$7 x + 2 y & = 3 \ 2 y & = − 7 x + 3 \ frac(2 y, 2) & = frac(− 7 x + 3, 2) \ y & = − frac(7, 2) x + frac(3, 2)$]], [#math.equation(block: false, alt: "2 x plus 7 y, equals 5; 7 y, equals minus 2 x plus 5; the fraction 7 y over 7, equals the fraction minus 2 x plus 5 over 7; y, equals minus the fraction 2 over 7 x plus the fraction 5 over 7")[$#h(4em) 2 x + 7 y & = 5 \ 7 y & = − 2 x + 5 \ frac(7 y, 7) & = frac(− 2 x + 5, 7) \ y & = − frac(2, 7) x + frac(5, 7)$]]), [Identify the slope of each line.], [#math.equation(block: false, alt: "y, equals m x plus b; m sub 1, equals minus the fraction 7 over 2")[$#h(1.9em) y & = m x + b \ m_(1) & = − frac(7, 2)$]], [#math.equation(block: false, alt: "y, equals m x plus b; m sub 1, equals minus the fraction 2 over 7")[$#h(5.9em) y & = m x + b \ m_(1) & = − frac(2, 7)$]], )) The slopes are reciprocals of each other, but they have the same sign. Since they are not negative reciprocals, the lines are not perpendicular. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use slopes to determine if the lines are perpendicular: ⓐ #math.equation(block: false, alt: "y equals −3 x plus 2")[$y = −3 x + 2$] and #math.equation(block: false, alt: "x minus 3 y equals 4")[$x − 3 y = 4$] ⓑ #math.equation(block: false, alt: "5 x plus 4 y equals 1")[$5 x + 4 y = 1$] and #math.equation(block: false, alt: "4 x plus 5 y equals 3 .")[$4 x + 5 y = 3 .$] #solutionbox[ ⓐ perpendicular ⓑ not perpendicular ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use slopes to determine if the lines are perpendicular: ⓐ #math.equation(block: false, alt: "y equals 2 x minus 5")[$y = 2 x − 5$] and #math.equation(block: false, alt: "x plus 2 y equals −6")[$x + 2 y = −6$] ⓑ #math.equation(block: false, alt: "2 x minus 9 y equals 3")[$2 x − 9 y = 3$] and #math.equation(block: false, alt: "9 x minus 2 y equals 1 .")[$9 x − 2 y = 1 .$] #solutionbox[ ⓐ perpendicular ⓑ not perpendicular ] ] === Key Concepts - #strong[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 rise measures the vertical change and the run measures the horizontal change. #linebreak() #linebreak() #linebreak() #linebreak() #linebreak() - #strong[How to find the slope of a line from its graph using] #math.equation(block: false, alt: "m equals the fraction rise over run .")[$m = frac("rise", "run") .$] + Locate two points on the line whose coordinates are integers. + Starting with one point, sketch a right triangle, 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 .")[$m = frac("rise", "run") .$] - #strong[Slope of a line between two points.] - 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: #linebreak() #math.equation(block: true, alt: "m equals the fraction y sub 2 minus y sub 1 over x sub 2 minus x sub 1 .")[$m = frac(y_(2) − y_(1), x_(2) − x_(1)) .$] - #strong[How to graph a line given a point and the slope.] + Plot the given point. + 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. + Starting at the given point, count out the rise and run to mark the second point. + Connect the points with a line. - #strong[Slope Intercept Form of an Equation of a Line] - The slope–intercept form of an equation of a line with slope #emph[m] and #emph[y]-intercept, #math.equation(block: false, alt: "open parenthesis 0 , b close parenthesis")[$( 0 , b )$] is #math.equation(block: false, alt: "y equals m x plus b")[$y = m x + b$]#figure(figph[The table has a title row that reads “Methods to Graph Lines”. Below this are four columns. The first column contains the following: Point Plotting. A blank table with two columns and four rows. The first row is a header row with the headers “x” and “y”. Find three points. Plot the points, make sure they line up, them draw the line. The second column contains: Slope-Intercept. Y equals m x plus b. Find the slope and y-intercept, then count the slope to get a second point. The third column: Intercepts. A table with two columns and four rows. The first row is a header row with the headers “x” and “y”. In the first row there is a 0 in the x column. In the second row there is a 0 in the y column. The remaining spaces are blank. Fourth column. Recognize vertical and horizontal lines. The equation has only one variable. X equals a vertical. Y equals b horizontal.], alt: "The table has a title row that reads “Methods to Graph Lines”. Below this are four columns. The first column contains the following: Point Plotting. A blank table with two columns and four rows. The first row is a header row with the headers “x” and “y”. Find three points. Plot the points, make sure they line up, them draw the line. The second column contains: Slope-Intercept. Y equals m x plus b. Find the slope and y-intercept, then count the slope to get a second point. The third column: Intercepts. A table with two columns and four rows. The first row is a header row with the headers “x” and “y”. In the first row there is a 0 in the x column. In the second row there is a 0 in the y column. The remaining spaces are blank. Fourth column. Recognize vertical and horizontal lines. The equation has only one variable. X equals a vertical. Y equals b horizontal.", caption: none) - #strong[Parallel Lines] - Parallel lines are lines in the same plane that do not intersect. #linebreak() Parallel lines have the same slope and different #emph[y]-intercepts. #linebreak() If #math.equation(block: false, alt: "m sub 1")[$m_(1)$] and #math.equation(block: false, alt: "m sub 2")[$m_(2)$] are the slopes of two parallel lines then #math.equation(block: false, alt: "m sub 1 equals m sub 2 .")[$m_(1) = m_(2) .$] #linebreak() Parallel vertical lines have different #emph[x]-intercepts. - #strong[Perpendicular Lines] - Perpendicular lines are lines in the same plane that form a right angle. - If #math.equation(block: false, alt: "m sub 1")[$m_(1)$] and #math.equation(block: false, alt: "m sub 2")[$m_(2)$] are the slopes of two perpendicular lines, then: #linebreak() their slopes are negative reciprocals of each other, #math.equation(block: false, alt: "m sub 1 equals minus the fraction 1 over m sub 2 .")[$m_(1) = − frac(1, m_(2)) .$] #linebreak() the product of their slopes is #math.equation(block: false, alt: "−1 ,")[$−1 ,$]#math.equation(block: false, alt: "m sub 1 times m sub 2 equals −1 .")[$m_(1) · m_(2) = −1 .$] - A vertical line and a horizontal line are always perpendicular to each other. ==== Practice Makes Perfect #strong[Find the Slope of a Line] In the following exercises, find the slope of each line shown. #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, negative 4) and (5, negative 2).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, negative 4) and (5, negative 2).", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 2 over 5")[$frac(2, 5)$] ] #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, negative 5) and (2, negative 2).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, negative 5) and (2, negative 2).", caption: none) #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, negative 1) and (4, 4).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, negative 1) and (4, 4).", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 5 over 4")[$frac(5, 4)$] ] #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, negative 2) and (3, 3).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, negative 2) and (3, 3).", caption: none) #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, 2) and (3, 1).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, 2) and (3, 1).", caption: none) #solutionbox[ #math.equation(block: true, alt: "minus the fraction 1 over 3")[$− frac(1, 3)$] ] #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, negative 1) and (3, negative 3).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, negative 1) and (3, negative 3).", caption: none) #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, 4) and (2, negative 1).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, 4) and (2, negative 1).", caption: none) #solutionbox[ #math.equation(block: true, alt: "minus the fraction 5 over 2")[$− frac(5, 2)$] ] #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, 2) and (4, negative 1).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, 2) and (4, negative 1).", caption: none) 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 −2")[$y = −2$] #math.equation(block: true, alt: "x equals −5")[$x = −5$] #solutionbox[ undefined ] #math.equation(block: true, alt: "x equals 4")[$x = 4$] 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 , 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 4 , −5 close parenthesis")[$( −3 , 3 ) , ( 4 , −5 )$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 8 over 7")[$− frac(8, 7)$] ] #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[ #math.equation(block: true, alt: "−1")[$−1$] ] #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 each line with the given point and slope. #math.equation(block: true, alt: "open parenthesis 2 , 5 close parenthesis ;")[$( 2 , 5 ) ;$]#math.equation(block: true, alt: "m equals minus the fraction 1 over 3")[$m = − frac(1, 3)$] #solutionbox[ #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (2, 5) and (5, 4).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (2, 5) and (5, 4).", caption: none) ] #math.equation(block: true, alt: "open parenthesis 1 , 4 close parenthesis")[$( 1 , 4 )$]; #math.equation(block: true, alt: "m equals minus the fraction 1 over 2")[$m = − frac(1, 2)$] #math.equation(block: true, alt: "open parenthesis −1 , −4 close parenthesis")[$( −1 , −4 )$]; #math.equation(block: true, alt: "m equals the fraction 4 over 3")[$m = frac(4, 3)$] #solutionbox[ #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (negative 1, negative 4) and (2, 0).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (negative 1, negative 4) and (2, 0).", caption: none) ] #math.equation(block: true, alt: "open parenthesis −3 , −5 close parenthesis")[$( −3 , −5 )$]; #math.equation(block: true, alt: "m equals the fraction 3 over 2")[$m = frac(3, 2)$] #emph[y]-intercept 3; #math.equation(block: false, alt: "m equals minus the fraction 2 over 5")[$m = − frac(2, 5)$] #solutionbox[ #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (0, 3) and (5, 1).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (0, 3) and (5, 1).", caption: none) ] #emph[x]-intercept #math.equation(block: false, alt: "−2")[$−2$]; #math.equation(block: false, alt: "m equals the fraction 3 over 4")[$m = frac(3, 4)$] #math.equation(block: true, alt: "open parenthesis −4 , 2 close parenthesis")[$( −4 , 2 )$]; #math.equation(block: true, alt: "m equals 4")[$m = 4$] #solutionbox[ #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (negative 4, 2) and (negative 3, 6).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (negative 4, 2) and (negative 3, 6).", caption: none) ] #math.equation(block: true, alt: "open parenthesis 1 , 5 close parenthesis")[$( 1 , 5 )$]; #math.equation(block: true, alt: "m equals −3")[$m = −3$] #strong[Graph a Line Using Its Slope and Intercept] In the following exercises, identify the slope and #emph[y]-intercept of each line. #math.equation(block: true, alt: "y equals −7 x plus 3")[$y = −7 x + 3$] #solutionbox[ #math.equation(block: true, alt: "m equals −7 ; open parenthesis 0 , 3 close parenthesis")[$m = −7 ; ( 0 , 3 )$] ] #math.equation(block: true, alt: "y equals 4 x minus 10")[$y = 4 x − 10$] #math.equation(block: true, alt: "3 x plus y equals 5")[$3 x + y = 5$] #solutionbox[ #math.equation(block: true, alt: "m equals −3 ; open parenthesis 0 , 5 close parenthesis")[$m = −3 ; ( 0 , 5 )$] ] #math.equation(block: true, alt: "4 x plus y equals 8")[$4 x + y = 8$] #math.equation(block: true, alt: "6 x plus 4 y equals 12")[$6 x + 4 y = 12$] #solutionbox[ #math.equation(block: true, alt: "m equals minus the fraction 3 over 2 ; open parenthesis 0 , 3 close parenthesis")[$m = − frac(3, 2) ; ( 0 , 3 )$] ] #math.equation(block: true, alt: "8 x plus 3 y equals 12")[$8 x + 3 y = 12$] #math.equation(block: true, alt: "5 x minus 2 y equals 6")[$5 x − 2 y = 6$] #solutionbox[ #math.equation(block: true, alt: "m equals the fraction 5 over 2 ; open parenthesis 0 , −3 close parenthesis")[$m = frac(5, 2) ; ( 0 , −3 )$] ] #math.equation(block: true, alt: "7 x minus 3 y equals 9")[$7 x − 3 y = 9$] In the following exercises, graph the line of each equation using its slope and #emph[y]-intercept. #math.equation(block: true, alt: "y equals 3 x minus 1")[$y = 3 x − 1$] #solutionbox[ #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (0, negative 1) and (1, 2).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (0, negative 1) and (1, 2).", caption: none) ] #math.equation(block: true, alt: "y equals 2 x minus 3")[$y = 2 x − 3$] #math.equation(block: true, alt: "y equals − x plus 3")[$y = "−" x + 3$] #solutionbox[ #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (0, 3) and (1, 2).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (0, 3) and (1, 2).", caption: none) ] #math.equation(block: true, alt: "y equals − x minus 4")[$y = "−" x − 4$] #math.equation(block: true, alt: "y equals minus the fraction 2 over 5 x minus 3")[$y = − frac(2, 5) x − 3$] #solutionbox[ #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (0, negative 3) and (5, negative 5).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (0, negative 3) and (5, negative 5).", caption: none) ] #math.equation(block: true, alt: "y equals minus the fraction 3 over 5 x plus 2")[$y = − frac(3, 5) x + 2$] #math.equation(block: true, alt: "3 x minus 2 y equals 4")[$3 x − 2 y = 4$] #solutionbox[ #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (0, negative 2) and (2, 1).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (0, negative 2) and (2, 1).", caption: none) ] #math.equation(block: true, alt: "3 x minus 4 y equals 8")[$3 x − 4 y = 8$] #strong[Choose the Most Convenient Method to Graph a Line] In the following exercises, determine the most convenient method to graph each line. #math.equation(block: true, alt: "x equals 2")[$x = 2$] #solutionbox[ vertical line ] #math.equation(block: true, alt: "y equals 5")[$y = 5$] #math.equation(block: true, alt: "y equals −3 x plus 4")[$y = −3 x + 4$] #solutionbox[ slope-intercept ] #math.equation(block: true, alt: "x minus y equals 5")[$x − y = 5$] #math.equation(block: true, alt: "x minus y equals 1")[$x − y = 1$] #solutionbox[ intercepts ] #math.equation(block: true, alt: "y equals the fraction 2 over 3 x minus 1")[$y = frac(2, 3) x − 1$] #math.equation(block: true, alt: "3 x minus 2 y equals −12")[$3 x − 2 y = −12$] #solutionbox[ intercepts ] #math.equation(block: true, alt: "2 x minus 5 y equals −10")[$2 x − 5 y = −10$] #strong[Graph and Interpret Applications of Slope–Intercept] The equation #math.equation(block: false, alt: "P equals 31 plus 1.75 w")[$P = 31 + 1.75 w$] models the relation between the amount of Tuyet’s monthly water bill payment, #emph[P], in dollars, and the number of units of water, #emph[w], used. ⓐ Find Tuyet’s payment for a month when 0 units of water are used. ⓑ Find Tuyet’s payment for a month when 12 units of water are used. ⓒ Interpret the slope and #emph[P]-intercept of the equation. ⓓ Graph the equation. #solutionbox[ ⓐ \$31 #linebreak() ⓑ \$52 #linebreak() ⓒ The slope, #math.equation(block: false, alt: "1.75 ,")[$1.75 ,$] means that the payment, #emph[P], increases by #math.equation(block: false, alt: "$ 1.75")[$"$" 1.75$] when the number of units of water used, #emph[w,] increases by 1. The #emph[P]-intercept means that when the number units of water Tuyet used is 0, the payment is \$31. #linebreak() ⓓ #linebreak() #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 1 to 21. The y-axis runs from negative 1 to 80. The line goes through the points (0, 31) and (12, 52).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 1 to 21. The y-axis runs from negative 1 to 80. The line goes through the points (0, 31) and (12, 52).", caption: none) ] The equation #math.equation(block: false, alt: "P equals 28 plus 2.54 w")[$P = 28 + 2.54 w$] models the relation between the amount of R and y’s monthly water bill payment, #emph[P], in dollars, and the number of units of water, #emph[w], used. ⓐ Find the payment for a month when R and y used 0 units of water. ⓑ Find the payment for a month when R and y used 15 units of water. ⓒ Interpret the slope and #emph[P]-intercept of the equation. ⓓ Graph the equation. Bruce drives his car for his job. The equation #math.equation(block: false, alt: "R equals 0.575 m plus 42")[$R = 0.575 m + 42$] models the relation between the amount in dollars, #emph[R], that he is reimbursed and the number of miles, #emph[m], he drives in one day. ⓐ Find the amount Bruce is reimbursed on a day when he drives 0 miles. ⓑ Find the amount Bruce is reimbursed on a day when he drives 220 miles. ⓒ Interpret the slope and #emph[R]-intercept of the equation. ⓓ Graph the equation. #solutionbox[ ⓐ \$42 #linebreak() ⓑ \$168.50 #linebreak() ⓒ The slope, 0.575 means that the amount he is reimbursed, #emph[R], increases by \$0.575 when the number of miles driven, #emph[m,] increases by 1. The #emph[R]-intercept means that when the number miles driven is 0, the amount reimbursed is \$42. #linebreak() ⓓ #linebreak() #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 50 to 250. The y-axis runs from negative 50 to 300. The line goes through the points (0, 42) and (220, 168.5).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 50 to 250. The y-axis runs from negative 50 to 300. The line goes through the points (0, 42) and (220, 168.5).", caption: none) ] Janelle is planning to rent a car while on vacation. The equation #math.equation(block: false, alt: "C equals 0.32 m plus 15")[$C = 0.32 m + 15$] models the relation between the cost in dollars, #emph[C], per day and the number of miles, #emph[m], she drives in one day. ⓐ Find the cost if Janelle drives the car 0 miles one day. ⓑ Find the cost on a day when Janelle drives the car 400 miles. ⓒ Interpret the slope and #emph[C]-intercept of the equation. ⓓ Graph the equation. Cherie works in retail and her weekly salary includes commission for the amount she sells. The equation #math.equation(block: false, alt: "S equals 400 plus 0.15 c")[$S = 400 + 0.15 c$] models the relation between her weekly salary, #emph[S], in dollars and the amount of her sales, #emph[c], in dollars. ⓐ Find Cherie’s salary for a week when her sales were \$0. ⓑ Find Cherie’s salary for a week when her sales were \$3,600. ⓒ Interpret the slope and #emph[S]-intercept of the equation. ⓓ Graph the equation. #solutionbox[ ⓐ \$400 #linebreak() ⓑ \$940 #linebreak() ⓒ The slope, #math.equation(block: false, alt: "0.15 ,")[$0.15 ,$] means that Cherie’s salary, #emph[S], increases by \$0.15 for every \$1 increase in her sales. The #emph[S]-intercept means that when her sales are \$0, her salary is \$400. #linebreak() ⓓ #linebreak() #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 500 to 3500. The y-axis runs from negative 200 to 1000. The line goes through the points (0, 400) and (3600, 940).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 500 to 3500. The y-axis runs from negative 200 to 1000. The line goes through the points (0, 400) and (3600, 940).", caption: none) ] Patel’s weekly salary includes a base pay plus commission on his sales. The equation #math.equation(block: false, alt: "S equals 750 plus 0.09 c")[$S = 750 + 0.09 c$] models the relation between his weekly salary, #emph[S], in dollars and the amount of his sales, #emph[c], in dollars. ⓐ Find Patel’s salary for a week when his sales were 0. ⓑ Find Patel’s salary for a week when his sales were 18,540. ⓒ Interpret the slope and #emph[S]-intercept of the equation. ⓓ Graph the equation. Costa is planning a lunch banquet. The equation #math.equation(block: false, alt: "C equals 450 plus 28 g")[$C = 450 + 28 g$] models the relation between the cost in dollars, #emph[C], of the banquet and the number of guests, #emph[g]. ⓐ Find the cost if the number of guests is 40. ⓑ Find the cost if the number of guests is 80. ⓒ Interpret the slope and #emph[C]-intercept of the equation. ⓓ Graph the equation. #solutionbox[ ⓐ \$1570 #linebreak() ⓑ \$2690 #linebreak() ⓒ The slope gives the cost per guest. The slope, 28, means that the cost, #emph[C], increases by \$28 when the number of guests increases by 1. The #emph[C]-intercept means that if the number of guests was 0, the cost would be \$450. #linebreak() ⓓ #linebreak() #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 20 to 100. The y-axis runs from negative 1000 to 7000. The line goes through the points (0, 450) and (40, 1570).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 20 to 100. The y-axis runs from negative 1000 to 7000. The line goes through the points (0, 450) and (40, 1570).", caption: none) ] Margie is planning a dinner banquet. The equation #math.equation(block: false, alt: "C equals 750 plus 42 g")[$C = 750 + 42 g$] models the relation between the cost in dollars, #emph[C] of the banquet and the number of guests, #emph[g]. ⓐ Find the cost if the number of guests is 50. ⓑ Find the cost if the number of guests is 100. ⓒ Interpret the slope and #emph[C]-intercept of the equation. ⓓ Graph the equation. #strong[Use Slopes to Identify Parallel and Perpendicular Lines] In the following exercises, use slopes and #emph[y]-intercepts to determine if the lines are parallel, perpendicular, or neither. #math.equation(block: true, alt: "y equals the fraction 3 over 4 x minus 3 ; 3 x minus 4 y equals −2")[$y = frac(3, 4) x − 3 ; #h(1em) 3 x − 4 y = −2$] #solutionbox[ parallel ] #math.equation(block: true, alt: "3 x plus 4 y equals minus 2 ; y equals the fraction 3 over 4 x minus 3")[$3 x + 4 y = − 2 ; #h(1em) y = frac(3, 4) x − 3$] #math.equation(block: true, alt: "2 x minus 4 y equals 6 ; x minus 2 y equals 3")[$2 x − 4 y = 6 ; #h(1em) x − 2 y = 3$] #solutionbox[ neither ] #math.equation(block: true, alt: "8 x plus 6 y equals 6 ; 12 x plus 9 y equals 12")[$8 x + 6 y = 6 ; #h(1em) 12 x + 9 y = 12$] #math.equation(block: true, alt: "x equals 5 ; x equals −6")[$x = 5 ; #h(1em) x = −6$] #solutionbox[ parallel ] #math.equation(block: true, alt: "x equals minus 3 ; x equals minus 2")[$x = − 3 ; #h(1em) x = − 2$] #math.equation(block: true, alt: "4 x minus 2 y equals 5 ; 3 x plus 6 y equals 8")[$4 x − 2 y = 5 ; #h(1em) 3 x + 6 y = 8$] #solutionbox[ perpendicular ] #math.equation(block: true, alt: "8 x minus 2 y equals 7 ; 3 x plus 12 y equals 9")[$8 x − 2 y = 7 ; #h(1em) 3 x + 12 y = 9$] #math.equation(block: true, alt: "3 x minus 6 y equals 12 ; 6 x minus 3 y equals 3")[$3 x − 6 y = 12 ; #h(1em) 6 x − 3 y = 3$] #solutionbox[ neither ] #math.equation(block: true, alt: "9 x minus 5 y equals 4 ; 5 x plus 9 y equals minus 1")[$9 x − 5 y = 4 ; #h(1em) 5 x + 9 y = − 1$] #math.equation(block: true, alt: "7 x minus 4 y equals 8 ; 4 x plus 7 y equals 14")[$7 x − 4 y = 8 ; #h(1em) 4 x + 7 y = 14$] #solutionbox[ perpendicular ] #math.equation(block: true, alt: "5 x minus 2 y equals 11 ; 5 x minus y equals 7")[$5 x − 2 y = 11 ; #h(1em) 5 x − y = 7$] #math.equation(block: true, alt: "3 x minus 2 y equals 8 ; 2 x plus 3 y equals 6")[$3 x − 2 y = 8 ; #h(1em) 2 x + 3 y = 6$] #solutionbox[ perpendicular ] #math.equation(block: true, alt: "2 x plus 3 y equals 5 ; 3 x minus 2 y equals 7")[$2 x + 3 y = 5 ; #h(1em) 3 x − 2 y = 7$] #math.equation(block: true, alt: "3 x minus 2 y equals 1 ; 2 x minus 3 y equals 2")[$3 x − 2 y = 1 ; #h(1em) 2 x − 3 y = 2$] #solutionbox[ neither ] #math.equation(block: true, alt: "2 x plus 4 y equals 3 ; 6 x plus 3 y equals 2")[$2 x + 4 y = 3 ; #h(1em) 6 x + 3 y = 2$] #math.equation(block: true, alt: "y equals 2 ; y equals 6")[$y = 2 ; #h(1em) y = 6$] #solutionbox[ parallel ] #math.equation(block: true, alt: "y equals minus 1 ; y equals 2")[$y = − 1 ; #h(1em) y = 2$] ==== Writing Exercises How does the graph of a line with slope #math.equation(block: false, alt: "m equals minus 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 ?")[$m = 2 ?$] #solutionbox[ It is perpendicular to the second line. ] Why is the slope of a vertical line “undefined”? Explain how you can graph a line given a point and its slope. #solutionbox[ Answers will vary. ] Explain in your own words how to decide which method to use to graph a line. ==== Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. #figure(figph[This table has 7 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “find the slope of a line”, “graph a line given a point and the slope”, “graph a line using its slope and intercept”, “choose the most convenient method to graph a line”, “graph and interpret applications of slope-intercept”, and “use slopes to identify parallel and perpendicular lines”. The other columns are left blank so that the learner may indicate their mastery level for each topic.], alt: "This table has 7 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “find the slope of a line”, “graph a line given a point and the slope”, “graph a line using its slope and intercept”, “choose the most convenient method to graph a line”, “graph and interpret applications of slope-intercept”, and “use slopes to identify parallel and perpendicular lines”. The other columns are left blank so that the learner may indicate their mastery level for each topic.", caption: none) ⓑ After reviewing this checklist, what will you do to become confident for all objectives?