#set document(title: "11.2 Use the Rectangular Coordinate System", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 11.2#h(0.6em)Use the Rectangular Coordinate System #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Before you get started, take this readiness quiz. Evaluate: #math.equation(block: false, alt: "x plus 3")[$x + 3$] when #math.equation(block: false, alt: "x equals −1 .")[$x = −1 .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "2")[$2$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate: #math.equation(block: false, alt: "2 x minus 5 y")[$2 x − 5 y$] when #math.equation(block: false, alt: "x equals 3 , y equals −2 .")[$x = 3 , y = −2 .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "16")[$16$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve for #math.equation(block: false, alt: "y : 40 minus 4 y equals 20 .")[$y ":" #h(0.2em) 40 − 4 y = 20 .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "5")[$5$] ] ] === Plot Points on a Rectangular Coordinate System Many maps, such as the Campus Map shown in , use a grid system to identify locations. Do you see the numbers #math.equation(block: false, alt: "1 , 2 , 3 ,")[$1 , 2 , 3 ,$] and #math.equation(block: false, alt: "4")[$4$] across the top and bottom of the map and the letters A, B, C, and D along the sides? Every location on the map can be identified by a number and a letter. For example, the Student Center is in section 2B. It is located in the grid section above the number #math.equation(block: false, alt: "2")[$2$] and next to the letter B. In which grid section is the Stadium? The Stadium is in section 4D. #figure(figph[The figure shows a labeled grid representing the Campus Map. The columns are labeled 1 through 4 and the rows are labeled A through D. At position A-1 is the title Parking Garage. At position A-4 is a rectangle labeled Residence Halls. At position B-2 is a rectangle labeled Student Center. At position B-3 is a rectangle labeled Engineering Building. At position C-1 is a rectangle labeled Taylor Hall. At position C-2 is a rectangle labeled Library. At position C-4 is a rectangle labeled Tiger Field. At position D-4 is a rectangle labeled Stadium.], alt: "The figure shows a labeled grid representing the Campus Map. The columns are labeled 1 through 4 and the rows are labeled A through D. At position A-1 is the title Parking Garage. At position A-4 is a rectangle labeled Residence Halls. At position B-2 is a rectangle labeled Student Center. At position B-3 is a rectangle labeled Engineering Building. At position C-1 is a rectangle labeled Taylor Hall. At position C-2 is a rectangle labeled Library. At position C-4 is a rectangle labeled Tiger Field. At position D-4 is a rectangle labeled Stadium.", caption: none) #examplebox("Example 1")[][ Use the map in . + ⓐ Find the grid section of the Residence Halls. + ⓑ What is located in grid section 4C? #solutionbox[ + ⓐ Read the number below the Residence Halls, #math.equation(block: false, alt: "4 ,")[$4 ,$] and the letter to the side, A. So the Residence Halls are in grid section 4A. + ⓑ Find #math.equation(block: false, alt: "4")[$4$] across the bottom of the map and C along the side. Look below the #math.equation(block: false, alt: "4")[$4$] and next to the C. Tiger Field is in grid section 4C. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use the map in . + ⓐ Find the grid section of Taylor Hall. + ⓑ What is located in section 3B? #solutionbox[ + ⓐ 1C + ⓑ Engineering Building ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use the map in . + ⓐ Find the grid section of the Parking Garage. + ⓑ What is located in section 2C? #solutionbox[ + ⓐ 1A + ⓑ Library ] ] Just as maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a rectangular coordinate system. To create a rectangular coordinate system, start with a horizontal number line. Show both positive and negative numbers as you did before, using a convenient scale unit. This horizontal number line is called the #strong[#emph[x]-axis]. #figure(figph[The figure shows a number line with integer values labeled from -5 to 5.], alt: "The figure shows a number line with integer values labeled from -5 to 5.", caption: none) Now, make a vertical number line passing through the #math.equation(block: false, alt: "x -axis")[$x "-axis"$] at #math.equation(block: false, alt: "0 .")[$0 .$] Put the positive numbers above #math.equation(block: false, alt: "0")[$0$] and the negative numbers below #math.equation(block: false, alt: "0 .")[$0 .$] This vertical line is called the #strong[#emph[y]-axis]. Vertical grid lines pass through the integers marked on the #math.equation(block: false, alt: "x -axis .")[$x "-axis" .$] Horizontal grid lines pass through the integers marked on the #math.equation(block: false, alt: "y -axis .")[$y "-axis" .$] The resulting grid is the rectangular coordinate system. The rectangular coordinate system is also called the #math.equation(block: false, alt: "x - y")[$x "-" y$] plane, the coordinate plane, or the Cartesian coordinate system (since it was developed by a mathematician named René Descartes.) #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. An arrow points to the horizontal axis with the label “x-axis”. An arrow points to the vertical axis with label “y-axis”. An arrow points to the intersection of the axes with label “origin”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. An arrow points to the horizontal axis with the label “x-axis”. An arrow points to the vertical axis with label “y-axis”. An arrow points to the intersection of the axes with label “origin”.", caption: [The rectangular coordinate system.]) The #math.equation(block: false, alt: "x -axis")[$x "-axis"$] and the #math.equation(block: false, alt: "y -axis")[$y "-axis"$] form the rectangular coordinate system. These axes divide a plane into four areas, called #strong[quadrants]. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise. #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The top-right portion of the plane is labeled “I”, the top-left portion of the plane is labeled “II”, the bottom-left portion of the plane is labelled “III” and the bottom-right portion of the plane is labeled “IV”], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The top-right portion of the plane is labeled “I”, the top-left portion of the plane is labeled “II”, the bottom-left portion of the plane is labelled “III” and the bottom-right portion of the plane is labeled “IV”", caption: [The four quadrants of the rectangular coordinate system]) In the rectangular coordinate system, every point is represented by an #strong[ordered pair]. The first number in the ordered pair is the #emph[x]-coordinate of the point, and the second number is the #emph[y]-coordinate of the point. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Ordered Pair] An ordered pair, #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$] gives the coordinates of a point in a rectangular coordinate system. #math.equation(block: true, alt: "The first number is the x -coordinate .; The second number is the y -coordinate .")[$"The first number is the" #h(0.2em) x "-coordinate" . \ "The second number is the" #h(0.2em) y "-coordinate" .$]#figure(figph[The ordered pair x y is labeled with the first coordinate x labeled as “x-coordinate” and the second coordinate y labeled as “y-coordinate”], alt: "The ordered pair x y is labeled with the first coordinate x labeled as “x-coordinate” and the second coordinate y labeled as “y-coordinate”", caption: none) ] So how do the coordinates of a point help you locate a point on the #math.equation(block: false, alt: "x - y")[$x "-" y$] plane? Let’s try locating the point #math.equation(block: false, alt: "open parenthesis 2 , 5 close parenthesis")[$( 2 , 5 )$]. In this ordered pair, the #math.equation(block: false, alt: "x")[$x$]-coordinate is #math.equation(block: false, alt: "2")[$2$] and the #math.equation(block: false, alt: "y")[$y$]-coordinate is #math.equation(block: false, alt: "5")[$5$]. We start by locating the #math.equation(block: false, alt: "x")[$x$] value, #math.equation(block: false, alt: "2 ,")[$2 ,$] on the #math.equation(block: false, alt: "x -axis.")[$x "-axis."$] Then we lightly sketch a vertical line through #math.equation(block: false, alt: "x equals 2 ,")[$x = 2 ,$] as shown. #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. There is a vertical dotted line passing through 2 on the x-axis.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. There is a vertical dotted line passing through 2 on the x-axis.", caption: none) Now we locate the #math.equation(block: false, alt: "y")[$y$] value, #math.equation(block: false, alt: "5 ,")[$5 ,$] on the #math.equation(block: false, alt: "y")[$y$]-axis and sketch a horizontal line through #math.equation(block: false, alt: "y equals 5")[$y = 5$]. The point where these two lines meet is the point with coordinates #math.equation(block: false, alt: "open parenthesis 2 , 5 close parenthesis .")[$( 2 , 5 ) .$] We plot the point there, as shown. #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. An arrow starts at the origin and extends right to the number 2 on the x-axis. An arrow starts at the end of the first arrow at 2 on the x-axis and goes vertically 5 units to a point labeled “2, 5” in parentheses.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. An arrow starts at the origin and extends right to the number 2 on the x-axis. An arrow starts at the end of the first arrow at 2 on the x-axis and goes vertically 5 units to a point labeled “2, 5” in parentheses.", caption: none) #examplebox("Example 2")[][ Plot #math.equation(block: false, alt: "open parenthesis 1 , 3 close parenthesis")[$( 1 , 3 )$] and #math.equation(block: false, alt: "open parenthesis 3 , 1 close parenthesis")[$( 3 , 1 )$] in the same rectangular coordinate system. #solutionbox[ The coordinate values are the same for both points, but the #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$] values are reversed. Let’s begin with point #math.equation(block: false, alt: "open parenthesis 1 , 3 close parenthesis .")[$( 1 , 3 ) .$] The #math.equation(block: false, alt: "x -coordinate")[$x "-coordinate"$] is #math.equation(block: false, alt: "1")[$1$] so find #math.equation(block: false, alt: "1")[$1$] on the #math.equation(block: false, alt: "x -axis")[$x "-axis"$] and sketch a vertical line through #math.equation(block: false, alt: "x equals 1 .")[$x = 1 .$] The #math.equation(block: false, alt: "y -coordinate")[$y "-coordinate"$] is #math.equation(block: false, alt: "3")[$3$] so we find #math.equation(block: false, alt: "3")[$3$] on the #math.equation(block: false, alt: "y -axis")[$y "-axis"$] and sketch a horizontal line through #math.equation(block: false, alt: "y equals 3 .")[$y = 3 .$] Where the two lines meet, we plot the point #math.equation(block: false, alt: "open parenthesis 1 , 3 close parenthesis .")[$( 1 , 3 ) .$] #linebreak() #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. A horizontal dotted line passes through 3 on the y axis. A vertical dotted line passes through 1 on the x axis. The dotted lines intersect at a point labeled “ordered pair 1, 3”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. A horizontal dotted line passes through 3 on the y axis. A vertical dotted line passes through 1 on the x axis. The dotted lines intersect at a point labeled “ordered pair 1, 3”.", caption: none) To plot the point #math.equation(block: false, alt: "open parenthesis 3 , 1 close parenthesis ,")[$( 3 , 1 ) ,$] we start by locating #math.equation(block: false, alt: "3")[$3$] on the #math.equation(block: false, alt: "x -axis")[$x "-axis"$] and sketch a vertical line through #math.equation(block: false, alt: "x equals 3 .")[$x = 3 .$] Then we find #math.equation(block: false, alt: "1")[$1$] on the #math.equation(block: false, alt: "y -axis")[$y "-axis"$] and sketch a horizontal line through #math.equation(block: false, alt: "y equals 1 .")[$y = 1 .$] Where the two lines meet, we plot the point #math.equation(block: false, alt: "open parenthesis 3 , 1 close parenthesis .")[$( 3 , 1 ) .$] #linebreak() #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. A horizontal dotted line passes through 1 on the y-axis. A vertical dotted line passes through 3 on the x axis. The dotted line intersects at a point labeled “ordered pair 3, 1”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. A horizontal dotted line passes through 1 on the y-axis. A vertical dotted line passes through 3 on the x axis. The dotted line intersects at a point labeled “ordered pair 3, 1”.", caption: none) Notice that the order of the coordinates does matter, so, #math.equation(block: false, alt: "open parenthesis 1 , 3 close parenthesis")[$( 1 , 3 )$] is not the same point as #math.equation(block: false, alt: "open parenthesis 3 , 1 close parenthesis .")[$( 3 , 1 ) .$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Plot each point on the same rectangular coordinate system: #math.equation(block: false, alt: "open parenthesis 5 , 2 close parenthesis , open parenthesis 2 , 5 close parenthesis .")[$( 5 , 2 ) , ( 2 , 5 ) .$] #solutionbox[ #figure(figph[This answer graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. There are two labeled points: the first is ordered pair (5, 2), and the second is (2, 5)], alt: "This answer graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. There are two labeled points: the first is ordered pair (5, 2), and the second is (2, 5)", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Plot each point on the same rectangular coordinate system: #math.equation(block: false, alt: "open parenthesis 4 , 2 close parenthesis , open parenthesis 2 , 4 close parenthesis .")[$( 4 , 2 ) , ( 2 , 4 ) .$] #solutionbox[ #figure(figph[This answer graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. There are two labeled points: the first is ordered pair (2, 4), and the second is (4, 2)], alt: "This answer graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. There are two labeled points: the first is ordered pair (2, 4), and the second is (4, 2)", caption: none) ] ] #examplebox("Example 3")[][ Plot each point in the rectangular coordinate system and identify the quadrant in which the point is located: + ⓐ #math.equation(block: false, alt: "open parenthesis −1 , 3 close parenthesis")[$#h(0.2em) ( −1 , 3 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis −3 , −4 close parenthesis")[$#h(0.2em) ( −3 , −4 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 2 , −3 close parenthesis")[$#h(0.2em) ( 2 , −3 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis 3 , the fraction 5 over 2 close parenthesis")[$( 3 , frac(5, 2) )$] #solutionbox[ The first number of the coordinate pair is the #math.equation(block: false, alt: "x -coordinate ,")[$x "-coordinate" ,$] and the second number is the #math.equation(block: false, alt: "y -coordinate .")[$y "-coordinate" .$] ⓐ Since #math.equation(block: false, alt: "x equals −1 , y equals 3 ,")[$x = −1 , y = 3 ,$] the point #math.equation(block: false, alt: "open parenthesis −1 , 3 close parenthesis")[$( −1 , 3 )$] is in Quadrant II. ⓑ Since #math.equation(block: false, alt: "x equals −3 , y equals −4 ,")[$x = −3 , y = −4 ,$] the point #math.equation(block: false, alt: "open parenthesis −3 , −4 close parenthesis")[$( −3 , −4 )$] is in Quadrant III. ⓒ Since #math.equation(block: false, alt: "x equals 2 , y equals −1 ,")[$x = 2 , y = −1 ,$] the point #math.equation(block: false, alt: "open parenthesis 2 , −1 close parenthesis")[$( 2 , −1 )$] is in Quadrant lV. ⓓ Since #math.equation(block: false, alt: "x equals 3 , y equals the fraction 5 over 2 ,")[$x = 3 , y = frac(5, 2) ,$] the point #math.equation(block: false, alt: "open parenthesis 3 , the fraction 5 over 2 close parenthesis")[$( 3 , frac(5, 2) )$] is in Quadrant I. It may be helpful to write #math.equation(block: false, alt: "the fraction 5 over 2")[$frac(5, 2)$] as the mixed number, #math.equation(block: false, alt: "2 the fraction 1 over 2 ,")[$2 frac(1, 2) ,$] or decimal, #math.equation(block: false, alt: "2.5 .")[$2.5 .$] Then we know that the point is halfway between #math.equation(block: false, alt: "2")[$2$] and #math.equation(block: false, alt: "3")[$3$] on the #math.equation(block: false, alt: "y -axis .")[$y "-axis" .$] #linebreak() #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point “ordered pair 3, 5 over 2” is labeled “ordered pair “3,5 over 2”. The point “ordered pair -1, 3” is labeled “ordered pair -1, 3”. The point “ordered pair -3, -4” is labeled “ordered pair -3, -4”. The point “ordered pair 2, -1” is labeled “ordered pair 2, -1”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point “ordered pair 3, 5 over 2” is labeled “ordered pair “3,5 over 2”. The point “ordered pair -1, 3” is labeled “ordered pair -1, 3”. The point “ordered pair -3, -4” is labeled “ordered pair -3, -4”. The point “ordered pair 2, -1” is labeled “ordered pair 2, -1”.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Plot each point on a rectangular coordinate system and identify the quadrant in which the point is located. + ⓐ #math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis")[$#h(0.2em) ( −2 , 1 ) #h(0.2em)$] + ⓑ #math.equation(block: false, alt: "open parenthesis −3 , −1 close parenthesis")[$#h(0.2em) ( −3 , −1 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 4 , −4 close parenthesis")[$#h(0.2em) ( 4 , −4 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis −4 , the fraction 3 over 2 close parenthesis")[$#h(0.2em) ( −4 , frac(3, 2) )$] #solutionbox[ (a) Quadrant II, (b) Quadrant III, (c) Quadrant IV, (d) Quadrant II #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair -2, 1” is labeled “a”. The point “ordered pair -3, 1” is labeled “b”. The point “ordered pair 4, -4 is labeled “c”. The point “ordered pair -4, 3/2” is labeled “d”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair -2, 1” is labeled “a”. The point “ordered pair -3, 1” is labeled “b”. The point “ordered pair 4, -4 is labeled “c”. The point “ordered pair -4, 3/2” is labeled “d”.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Plot each point on a rectangular coordinate system and identify the quadrant in which the point is located. + ⓐ #math.equation(block: false, alt: "open parenthesis −4 , 1 close parenthesis")[$#h(0.2em) ( −4 , 1 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis −2 , 3 close parenthesis")[$#h(0.2em) ( −2 , 3 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 2 , −5 close parenthesis")[$#h(0.2em) ( 2 , −5 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis −3 , the fraction 5 over 2 close parenthesis")[$#h(0.2em) ( −3 , frac(5, 2) )$] #solutionbox[ (a) Quadrant II, (b) Quadrant II, (c) Quadrant IV, (d) Quadrant II #figure(figph[This image is an answer graph and shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair -4, 1” is labeled “a”. The point “ordered pair -2, 3” is labeled “b”. The point “ordered pair 2, -5” is labeled “c”. The point “ordered pair -3, 5/2” is labeled “d”.], alt: "This image is an answer graph and shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair -4, 1” is labeled “a”. The point “ordered pair -2, 3” is labeled “b”. The point “ordered pair 2, -5” is labeled “c”. The point “ordered pair -3, 5/2” is labeled “d”.", caption: none) ] ] How do the signs affect the location of the points? #examplebox("Example 4")[][ Plot each point: + ⓐ #math.equation(block: false, alt: "open parenthesis −5 , 2 close parenthesis")[$#h(0.2em) ( −5 , 2 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis −5 , −2 close parenthesis")[$#h(0.2em) ( −5 , −2 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 5 , 2 close parenthesis")[$#h(0.2em) ( 5 , 2 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis 5 , −2 close parenthesis")[$#h(0.2em) ( 5 , −2 )$] #solutionbox[ As we locate the #math.equation(block: false, alt: "x -coordinate")[$x "-coordinate"$] and the #math.equation(block: false, alt: "y -coordinate ,")[$y "-coordinate" ,$] we must be careful with the signs. #linebreak() #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point “ordered pair 5, 2” is labeled “ordered pair 5, 2”. The point “ordered pair -5, 2” is labeled “ordered pair -5, 2”. The point “ordered pair -5, -2” is labeled “ordered pair -5, -2”. The point “ordered pair 5, -2” is labeled “ordered pair 5, -2”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point “ordered pair 5, 2” is labeled “ordered pair 5, 2”. The point “ordered pair -5, 2” is labeled “ordered pair -5, 2”. The point “ordered pair -5, -2” is labeled “ordered pair -5, -2”. The point “ordered pair 5, -2” is labeled “ordered pair 5, -2”.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Plot each point: + ⓐ #math.equation(block: false, alt: "open parenthesis 4 , −3 close parenthesis")[$#h(0.2em) ( 4 , −3 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 4 , 3 close parenthesis")[$#h(0.2em) ( 4 , 3 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis −4 , −3 close parenthesis")[$#h(0.2em) ( −4 , −3 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis −4 , 3 close parenthesis")[$#h(0.2em) ( −4 , 3 )$] #solutionbox[ #figure(figph[This image is an answer graph and shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair 4, -3” is labeled “a”. The point “ordered pair 4, 3” is labeled “b”. The point “ordered pair -4, -3” is labeled “c”. The point “ordered pair -4, 3” is labeled “d”.], alt: "This image is an answer graph and shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair 4, -3” is labeled “a”. The point “ordered pair 4, 3” is labeled “b”. The point “ordered pair -4, -3” is labeled “c”. The point “ordered pair -4, 3” is labeled “d”.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Plot each point: + ⓐ #math.equation(block: false, alt: "open parenthesis −1 , 4 close parenthesis")[$#h(0.2em) ( −1 , 4 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 1 , 4 close parenthesis")[$#h(0.2em) ( 1 , 4 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 1 , −4 close parenthesis")[$#h(0.2em) ( 1 , −4 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis −1 , −4 close parenthesis")[$#h(0.2em) ( −1 , −4 )$] #solutionbox[ #figure(figph[This image is an answer graph and shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair -1, -4” is labeled “a”. The point “ordered pair 1, 4” is labeled “b”. The point “ordered pair 1, -4” is labeled “c”. The point “ordered pair -1, -4” is labeled “d”.], alt: "This image is an answer graph and shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair -1, -4” is labeled “a”. The point “ordered pair 1, 4” is labeled “b”. The point “ordered pair 1, -4” is labeled “c”. The point “ordered pair -1, -4” is labeled “d”.", caption: none) ] ] You may have noticed some patterns as you graphed the points in the two previous examples. For each point in Quadrant IV, what do you notice about the signs of the coordinates? What about the signs of the coordinates of the points in the third quadrant? The second quadrant? The first quadrant? Can you tell just by looking at the coordinates in which quadrant the point (−2, 5) is located? In which quadrant is (2, −5) located? #figure(figph[A Cartesian coordinate system shows points (-2, 5) in Quadrant II and (2, -5) in Quadrant IV. The axes range from -7 to 7, with grid lines every unit.], alt: "A Cartesian coordinate system shows points (-2, 5) in Quadrant II and (2, -5) in Quadrant IV. The axes range from -7 to 7, with grid lines every unit.", caption: none) We can summarize sign patterns of the quadrants as follows. Also see . #figure(table( columns: 4, align: left, inset: 6pt, table.header([Quadrant I], [Quadrant II], [Quadrant III], [Quadrant IV]), [(#emph[x],#emph[y])], [(#emph[x],#emph[y])], [(#emph[x],#emph[y])], [(#emph[x],#emph[y])], [(+,+)], [(−,+)], [(−,−)], [(+,−)], )) #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The top-right portion of the plane is labeled “I” and “ordered pair +, +”, the top-left portion of the plane is labeled “II” and “ordered pair -, +”, the bottom-left portion of the plane is labelled “III” “ordered pair -, -” and the bottom-right portion of the plane is labeled “IV” and “ordered pair +, -”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The top-right portion of the plane is labeled “I” and “ordered pair +, +”, the top-left portion of the plane is labeled “II” and “ordered pair -, +”, the bottom-left portion of the plane is labelled “III” “ordered pair -, -” and the bottom-right portion of the plane is labeled “IV” and “ordered pair +, -”.", caption: none) What if one coordinate is zero? Where is the point #math.equation(block: false, alt: "open parenthesis 0 , 4 close parenthesis")[$( 0 , 4 )$] located? Where is the point #math.equation(block: false, alt: "open parenthesis −2 , 0 close parenthesis")[$( −2 , 0 )$] located? The point #math.equation(block: false, alt: "open parenthesis 0 , 4 close parenthesis")[$( 0 , 4 )$] is on the #emph[y]-axis and the point #math.equation(block: false, alt: "open parenthesis minus 2 , 0 close parenthesis")[$( − 2 , 0 )$] is on the #emph[x]-axis. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Points on the Axes] Points with a #math.equation(block: false, alt: "y -coordinate")[$y "-coordinate"$] equal to #math.equation(block: false, alt: "0")[$0$] are on the #math.equation(block: false, alt: "x -axis ,")[$x "-axis" ,$] and have coordinates #math.equation(block: false, alt: "open parenthesis a , 0 close parenthesis .")[$( a , 0 ) .$] Points with an #math.equation(block: false, alt: "x -coordinate")[$x "-coordinate"$] equal to #math.equation(block: false, alt: "0")[$0$] are on the #math.equation(block: false, alt: "y -axis ,")[$y "-axis" ,$] and have coordinates #math.equation(block: false, alt: "open parenthesis 0 , b close parenthesis .")[$( 0 , b ) .$] What is the ordered pair of the point where the axes cross? At that point both coordinates are zero, so its ordered pair is #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]. The point has a special name. It is called the #emph[origin]. ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[The Origin] #math.equation(block: true, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] ] #examplebox("Example 5")[][ Plot each point on a coordinate grid: + ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis")[$#h(0.2em) ( 0 , 5 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 4 , 0 close parenthesis")[$#h(0.2em) ( 4 , 0 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis −3 , 0 close parenthesis")[$#h(0.2em) ( −3 , 0 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$#h(0.2em) ( 0 , 0 )$] + ⓔ #math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$#h(0.2em) ( 0 , −1 )$] #solutionbox[ + ⓐ Since #math.equation(block: false, alt: "x equals 0 ,")[$x = 0 ,$] the point whose coordinates are #math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis")[$( 0 , 5 )$] is on the #math.equation(block: false, alt: "y -axis .")[$y "-axis" .$] + ⓑ Since #math.equation(block: false, alt: "y equals 0 ,")[$y = 0 ,$] the point whose coordinates are #math.equation(block: false, alt: "open parenthesis 4 , 0 close parenthesis")[$( 4 , 0 )$] is on the #math.equation(block: false, alt: "x -axis .")[$x "-axis" .$] + ⓒ Since #math.equation(block: false, alt: "y equals 0 ,")[$y = 0 ,$] the point whose coordinates are #math.equation(block: false, alt: "open parenthesis −3 , 0 close parenthesis")[$( −3 , 0 )$] is on the #math.equation(block: false, alt: "x -axis .")[$x "-axis" .$] + ⓓ Since #math.equation(block: false, alt: "x equals 0")[$x = 0$] and #math.equation(block: false, alt: "y equals 0 ,")[$y = 0 ,$] the point whose coordinates are #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] is the origin. + ⓔ Since #math.equation(block: false, alt: "x equals 0 ,")[$x = 0 ,$] the point whose coordinates are #math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$( 0 , −1 )$] is on the #math.equation(block: false, alt: "y -axis .")[$y "-axis" .$] #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point “ordered pair 0, 0” is labeled “0, 0” in parentheses. The point “ordered pair 4, 0” is labeled “4, 0” in parentheses. The point “ordered pair 0, 5” is labeled “0, 5” in parentheses. The point “ordered pair 0, -1” is labeled “ 0, -1” in parentheses.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point “ordered pair 0, 0” is labeled “0, 0” in parentheses. The point “ordered pair 4, 0” is labeled “4, 0” in parentheses. The point “ordered pair 0, 5” is labeled “0, 5” in parentheses. The point “ordered pair 0, -1” is labeled “ 0, -1” in parentheses.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Plot each point on a coordinate grid: + ⓐ #math.equation(block: false, alt: "open parenthesis 4 , 0 close parenthesis")[$#h(0.2em) ( 4 , 0 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis −2 , 0 close parenthesis")[$#h(0.2em) ( −2 , 0 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$#h(0.2em) ( 0 , 0 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$#h(0.2em) ( 0 , 2 )$] + ⓔ #math.equation(block: false, alt: "open parenthesis 0 , −3 close parenthesis")[$#h(0.2em) ( 0 , −3 )$] #solutionbox[ #figure(figph[This image is an answer graph and shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point for ordered pair 4, 0 is plotted. The point for ordered pair -2, 0 is plotted. The point for ordered pair 0,0 is plotted. The point for ordered pair 0, 2 is plotted. The point for ordered pair 0,-3 is plotted.], alt: "This image is an answer graph and shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point for ordered pair 4, 0 is plotted. The point for ordered pair -2, 0 is plotted. The point for ordered pair 0,0 is plotted. The point for ordered pair 0, 2 is plotted. The point for ordered pair 0,-3 is plotted.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Plot each point on a coordinate grid: + ⓐ #math.equation(block: false, alt: "open parenthesis −5 , 0 close parenthesis")[$#h(0.2em) ( −5 , 0 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis")[$#h(0.2em) ( 3 , 0 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$#h(0.2em) ( 0 , 0 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$#h(0.2em) ( 0 , −1 )$] + ⓔ #math.equation(block: false, alt: "open parenthesis 0 , 4 close parenthesis")[$#h(0.2em) ( 0 , 4 )$] #solutionbox[ #figure(figph[This image is an answer graph and shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point for ordered pair -5, 0 is plotted. The point for ordered pair 3, 0 is plotted. The point for ordered pair 0,0 is plotted. The point for ordered pair 0, -1 is plotted. The point for ordered pair 0,4 is plotted.], alt: "This image is an answer graph and shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point for ordered pair -5, 0 is plotted. The point for ordered pair 3, 0 is plotted. The point for ordered pair 0,0 is plotted. The point for ordered pair 0, -1 is plotted. The point for ordered pair 0,4 is plotted.", caption: none) ] ] === Identify Points on a Graph In algebra, being able to identify the coordinates of a point shown on a graph is just as important as being able to plot points. To identify the #strong[#emph[x]-coordinate] of a point on a graph, read the number on the #strong[#emph[x]-axis] directly above or below the point. To identify the #strong[#emph[y]-coordinate] of a point, read the number on the #strong[#emph[y]-axis] directly to the left or right of the point. Remember, to write the ordered pair using the correct order #math.equation(block: false, alt: "open parenthesis x , y close parenthesis .")[$( x , y ) .$] #examplebox("Example 6")[][ Name the ordered pair of each point shown: #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point “ordered pair 2, 4” is labeled C. The point “ordered pair -3, 3” is labeled A. The point “ordered pair -1, -3” is labeled B. The point “ordered pair 4, -4” is labeled D.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point “ordered pair 2, 4” is labeled C. The point “ordered pair -3, 3” is labeled A. The point “ordered pair -1, -3” is labeled B. The point “ordered pair 4, -4” is labeled D.", caption: none) #solutionbox[ Point A is above #math.equation(block: false, alt: "−3")[$−3$] on the #math.equation(block: false, alt: "x -axis ,")[$x "-axis" ,$] so the #math.equation(block: false, alt: "x -coordinate")[$x "-coordinate"$] of the point is #math.equation(block: false, alt: "−3 .")[$−3 .$] The point is to the left of #math.equation(block: false, alt: "3")[$3$] on the #math.equation(block: false, alt: "y -axis ,")[$y "-axis" ,$] so the #math.equation(block: false, alt: "y -coordinate")[$y "-coordinate"$] of the point is #math.equation(block: false, alt: "3 .")[$3 .$] The coordinates of the point are #math.equation(block: false, alt: "open parenthesis −3 , 3 close parenthesis .")[$( −3 , 3 ) .$] Point B is below #math.equation(block: false, alt: "−1")[$−1$] on the #math.equation(block: false, alt: "x -axis ,")[$x "-axis" ,$] so the #math.equation(block: false, alt: "x -coordinate")[$x "-coordinate"$] of the point is #math.equation(block: false, alt: "−1 .")[$−1 .$] The point is to the left of #math.equation(block: false, alt: "−3")[$−3$] on the #math.equation(block: false, alt: "y -axis ,")[$y "-axis" ,$] so the #math.equation(block: false, alt: "y -coordinate")[$y "-coordinate"$] of the point is #math.equation(block: false, alt: "−3 .")[$−3 .$] The coordinates of the point are #math.equation(block: false, alt: "open parenthesis −1 , −3 close parenthesis .")[$( −1 , −3 ) .$] Point C is above #math.equation(block: false, alt: "2")[$2$] on the #math.equation(block: false, alt: "x -axis ,")[$x "-axis" ,$] so the #math.equation(block: false, alt: "x -coordinate")[$x "-coordinate"$] of the point is #math.equation(block: false, alt: "2 .")[$2 .$] The point is to the right of #math.equation(block: false, alt: "4")[$4$] on the #math.equation(block: false, alt: "y -axis ,")[$y "-axis" ,$] so the #math.equation(block: false, alt: "y -coordinate")[$y "-coordinate"$] of the point is #math.equation(block: false, alt: "4 .")[$4 .$] The coordinates of the point are #math.equation(block: false, alt: "open parenthesis 2 , 4 close parenthesis .")[$( 2 , 4 ) .$] Point D is below #math.equation(block: false, alt: "4")[$4$] on the #math.equation(block: false, alt: "x minus axis ,")[$x − "axis" ,$] so the #math.equation(block: false, alt: "x -coordinate")[$x "-coordinate"$] of the point is #math.equation(block: false, alt: "4 .")[$4 .$] The point is to the right of #math.equation(block: false, alt: "−4")[$−4$] on the #math.equation(block: false, alt: "y -axis ,")[$y "-axis" ,$] so the #math.equation(block: false, alt: "y -coordinate")[$y "-coordinate"$] of the point is #math.equation(block: false, alt: "−4 .")[$−4 .$] The coordinates of the point are #math.equation(block: false, alt: "open parenthesis 4 , −4 close parenthesis .")[$( 4 , −4 ) .$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Name the ordered pair of each point shown: #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point “ordered pair 5, 1” is labeled A. The point “ordered pair -2, 4” is labeled B. The point “ordered pair -5, -1” is labeled C. The point “ordered pair 3, -2” is labeled D.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point “ordered pair 5, 1” is labeled A. The point “ordered pair -2, 4” is labeled B. The point “ordered pair -5, -1” is labeled C. The point “ordered pair 3, -2” is labeled D.", caption: none) #solutionbox[ + A: (5,1) + B: (−2,4) + C: (−5,−1) + D: (3,−2) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Name the ordered pair of each point shown: #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point “ordered pair 4, 2” is labeled A. The point “ordered pair -2, 3” is labeled B. The point “ordered pair -4, -4” is labeled C. The point “ordered pair 3, -5” is labeled D.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point “ordered pair 4, 2” is labeled A. The point “ordered pair -2, 3” is labeled B. The point “ordered pair -4, -4” is labeled C. The point “ordered pair 3, -5” is labeled D.", caption: none) #solutionbox[ + A: (4,2) + B: (−2,3) + C: (−4,−4) + D: (3,−5) ] ] #examplebox("Example 7")[][ Name the ordered pair of each point shown: #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point “ordered pair 3, 0” is labeled C. The point “ordered pair 0, 1” is labeled D. The point “ordered pair -4, 0” is labeled A. The point “ordered pair 0, -2” is labeled B.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point “ordered pair 3, 0” is labeled C. The point “ordered pair 0, 1” is labeled D. The point “ordered pair -4, 0” is labeled A. The point “ordered pair 0, -2” is labeled B.", caption: none) #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Point A is on the #emph[x]-axis at #math.equation(block: false, alt: "x equals minus 4")[$x = − 4$].], [The coordinates of point A are #math.equation(block: false, alt: "open parenthesis minus 4 , 0 close parenthesis")[$( − 4 , 0 )$].]), [Point B is on the #emph[y]-axis at #math.equation(block: false, alt: "y equals minus 2")[$y = − 2$]], [The coordinates of point B are #math.equation(block: false, alt: "open parenthesis 0 , minus 2 close parenthesis")[$( 0 , − 2 )$].], [Point C is on the #emph[x]-axis at #math.equation(block: false, alt: "x equals 3")[$x = 3$].], [The coordinates of point C are #math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis")[$( 3 , 0 )$].], [Point D is on the #emph[y-]axis at #math.equation(block: false, alt: "y equals 1")[$y = 1$].], [The coordinates of point D are #math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$].], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Name the ordered pair of each point shown: #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point “ordered pair 4, 0” is labeled A. The point “ordered pair 0, 3” is labeled B. The point “ordered pair -3, 0” is labeled C. The point “ordered pair 0, -5” is labeled D.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point “ordered pair 4, 0” is labeled A. The point “ordered pair 0, 3” is labeled B. The point “ordered pair -3, 0” is labeled C. The point “ordered pair 0, -5” is labeled D.", caption: none) #solutionbox[ + A: (4,0) + B: (0,3) + C: (−3,0) + D: (0,−5) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Name the ordered pair of each point shown: #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point “ordered pair 5, 0” is labeled C. The point “ordered pair 0, 2” is labeled D. The point “ordered pair -3, 0” is labeled A. The point “ordered pair 0,-3” is labeled B.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point “ordered pair 5, 0” is labeled C. The point “ordered pair 0, 2” is labeled D. The point “ordered pair -3, 0” is labeled A. The point “ordered pair 0,-3” is labeled B.", caption: none) #solutionbox[ + A: (−3,0) + B: (0,−3) + C: (5,0) + D: (0,2) ] ] === Verify Solutions to an Equation in Two Variables All the equations we solved so far have been equations with one variable. In almost every case, when we solved the equation we got exactly one #strong[solution]. The process of solving an equation ended with a statement such as #math.equation(block: false, alt: "x equals 4 .")[$x = 4 .$] Then we checked the solution by substituting back into the equation. Here’s an example of a #strong[linear equation in one variable], and its one solution. #math.equation(block: true, alt: "3 x plus 5 equals 17; 3 x equals 12; x equals 4")[$3 x + 5 = 17 \ \ #h(1.6em) 3 x = 12 \ #h(2.1em) x = 4$]But equations can have more than one variable. Equations with two variables can be written in the general form #math.equation(block: false, alt: "A x plus B y equals C .")[$A x + B y = C .$] An equation of this form is called a linear equation in two variables. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Linear Equation] An equation of the form #math.equation(block: false, alt: "A x plus B y equals C ,")[$A x + B y = C ,$] where #math.equation(block: false, alt: "A and B")[$A #h(0.2em) "and" #h(0.2em) B$] are not both zero, is called a linear equation in two variables. ] Notice that the word “line” is in linear. Here is an example of a linear equation in two variables, #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y :")[$y ":"$] #figure(figph[A series of equations is shown. The first line shows A x + B x = C. The “A” is red, the “B” is blue, and the “C” is turquoise. The second line shows x + 4 y = 8. The “4” is blue and the “8” is turquoise. The last line shows A =1 in red, B = 4 in blue, and C =8 in turquoise.], alt: "A series of equations is shown. The first line shows A x + B x = C. The “A” is red, the “B” is blue, and the “C” is turquoise. The second line shows x + 4 y = 8. The “4” is blue and the “8” is turquoise. The last line shows A =1 in red, B = 4 in blue, and C =8 in turquoise.", caption: none) Is #math.equation(block: false, alt: "y equals −5 x plus 1")[$y = −5 x + 1$] a linear equation? It does not appear to be in the form #math.equation(block: false, alt: "A x plus B y equals C .")[$A x + B y = C .$] But we could rewrite it in this form. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The image displays the linear equation y = -5x + 1, written in a standard mathematical notation on a white background. This represents a line with a negative slope and a positive y-intercept.], alt: "The image displays the linear equation y = -5x + 1, written in a standard mathematical notation on a white background. This represents a line with a negative slope and a positive y-intercept.", caption: none)]), [Add #math.equation(block: false, alt: "5 x")[$5 x$] to both sides.], [#figure(figph[The image shows the algebraic equation y + 5x = -5x + 1 + 5x.], alt: "The image shows the algebraic equation y + 5x = -5x + 1 + 5x.", caption: none)], [Simplify.], [#figure(figph[A linear equation is displayed as y + 5x = 1, featuring variables y and x, constants 5 and 1, and arithmetic operators addition and equality. The text is in a clear, dark font against a white background.], alt: "A linear equation is displayed as y + 5x = 1, featuring variables y and x, constants 5 and 1, and arithmetic operators addition and equality. The text is in a clear, dark font against a white background.", caption: none)], [Use the Commutative Property to put it in #math.equation(block: false, alt: "A x plus B y equals C .")[$A x + B y = C .$]], [#figure(figph[Two linear equations are displayed: Ax + By = C (where A is red and B is light blue) and 5x + y = 1. The first equation shows a general form, while the second is a specific example.], alt: "Two linear equations are displayed: Ax + By = C (where A is red and B is light blue) and 5x + y = 1. The first equation shows a general form, while the second is a specific example.", caption: none)], )) By rewriting #math.equation(block: false, alt: "y equals −5 x plus 1")[$y = −5 x + 1$] as #math.equation(block: false, alt: "5 x plus y equals 1 ,")[$5 x + y = 1 ,$] we can see that it is a linear equation in two variables because it can be written in the form #math.equation(block: false, alt: "A x plus B y equals C .")[$A x + B y = C .$] Linear equations in two variables have infinitely many solutions. For every number that is substituted for #math.equation(block: false, alt: "x ,")[$x ,$] there is a corresponding #math.equation(block: false, alt: "y")[$y$] value. This pair of values is a #strong[solution to the linear equation] and is represented by the ordered pair #math.equation(block: false, alt: "open parenthesis x , y close parenthesis .")[$( x , y ) .$] When we substitute these values of #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$] into the equation, the result is a true statement because the value on the left side is equal to the value on the right side. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Solution to a Linear Equation in Two Variables] An ordered pair #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$] is a solution to the linear equation #math.equation(block: false, alt: "A x plus B y equals C ,")[$A x + B y = C ,$] if the equation is a true statement when the #math.equation(block: false, alt: "x -")[$x "-"$] and #math.equation(block: false, alt: "y -values")[$y "-values"$] of the ordered pair are substituted into the equation. ] #examplebox("Example 8")[][ Determine which ordered pairs are solutions of the equation #math.equation(block: false, alt: "x plus 4 y equals 8 :")[$x + 4 y = 8 ":"$] + ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$#h(0.2em) ( 0 , 2 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 2 , −4 close parenthesis")[$#h(0.2em) ( 2 , −4 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis −4 , 3 close parenthesis")[$#h(0.2em) ( −4 , 3 )$] #solutionbox[ Substitute the #math.equation(block: false, alt: "x - and y -values")[$x "- and" #h(0.2em) y "-values"$] from each ordered pair into the equation and determine if the result is a true statement. #figure(table( columns: 3, align: left, inset: 6pt, table.header([ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$#h(0.2em) ( 0 , 2 )$]], [ⓑ #math.equation(block: false, alt: "open parenthesis 2 , −4 close parenthesis")[$#h(0.2em) ( 2 , −4 )$]], [ⓒ #math.equation(block: false, alt: "open parenthesis −4 , 3 close parenthesis")[$#h(0.2em) ( −4 , 3 )$]]), [#figure(figph[This image demonstrates how to verify that x=0 and y=2 is a solution to the equation x+4y=8 by substituting the values, which results in the true statement 8=8.], alt: "This image demonstrates how to verify that x=0 and y=2 is a solution to the equation x+4y=8 by substituting the values, which results in the true statement 8=8.", caption: none)], [#figure(figph[The image demonstrates checking if x=2 and y=-4 satisfy the equation x + 4y = 8, showing that -14 does not equal 8, thus proving they are not solutions.], alt: "The image demonstrates checking if x=2 and y=-4 satisfy the equation x + 4y = 8, showing that -14 does not equal 8, thus proving they are not solutions.", caption: none)], [#figure(figph[A step-by-step verification demonstrating that the values x = -4 and y = 3 satisfy the equation x + 4y = 8, leading to 8 = 8, confirmed with a checkmark.], alt: "A step-by-step verification demonstrating that the values x = -4 and y = 3 satisfy the equation x + 4y = 8, leading to 8 = 8, confirmed with a checkmark.", caption: none)], [#math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$] is a solution.], [#math.equation(block: false, alt: "open parenthesis 2 , −4 close parenthesis")[$( 2 , −4 )$] is not a solution.], [#math.equation(block: false, alt: "open parenthesis −4 , 3 close parenthesis")[$( −4 , 3 )$] is a solution.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Determine which ordered pairs are solutions to the given equation: #math.equation(block: false, alt: "2 x plus 3 y equals 6")[$2 x + 3 y = 6$] + ⓐ #math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis")[$#h(0.2em) ( 3 , 0 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$#h(0.2em) ( 2 , 0 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 6 , −2 close parenthesis")[$#h(0.2em) ( 6 , −2 )$] #solutionbox[ ⓐ , ⓒ ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Determine which ordered pairs are solutions to the given equation: #math.equation(block: false, alt: "4 x minus y equals 8")[$4 x − y = 8$] + ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 8 close parenthesis")[$#h(0.2em) ( 0 , 8 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$#h(0.2em) ( 2 , 0 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 1 , −4 close parenthesis")[$#h(0.2em) ( 1 , −4 )$] #solutionbox[ ⓑ , ⓒ ] ] #examplebox("Example 9")[][ Determine which ordered pairs are solutions of the equation. #math.equation(block: false, alt: "y equals 5 x minus 1 :")[$y = 5 x − 1 ":"$] + ⓐ #math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$#h(0.2em) ( 0 , −1 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 1 , 4 close parenthesis")[$#h(0.2em) ( 1 , 4 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis −2 , −7 close parenthesis")[$#h(0.2em) ( −2 , −7 )$] #solutionbox[ Substitute the #math.equation(block: false, alt: "x -")[$x "-"$] and #math.equation(block: false, alt: "y -values")[$y "-values"$] from each ordered pair into the equation and determine if it results in a true statement. #figure(table( columns: 3, align: left, inset: 6pt, table.header([ⓐ #math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$#h(0.2em) ( 0 , −1 )$]], [ⓑ #math.equation(block: false, alt: "open parenthesis 1 , 4 close parenthesis")[$#h(0.2em) ( 1 , 4 )$]], [ⓒ #math.equation(block: false, alt: "open parenthesis −2 , −7 close parenthesis")[$#h(0.2em) ( −2 , −7 )$]]), [#figure(figph[Mathematical steps verify if x=0, y=-1 satisfies y=5x-1. Substitution leads to -1 ?= 5(0)-1, simplifying to -1 = -1, marked with a check. The point is a solution.], alt: "Mathematical steps verify if x=0, y=-1 satisfies y=5x-1. Substitution leads to -1 ?= 5(0)-1, simplifying to -1 = -1, marked with a check. The point is a solution.", caption: none)], [#figure(figph[A step-by-step mathematical verification is shown, starting with x=1 and y=4, and the equation y=5x-1. The values are substituted into the equation, leading to the confirmation 4=4 with a checkmark.], alt: "A step-by-step mathematical verification is shown, starting with x=1 and y=4, and the equation y=5x-1. The values are substituted into the equation, leading to the confirmation 4=4 with a checkmark.", caption: none)], [#figure(figph[A step-by-step verification of whether the point (-2, -7) satisfies the equation y = 5x - 1, demonstrating that -7 is not equal to -11, thus the point is not on the line.], alt: "A step-by-step verification of whether the point (-2, -7) satisfies the equation y = 5x - 1, demonstrating that -7 is not equal to -11, thus the point is not on the line.", caption: none)], [#math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$( 0 , −1 )$] is a solution.], [#math.equation(block: false, alt: "open parenthesis 1 , 4 close parenthesis")[$( 1 , 4 )$] is a solution.], [#math.equation(block: false, alt: "open parenthesis −2 , −7 close parenthesis")[$( −2 , −7 )$] is not a solution.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Determine which ordered pairs are solutions of the given equation: #math.equation(block: false, alt: "y equals 4 x minus 3")[$y = 4 x − 3$] + ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$#h(0.2em) ( 0 , 3 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 1 , 1 close parenthesis")[$#h(0.2em) ( 1 , 1 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 1 , 0 close parenthesis")[$#h(0.2em) ( 1 , 0 )$] #solutionbox[ ⓑ ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Determine which ordered pairs are solutions of the given equation: #math.equation(block: false, alt: "y equals −2 x plus 6")[$y = −2 x + 6$] + ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 6 close parenthesis")[$#h(0.2em) ( 0 , 6 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 1 , 4 close parenthesis")[$#h(0.2em) ( 1 , 4 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis −2 , −2 close parenthesis")[$#h(0.2em) ( −2 , −2 )$] #solutionbox[ ⓐ , ⓑ ] ] === Complete a Table of Solutions to a Linear Equation In the previous examples, we substituted the #math.equation(block: false, alt: "x - and y -values")[$x "- and" #h(0.2em) y "-values"$] of a given #strong[ordered pair] to determine whether or not it was a solution to a linear equation. But how do we find the ordered pairs if they are not given? One way is to choose a value for #math.equation(block: false, alt: "x")[$x$] and then solve the equation for #math.equation(block: false, alt: "y .")[$y .$] Or, choose a value for #math.equation(block: false, alt: "y")[$y$] and then solve for #math.equation(block: false, alt: "x .")[$x .$] We’ll start by looking at the solutions to the equation #math.equation(block: false, alt: "y equals 5 x minus 1")[$y = 5 x − 1$] we found in . We can summarize this information in a table of solutions. #figure(table( columns: 3, align: left, inset: 6pt, [#math.equation(block: false, alt: "y equals 5 x minus 1")[$y = 5 x − 1$]], [], [], [#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "−1")[$−1$]], [#math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$( 0 , −1 )$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "open parenthesis 1 , 4 close parenthesis")[$( 1 , 4 )$]], [], [], [], )) To find a third solution, we’ll let #math.equation(block: false, alt: "x equals 2")[$x = 2$] and solve for #math.equation(block: false, alt: "y .")[$y .$] #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "y equals 5 x minus 1")[$y = 5 x − 1$]]), [#figure(figph[The image shows the text 'Substitute x = 2.' in a blue-green gradient font against a white background.], alt: "The image shows the text 'Substitute x = 2.' in a blue-green gradient font against a white background.", caption: none)], [#figure(figph[A mathematical equation shows 'y = 5(2) - 1', where the number 2 is highlighted in blue.], alt: "A mathematical equation shows 'y = 5(2) - 1', where the number 2 is highlighted in blue.", caption: none)], [Multiply.], [#math.equation(block: false, alt: "y equals 10 minus 1")[$y = 10 − 1$]], [Simplify.], [#math.equation(block: false, alt: "y equals 9")[$y = 9$]], )) The ordered pair is a solution to #math.equation(block: false, alt: "y equals 5 x minus 1")[$y = 5 x − 1$]. We will add it to the table. #figure(table( columns: 3, align: left, inset: 6pt, [#math.equation(block: false, alt: "y equals 5 x minus 1")[$y = 5 x − 1$]], [], [], [#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "−1")[$−1$]], [#math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$( 0 , −1 )$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "open parenthesis 1 , 4 close parenthesis")[$( 1 , 4 )$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "9")[$9$]], [#math.equation(block: false, alt: "open parenthesis 2 , 9 close parenthesis")[$( 2 , 9 )$]], )) We can find more solutions to the equation by substituting any value of #math.equation(block: false, alt: "x")[$x$] or any value of #math.equation(block: false, alt: "y")[$y$] and solving the resulting equation to get another ordered pair that is a solution. There are an infinite number of solutions for this equation. #examplebox("Example 10")[][ Complete the table to find three solutions to the equation #math.equation(block: false, alt: "y equals 4 x minus 2 :")[$y = 4 x − 2 ":"$] #figure(table( columns: 3, align: left, inset: 6pt, [#math.equation(block: false, alt: "y equals 4 x minus 2")[$y = 4 x − 2$]], [], [], [#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]], [#math.equation(block: false, alt: "0")[$0$]], [], [], [#math.equation(block: false, alt: "−1")[$−1$]], [], [], [#math.equation(block: false, alt: "2")[$2$]], [], [], )) #solutionbox[ Substitute #math.equation(block: false, alt: "x equals 0 , x equals −1 ,")[$x = 0 , x = −1 ,$] and #math.equation(block: false, alt: "x equals 2")[$x = 2$] into #math.equation(block: false, alt: "y equals 4 x minus 2 .")[$y = 4 x − 2 .$] #figure(table( columns: 3, align: left, inset: 6pt, table.header([#figure(figph[A mathematical equation displaying 'x = 0' with the number 0 highlighted in a light blue color, set against a plain white background.], alt: "A mathematical equation displaying 'x = 0' with the number 0 highlighted in a light blue color, set against a plain white background.", caption: none)], [#figure(figph[A mathematical equation on a white background, displaying 'x = -1'. The 'x' and '=' symbols are in a dark gray tone, while the '-1' is depicted in a lighter blue color, indicating a numerical value.], alt: "A mathematical equation on a white background, displaying 'x = -1'. The 'x' and '=' symbols are in a dark gray tone, while the '-1' is depicted in a lighter blue color, indicating a numerical value.", caption: none)], [#figure(figph[A simple mathematical equation shows the variable 'x' is equal to the number 2, displayed with 'x' in black and '2' in light blue.], alt: "A simple mathematical equation shows the variable 'x' is equal to the number 2, displayed with 'x' in black and '2' in light blue.", caption: none)]), [#math.equation(block: false, alt: "y equals 4 x minus 2")[$y = 4 x − 2$]], [#math.equation(block: false, alt: "y equals 4 x minus 2")[$y = 4 x − 2$]], [#math.equation(block: false, alt: "y equals 4 x minus 2")[$y = 4 x − 2$]], [#figure(figph[A mathematical equation: y = 4 \* 0 - 2, with the number 0 highlighted in blue. The equation evaluates to y = -2.], alt: "A mathematical equation: y = 4 * 0 - 2, with the number 0 highlighted in blue. The equation evaluates to y = -2.", caption: none)], [#figure(figph[A mathematical equation, y = 4(-1) - 2, is displayed.], alt: "A mathematical equation, y = 4(-1) - 2, is displayed.", caption: none)], [#figure(figph[A mathematical equation 'y = 4 \* 2 - 2' is displayed, with the second '2' highlighted in a light blue color.], alt: "A mathematical equation 'y = 4 * 2 - 2' is displayed, with the second '2' highlighted in a light blue color.", caption: none)], [#math.equation(block: false, alt: "y equals 0 minus 2")[$y = 0 − 2$]], [#math.equation(block: false, alt: "y equals −4 minus 2")[$y = −4 − 2$]], [#math.equation(block: false, alt: "y equals 8 minus 2")[$y = 8 − 2$]], [#math.equation(block: false, alt: "y equals −2")[$y = −2$]], [#math.equation(block: false, alt: "y equals −6")[$y = −6$]], [#math.equation(block: false, alt: "y equals 6")[$y = 6$]], [#math.equation(block: false, alt: "open parenthesis 0 , −2 close parenthesis")[$( 0 , −2 )$]], [#math.equation(block: false, alt: "open parenthesis −1 , −6 close parenthesis")[$( −1 , −6 )$]], [#math.equation(block: false, alt: "open parenthesis 2 , 6 close parenthesis")[$( 2 , 6 )$]], )) The results are summarized in the table. #figure(table( columns: 3, align: left, inset: 6pt, [#math.equation(block: false, alt: "y equals 4 x minus 2")[$y = 4 x − 2$]], [], [], [#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "−2")[$−2$]], [#math.equation(block: false, alt: "open parenthesis 0 , −2 close parenthesis")[$( 0 , −2 )$]], [#math.equation(block: false, alt: "−1")[$−1$]], [#math.equation(block: false, alt: "−6")[$−6$]], [#math.equation(block: false, alt: "open parenthesis −1 , −6 close parenthesis")[$( −1 , −6 )$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "6")[$6$]], [#math.equation(block: false, alt: "open parenthesis 2 , 6 close parenthesis")[$( 2 , 6 )$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Complete the table to find three solutions to the equation: #math.equation(block: false, alt: "y equals 3 x minus 1 .")[$y = 3 x − 1 .$] #figure(table( columns: 3, align: left, inset: 6pt, [#math.equation(block: false, alt: "y equals 3 x minus 1")[$y = 3 x − 1$]], [], [], [#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]], [#math.equation(block: false, alt: "0")[$0$]], [], [], [#math.equation(block: false, alt: "−1")[$−1$]], [], [], [#math.equation(block: false, alt: "2")[$2$]], [], [], )) #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, [#math.equation(block: false, alt: "y equals 3 x minus 1")[$y = 3 x − 1$]], [], [], [#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "−1")[$−1$]], [#math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$( 0 , −1 )$]], [#math.equation(block: false, alt: "−1")[$−1$]], [#math.equation(block: false, alt: "−4")[$−4$]], [#math.equation(block: false, alt: "open parenthesis −1 , −4 close parenthesis")[$( −1 , −4 )$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "open parenthesis 2 , 5 close parenthesis")[$( 2 , 5 )$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Complete the table to find three solutions to the equation: #math.equation(block: false, alt: "y equals 6 x plus 1")[$y = 6 x + 1$] #figure(table( columns: 3, align: left, inset: 6pt, [#math.equation(block: false, alt: "y equals 6 x plus 1")[$y = 6 x + 1$]], [], [], [#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]], [#math.equation(block: false, alt: "0")[$0$]], [], [], [#math.equation(block: false, alt: "1")[$1$]], [], [], [#math.equation(block: false, alt: "−2")[$−2$]], [], [], )) #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, [#math.equation(block: false, alt: "y equals 6 x plus 1")[$y = 6 x + 1$]], [], [], [#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "7")[$7$]], [#math.equation(block: false, alt: "open parenthesis 1 , 7 close parenthesis")[$( 1 , 7 )$]], [#math.equation(block: false, alt: "−2")[$−2$]], [#math.equation(block: false, alt: "−11")[$−11$]], [#math.equation(block: false, alt: "open parenthesis −2 , −11 close parenthesis")[$( −2 , −11 )$]], )) ] ] #examplebox("Example 11")[][ Complete the table to find three solutions to the equation #math.equation(block: false, alt: "5 x minus 4 y equals 20 :")[$5 x − 4 y = 20 ":"$] #figure(table( columns: 3, align: left, inset: 6pt, [#math.equation(block: false, alt: "5 x minus 4 y equals 20")[$5 x − 4 y = 20$]], [], [], [#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]], [#math.equation(block: false, alt: "0")[$0$]], [], [], [], [#math.equation(block: false, alt: "0")[$0$]], [], [], [#math.equation(block: false, alt: "5")[$5$]], [], )) #solutionbox[ #figure(figph[The figure shows three algebraic substitutions into an equation. The first substitution is x = 0, with 0 shown in blue. The next line is 5 x- 4 y = 20. The next line is 5 times 0, shown in blue - 4 y = 20. The next line is 0 - 4 y = 20. The next line is - 4 y = 20. The next line is y = -5. The last line is “ordered pair 0, -5”. The second substitution is y = 0, with 0 shown in red. The next line is 5 x- 4 y = 20. The next line is 5 x - 4 times 0, with 0 shown in red. The next line is 5 x - 0 = 20. The next line is 5 x = 20. The next line is x = 4. The last line is “ordered pair 4, 0”. The third substitution is y = 5, with 5 shown in red. The next line is 5 x- 4 y = 20. The next line is 5 x - 4 times 5, with 5 shown in blue. The next line is 5 x - 20 = 20. The next line is 5 x = 40. The next line is x = 8. The last line is “ordered pair 8, 5”.], alt: "The figure shows three algebraic substitutions into an equation. The first substitution is x = 0, with 0 shown in blue. The next line is 5 x- 4 y = 20. The next line is 5 times 0, shown in blue - 4 y = 20. The next line is 0 - 4 y = 20. The next line is - 4 y = 20. The next line is y = -5. The last line is “ordered pair 0, -5”. The second substitution is y = 0, with 0 shown in red. The next line is 5 x- 4 y = 20. The next line is 5 x - 4 times 0, with 0 shown in red. The next line is 5 x - 0 = 20. The next line is 5 x = 20. The next line is x = 4. The last line is “ordered pair 4, 0”. The third substitution is y = 5, with 5 shown in red. The next line is 5 x- 4 y = 20. The next line is 5 x - 4 times 5, with 5 shown in blue. The next line is 5 x - 20 = 20. The next line is 5 x = 40. The next line is x = 8. The last line is “ordered pair 8, 5”.", caption: none) The results are summarized in the table. #figure(table( columns: 3, align: left, inset: 6pt, [#math.equation(block: false, alt: "5 x minus 4 y equals 20")[$5 x − 4 y = 20$]], [], [], [#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "−5")[$−5$]], [#math.equation(block: false, alt: "open parenthesis 0 , −5 close parenthesis")[$( 0 , −5 )$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "open parenthesis 4 , 0 close parenthesis")[$( 4 , 0 )$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "open parenthesis 8 , 5 close parenthesis")[$( 8 , 5 )$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Complete the table to find three solutions to the equation: #math.equation(block: false, alt: "2 x minus 5 y equals 20 .")[$2 x − 5 y = 20 .$] #figure(table( columns: 3, align: left, inset: 6pt, [#math.equation(block: false, alt: "2 x minus 5 y equals 20")[$2 x − 5 y = 20$]], [], [], [#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]], [#math.equation(block: false, alt: "0")[$0$]], [], [], [], [#math.equation(block: false, alt: "0")[$0$]], [], [#math.equation(block: false, alt: "−5")[$−5$]], [], [], )) #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, [#math.equation(block: false, alt: "2 x minus 5 y equals 20")[$2 x − 5 y = 20$]], [], [], [#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "−4")[$−4$]], [#math.equation(block: false, alt: "open parenthesis 0 , −4 close parenthesis")[$( 0 , −4 )$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "open parenthesis 10 , 0 close parenthesis")[$( 10 , 0 )$]], [#math.equation(block: false, alt: "−5")[$−5$]], [#math.equation(block: false, alt: "−6")[$−6$]], [#math.equation(block: false, alt: "open parenthesis −5 , −6 close parenthesis")[$( −5 , −6 )$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Complete the table to find three solutions to the equation: #math.equation(block: false, alt: "3 x minus 4 y equals 12 .")[$3 x − 4 y = 12 .$] #figure(table( columns: 3, align: left, inset: 6pt, [#math.equation(block: false, alt: "3 x minus 4 y equals 12")[$3 x − 4 y = 12$]], [], [], [#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]], [#math.equation(block: false, alt: "0")[$0$]], [], [], [], [#math.equation(block: false, alt: "0")[$0$]], [], [#math.equation(block: false, alt: "−4")[$−4$]], [], [], )) #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, [#math.equation(block: false, alt: "3 x minus 4 y equals 12")[$3 x − 4 y = 12$]], [], [], [#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "−3")[$−3$]], [#math.equation(block: false, alt: "open parenthesis 0 , −3 close parenthesis")[$( 0 , −3 )$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "open parenthesis 4 , 0 close parenthesis")[$( 4 , 0 )$]], [#math.equation(block: false, alt: "−4")[$−4$]], [#math.equation(block: false, alt: "−6")[$−6$]], [#math.equation(block: false, alt: "open parenthesis −4 , −6 close parenthesis")[$( −4 , −6 )$]], )) ] ] === Find Solutions to Linear Equations in Two Variables To find a solution to a linear equation, we can choose any number we want to substitute into the equation for either #math.equation(block: false, alt: "x")[$x$] or #math.equation(block: false, alt: "y .")[$y .$] We could choose #math.equation(block: false, alt: "1 , 100 , 1,000 ,")[$1 , 100 , 1,000 ,$] or any other value we want. But it’s a good idea to choose a number that’s easy to work with. We’ll usually choose #math.equation(block: false, alt: "0")[$0$] as one of our values. #examplebox("Example 12")[][ Find a solution to the equation #math.equation(block: false, alt: "3 x plus 2 y equals 6 .")[$3 x + 2 y = 6 .$] #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, table.header([#strong[Step 1:] Choose any value for one of the variables in the equation.], [], [We can substitute any value we want for #math.equation(block: false, alt: "x")[$#h(0.2em) x #h(0.2em)$] or any value for #math.equation(block: false, alt: "y .")[$#h(0.2em) y .$] #linebreak() Let's pick #math.equation(block: false, alt: "x equals 0 .")[$x = 0 .$] #linebreak() What is the value of #math.equation(block: false, alt: "y")[$#h(0.2em) y #h(0.2em)$] if #math.equation(block: false, alt: "x equals 0")[$#h(0.2em) x = 0$]?]), [#strong[Step 2:] Substitute that value into the equation. #linebreak() Solve for the other variable.], [#linebreak() Substitute #math.equation(block: false, alt: "0")[$0$] for #math.equation(block: false, alt: "x .")[$#h(0.2em) x .$] #linebreak() Simplify. #linebreak() #linebreak() Divide both sides by 2.], [#figure(figph[Step-by-step calculation showing how to find the y-intercept (y=3) of the equation 3x + 2y = 6 by setting x to 0.], alt: "Step-by-step calculation showing how to find the y-intercept (y=3) of the equation 3x + 2y = 6 by setting x to 0.", caption: none)], [#strong[Step 3:] Write the solution as an ordered pair.], [So, when #math.equation(block: false, alt: "x equals 0 , y equals 3 .")[$x = 0 , y = 3 .$]], [This solution is represented by the ordered pair #math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis .")[$( 0 , 3 ) .$]], [#strong[Step 4:] Check.], [#figure(figph[Instruction to substitute x=0 and y=3 into the equation 3x + 2y = 6.], alt: "Instruction to substitute x=0 and y=3 into the equation 3x + 2y = 6.", caption: none) #linebreak() Is the result a true equation? #linebreak() Yes!], [#figure(figph[Verifying if the point (0, 3) satisfies the equation 3x + 2y = 6. Substituting x=0 and y=3 proves the equality 6=6, confirming it's a solution.], alt: "Verifying if the point (0, 3) satisfies the equation 3x + 2y = 6. Substituting x=0 and y=3 proves the equality 6=6, confirming it's a solution.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find a solution to the equation: #math.equation(block: false, alt: "4 x plus 3 y equals 12 .")[$4 x + 3 y = 12 .$] #solutionbox[ Answers will vary. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find a solution to the equation: #math.equation(block: false, alt: "2 x plus 4 y equals 8 .")[$2 x + 4 y = 8 .$] #solutionbox[ Answers will vary. ] ] We said that linear equations in two variables have infinitely many solutions, and we’ve just found one of them. Let’s find some other solutions to the equation #math.equation(block: false, alt: "3 x plus 2 y equals 6 .")[$3 x + 2 y = 6 .$] #examplebox("Example 13")[][ Find three more solutions to the equation #math.equation(block: false, alt: "3 x plus 2 y equals 6 .")[$3 x + 2 y = 6 .$] #solutionbox[ To find solutions to #math.equation(block: false, alt: "3 x plus 2 y equals 6 ,")[$3 x + 2 y = 6 ,$] choose a value for #math.equation(block: false, alt: "x")[$x$] or #math.equation(block: false, alt: "y .")[$y .$] Remember, we can choose any value we want for #math.equation(block: false, alt: "x")[$x$] or #math.equation(block: false, alt: "y .")[$y .$] Here we chose #math.equation(block: false, alt: "1")[$1$] for #math.equation(block: false, alt: "x ,")[$x ,$] and #math.equation(block: false, alt: "0")[$0$] and #math.equation(block: false, alt: "−3")[$−3$] for #math.equation(block: false, alt: "y .")[$y .$] #linebreak() #figure(table( columns: 4, align: left, inset: 6pt, table.header([Substitute it into the equation.], [#figure(figph[Algebraic substitution showing y=0 being substituted into the equation 3x+2y=6 to simplify it to 3x+2(0)=6.], alt: "Algebraic substitution showing y=0 being substituted into the equation 3x+2y=6 to simplify it to 3x+2(0)=6.", caption: none)], [#figure(figph[An image depicting a step in solving a system of equations, where the value x = 1 is substituted into the equation 3x + 2y = 6, leading to 3(1) + 2y = 6. The number 1 is highlighted.], alt: "An image depicting a step in solving a system of equations, where the value x = 1 is substituted into the equation 3x + 2y = 6, leading to 3(1) + 2y = 6. The number 1 is highlighted.", caption: none)], [#figure(figph[A step-by-step display of algebraic substitution, showing the value y = -3 being substituted into the equation 3x + 2y = 6, resulting in 3x + 2(-3) = 6. The -3 is highlighted in red in both its initial definition and its substituted form.], alt: "A step-by-step display of algebraic substitution, showing the value y = -3 being substituted into the equation 3x + 2y = 6, resulting in 3x + 2(-3) = 6. The -3 is highlighted in red in both its initial definition and its substituted form.", caption: none)]), [Simplify. #linebreak() Solve.], [#figure(figph[A mathematical equation shown in two steps: first, 3x + 0 = 6, and then simplified to 3x = 6. This illustrates the identity property of addition where adding zero does not change the value.], alt: "A mathematical equation shown in two steps: first, 3x + 0 = 6, and then simplified to 3x = 6. This illustrates the identity property of addition where adding zero does not change the value.", caption: none)], [#figure(figph[Mathematical problem showing the simplification of 3 + 2y = 6 to 2y = 3, illustrating a step in solving an algebraic equation.], alt: "Mathematical problem showing the simplification of 3 + 2y = 6 to 2y = 3, illustrating a step in solving an algebraic equation.", caption: none)], [#figure(figph[An algebra problem showing the steps to solve for x, with the initial equation 3x - 6 = 6 followed by 3x = 12.], alt: "An algebra problem showing the steps to solve for x, with the initial equation 3x - 6 = 6 followed by 3x = 12.", caption: none)], [], [#figure(figph[The mathematical equation 'x = 2' is displayed in black text on a white background, representing a simple algebraic solution or assignment.], alt: "The mathematical equation 'x = 2' is displayed in black text on a white background, representing a simple algebraic solution or assignment.", caption: none)], [#figure(figph[The equation y = 3/2 is displayed in black text on a white background.], alt: "The equation y = 3/2 is displayed in black text on a white background.", caption: none)], [#figure(figph[The image displays a simple algebraic equation, 'x = 4,' centered on a white background. The variable 'x' is set equal to the number '4', representing a straightforward mathematical statement.], alt: "The image displays a simple algebraic equation, 'x = 4,' centered on a white background. The variable 'x' is set equal to the number '4', representing a straightforward mathematical statement.", caption: none)], [Write the ordered pair.], [#math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$]], [#math.equation(block: false, alt: "open parenthesis 1 , the fraction 3 over 2 close parenthesis")[$( 1 , frac(3, 2) )$]], [#math.equation(block: false, alt: "open parenthesis 4 , −3 close parenthesis")[$( 4 , −3 )$]], )) Check your answers. #linebreak() #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$]], [#math.equation(block: false, alt: "open parenthesis 1 , the fraction 3 over 2 close parenthesis")[$( 1 , frac(3, 2) )$]], [#math.equation(block: false, alt: "open parenthesis 4 , −3 close parenthesis")[$( 4 , −3 )$]]), [#figure(figph[Checking if (2,0) is a solution to the equation 3x + 2y = 6. Substituting x=2 and y=0, the equation simplifies to 6 + 0 = 6, confirming that (2,0) is indeed a valid solution.], alt: "Checking if (2,0) is a solution to the equation 3x + 2y = 6. Substituting x=2 and y=0, the equation simplifies to 6 + 0 = 6, confirming that (2,0) is indeed a valid solution.", caption: none)], [#figure(figph[Solution verification for 3x + 2y = 6: substituting x=1 and y=3/2 correctly yields 6=6, confirming the values satisfy the equation.], alt: "Solution verification for 3x + 2y = 6: substituting x=1 and y=3/2 correctly yields 6=6, confirming the values satisfy the equation.", caption: none)], [#figure(figph[This image demonstrates how to check if the point (4, -3) is a solution to the equation 3x + 2y = 6. Substituting x=4 and y=-3 into the equation simplifies it to 6=6, confirming it's a valid solution.], alt: "This image demonstrates how to check if the point (4, -3) is a solution to the equation 3x + 2y = 6. Substituting x=4 and y=-3 into the equation simplifies it to 6=6, confirming it's a valid solution.", caption: none)], )) So #math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis , open parenthesis 1 , the fraction 3 over 2 close parenthesis")[$( 2 , 0 ) , ( 1 , frac(3, 2) )$] and #math.equation(block: false, alt: "open parenthesis 4 , −3 close parenthesis")[$( 4 , −3 )$] are all solutions to the equation #math.equation(block: false, alt: "3 x plus 2 y equals 6 .")[$3 x + 2 y = 6 .$] In the previous example, we found that #math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$( 0 , 3 )$] is a solution, too. We can list these solutions in a table. #figure(table( columns: 3, align: left, inset: 6pt, [#math.equation(block: false, alt: "3 x plus 2 y equals 6")[$3 x + 2 y = 6$]], [], [], [#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$( 0 , 3 )$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "the fraction 3 over 2")[$frac(3, 2)$]], [#math.equation(block: false, alt: "open parenthesis 1 , the fraction 3 over 2 close parenthesis")[$( 1 , frac(3, 2) )$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "−3")[$−3$]], [#math.equation(block: false, alt: "open parenthesis 4 , −3 close parenthesis")[$( 4 , −3 )$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find three solutions to the equation: #math.equation(block: false, alt: "2 x plus 3 y equals 6 .")[$2 x + 3 y = 6 .$] #solutionbox[ Answers will vary. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find three solutions to the equation: #math.equation(block: false, alt: "4 x plus 2 y equals 8 .")[$4 x + 2 y = 8 .$] #solutionbox[ Answers will vary. ] ] Let’s find some solutions to another equation now. #examplebox("Example 14")[][ Find three solutions to the equation #math.equation(block: false, alt: "x minus 4 y equals 8 .")[$x − 4 y = 8 .$] #solutionbox[ #figure(table( columns: 4, align: left, inset: 6pt, table.header([], [#figure(figph[The image displays the linear equation x - 4y = 8, written in a standard algebraic format.], alt: "The image displays the linear equation x - 4y = 8, written in a standard algebraic format.", caption: none)], [#figure(figph[The image displays the equation x - 4y = 8.], alt: "The image displays the equation x - 4y = 8.", caption: none)], [#figure(figph[A mathematical equation displays 'x - 4y = 8' in a bold, black font against a white background.], alt: "A mathematical equation displays 'x - 4y = 8' in a bold, black font against a white background.", caption: none)]), [Choose a value for #math.equation(block: false, alt: "x")[$x$] or #math.equation(block: false, alt: "y .")[$y .$]], [#figure(figph[A close-up image displaying the mathematical equation 'x = 0' in black and teal typography against a white background.], alt: "A close-up image displaying the mathematical equation 'x = 0' in black and teal typography against a white background.", caption: none)], [#figure(figph[The equation y = 0 is displayed, with the number 0 highlighted in red.], alt: "The equation y = 0 is displayed, with the number 0 highlighted in red.", caption: none)], [#figure(figph[The equation y = 3 is displayed with the number 3 highlighted in red, indicating a constant value for y.], alt: "The equation y = 3 is displayed with the number 3 highlighted in red, indicating a constant value for y.", caption: none)], [Substitute it into the equation.], [#figure(figph[The image displays the mathematical equation 0 - 4y = 8. The '0' is rendered in a light blue color, while the rest of the equation, including the minus sign, '4y', equals sign, and '8', is in black.], alt: "The image displays the mathematical equation 0 - 4y = 8. The '0' is rendered in a light blue color, while the rest of the equation, including the minus sign, '4y', equals sign, and '8', is in black.", caption: none)], [#figure(figph[A mathematical equation shows 'x - 4 \* 0 = 8', where the number 0 is highlighted in red.], alt: "A mathematical equation shows 'x - 4 * 0 = 8', where the number 0 is highlighted in red.", caption: none)], [#figure(figph[The mathematical equation x - 4 ⋅ 3 = 8, with the number 3 highlighted in red.], alt: "The mathematical equation x - 4 ⋅ 3 = 8, with the number 3 highlighted in red.", caption: none)], [Solve.], [#figure(figph[Solving the linear equation -4y = 8, which results in y = -2.], alt: "Solving the linear equation -4y = 8, which results in y = -2.", caption: none)], [#figure(figph[A straightforward algebraic problem showing x - 0 = 8, simplifying to x = 8. A clear example of the additive identity property.], alt: "A straightforward algebraic problem showing x - 0 = 8, simplifying to x = 8. A clear example of the additive identity property.", caption: none)], [#figure(figph[An algebraic equation x - 12 = 8 is shown, with its solution x = 20 presented below it. The image displays a simple mathematical problem and its answer.], alt: "An algebraic equation x - 12 = 8 is shown, with its solution x = 20 presented below it. The image displays a simple mathematical problem and its answer.", caption: none)], [Write the ordered pair.], [#math.equation(block: false, alt: "open parenthesis 0 , −2 close parenthesis")[$( 0 , −2 )$]], [#math.equation(block: false, alt: "open parenthesis 8 , 0 close parenthesis")[$( 8 , 0 )$]], [#math.equation(block: false, alt: "open parenthesis 20 , 3 close parenthesis")[$( 20 , 3 )$]], )) So #math.equation(block: false, alt: "open parenthesis 0 , −2 close parenthesis , open parenthesis 8 , 0 close parenthesis ,")[$( 0 , −2 ) , ( 8 , 0 ) ,$] and #math.equation(block: false, alt: "open parenthesis 20 , 3 close parenthesis")[$( 20 , 3 )$] are three solutions to the equation #math.equation(block: false, alt: "x minus 4 y equals 8 .")[$x − 4 y = 8 .$] #figure(table( columns: 3, align: left, inset: 6pt, [#math.equation(block: false, alt: "x minus 4 y equals 8")[$x − 4 y = 8$]], [], [], [#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "−2")[$−2$]], [#math.equation(block: false, alt: "open parenthesis 0 , −2 close parenthesis")[$( 0 , −2 )$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "open parenthesis 8 , 0 close parenthesis")[$( 8 , 0 )$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "open parenthesis 20 , 3 close parenthesis")[$( 20 , 3 )$]], )) Remember, there are an infinite number of solutions to each linear equation. Any point you find is a solution if it makes the equation true. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find three solutions to the equation: #math.equation(block: false, alt: "4 x plus y equals 8 .")[$4 x + y = 8 .$] #solutionbox[ Answers will vary. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find three solutions to the equation: #math.equation(block: false, alt: "x plus 5 y equals 10 .")[$x + 5 y = 10 .$] #solutionbox[ Answers will vary. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[ACCESS ADDITIONAL ONLINE RESOURCES] - #link("https://openstaxcollege.org/l/24plotpoints")[Plotting Points] - #link("https://openstaxcollege.org/l/24quadrants")[Identifying Quadrants] - #link("https://openstaxcollege.org/l/24verlineq")[Verifying Solution to Linear Equation] ] === Key Concepts - #emph[Sign Patterns of the Quadrants]#figure(table( columns: 4, align: left, inset: 6pt, table.header([Quadrant I], [Quadrant II], [Quadrant III], [Quadrant IV]), [(#emph[x],#emph[y])], [(#emph[x],#emph[y])], [(#emph[x],#emph[y])], [(#emph[x],#emph[y])], [(+,+)], [(−,+)], [(−,−)], [(+,−)], )) - #emph[Coordinates of Zero]- Points with a #emph[y-]coordinate equal to 0 are on the #emph[x-]axis, and have coordinates ( #emph[a], 0). - Points with a #emph[x-]coordinate equal to 0 are on the #emph[y-]axis, and have coordinates ( 0, #emph[b]). - The point (0, 0) is called the origin. It is the point where the #emph[x-]axis and #emph[y-]axis intersect. ==== Practice Makes Perfect #strong[Plot Points on a Rectangular Coordinate System] In the following exercises, plot each point on a coordinate grid. #math.equation(block: true, alt: "open parenthesis 3 , 2 close parenthesis")[$( 3 , 2 )$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair 3, 2” is labeled], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair 3, 2” is labeled", caption: none) ] #math.equation(block: true, alt: "open parenthesis 4 , 1 close parenthesis")[$( 4 , 1 )$] #math.equation(block: true, alt: "open parenthesis 1 , 5 close parenthesis")[$( 1 , 5 )$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair 1, 5” is labeled], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair 1, 5” is labeled", caption: none) ] #math.equation(block: true, alt: "open parenthesis 3 , 4 close parenthesis")[$( 3 , 4 )$] #math.equation(block: true, alt: "open parenthesis 4 , 1 close parenthesis , open parenthesis 1 , 4 close parenthesis")[$( 4 , 1 ) , ( 1 , 4 )$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair 1, 4” is labeled. The point “ordered pair 4, 1” is labeled.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair 1, 4” is labeled. The point “ordered pair 4, 1” is labeled.", caption: none) ] #math.equation(block: true, alt: "open parenthesis 3 , 2 close parenthesis , open parenthesis 2 , 3 close parenthesis")[$( 3 , 2 ) , ( 2 , 3 )$] #math.equation(block: true, alt: "open parenthesis 3 , 4 close parenthesis , open parenthesis 4 , 3 close parenthesis")[$( 3 , 4 ) , ( 4 , 3 )$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair 3, 4” is labeled. The point “ordered pair 4, 3” is labeled.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair 3, 4” is labeled. The point “ordered pair 4, 3” is labeled.", caption: none) ] In the following exercises, plot each point on a coordinate grid and identify the quadrant in which the point is located. + ⓐ #math.equation(block: false, alt: "open parenthesis −4 , 2 close parenthesis")[$#h(0.2em) ( −4 , 2 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis −1 , −2 close parenthesis")[$#h(0.2em) ( −1 , −2 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 3 , −5 close parenthesis")[$#h(0.2em) ( 3 , −5 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis 2 , the fraction 5 over 2 close parenthesis")[$#h(0.2em) ( 2 , frac(5, 2) )$] + ⓐ #math.equation(block: false, alt: "open parenthesis −2 , −3 close parenthesis")[$#h(0.2em) ( −2 , −3 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 3 , −3 close parenthesis")[$#h(0.2em) ( 3 , −3 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis −4 , 1 close parenthesis")[$#h(0.2em) ( −4 , 1 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis 1 , the fraction 3 over 2 close parenthesis")[$#h(0.2em) ( 1 , frac(3, 2) )$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The quadrants are labeled I, II, III, and IV. The point (-1, 1) is labeled a, the point (-2, -1) is labeled b. The point (1, -4) is labeled c, and the point (3, 7/2) is labeled d.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The quadrants are labeled I, II, III, and IV. The point (-1, 1) is labeled a, the point (-2, -1) is labeled b. The point (1, -4) is labeled c, and the point (3, 7/2) is labeled d.", caption: none) ] + ⓐ #math.equation(block: false, alt: "open parenthesis −1 , 1 close parenthesis")[$#h(0.2em) ( −1 , 1 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis −2 , −1 close parenthesis")[$#h(0.2em) ( −2 , −1 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 1 , −4 close parenthesis")[$#h(0.2em) ( 1 , −4 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis 3 , the fraction 7 over 2 close parenthesis")[$#h(0.2em) ( 3 , frac(7, 2) )$] In the following exercises, plot each point on a coordinate grid. + ⓐ #math.equation(block: false, alt: "open parenthesis 3 , −2 close parenthesis")[$#h(0.2em) ( 3 , −2 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis −3 , 2 close parenthesis")[$#h(0.2em) ( −3 , 2 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis −3 , −2 close parenthesis")[$#h(0.2em) ( −3 , −2 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis 3 , 2 close parenthesis")[$#h(0.2em) ( 3 , 2 )$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point (3, -2) is labeled a, the point (-3, 2) is labeled b. The point (-3, -2) is labeled c, and the point (3, 2) is labeled d.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point (3, -2) is labeled a, the point (-3, 2) is labeled b. The point (-3, -2) is labeled c, and the point (3, 2) is labeled d.", caption: none) ] + ⓐ #math.equation(block: false, alt: "open parenthesis 4 , −1 close parenthesis")[$#h(0.2em) ( 4 , −1 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis −4 , 1 close parenthesis")[$#h(0.2em) ( −4 , 1 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis −4 , −1 close parenthesis")[$#h(0.2em) ( −4 , −1 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis 4 , 1 close parenthesis")[$#h(0.2em) ( 4 , 1 )$] + ⓐ #math.equation(block: false, alt: "open parenthesis −2 , 0 close parenthesis")[$#h(0.2em) ( −2 , 0 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis −3 , 0 close parenthesis")[$#h(0.2em) ( −3 , 0 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 0 , 4 close parenthesis")[$#h(0.2em) ( 0 , 4 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$#h(0.2em) ( 0 , 2 )$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point (-2, 0) is labeled a, the point (-3, 0) is labeled b. The point (0, 4) is labeled c, and the point (0, 2) is labeled d.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point (-2, 0) is labeled a, the point (-3, 0) is labeled b. The point (0, 4) is labeled c, and the point (0, 2) is labeled d.", caption: none) ] #strong[Identify Points on a Graph] In the following exercises, name the ordered pair of each point shown. #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair -4, 1” is labeled “A”. The point “ordered pair -3, -4” is labeled “B”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair -4, 1” is labeled “A”. The point “ordered pair -3, -4” is labeled “B”.", caption: none) #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair 4, 3” is labeled “D”. The point “ordered pair 1, -3” is labeled “C”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair 4, 3” is labeled “D”. The point “ordered pair 1, -3” is labeled “C”.", caption: none) #solutionbox[ C(1, -3) D(4, 3) ] #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair -3, -2” is labeled “X”. The point “ordered pair 5, -1” is labeled “Y”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair -3, -2” is labeled “X”. The point “ordered pair 5, -1” is labeled “Y”.", caption: none) #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair -2, 4” is labeled “S”. The point “ordered pair -4, -2” is labeled “T”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair -2, 4” is labeled “S”. The point “ordered pair -4, -2” is labeled “T”.", caption: none) #solutionbox[ S(-2, 4) T(-4, -2) ] #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair -2, 0” is labeled “B”. The point “ordered pair 0, -2” is labeled “A”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair -2, 0” is labeled “B”. The point “ordered pair 0, -2” is labeled “A”.", caption: none) #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair -1, 0” is labeled “D”. The point “ordered pair 0, -1” is labeled “C”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair -1, 0” is labeled “D”. The point “ordered pair 0, -1” is labeled “C”.", caption: none) #solutionbox[ C(0, -1) D(-1, 0) ] #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair 3, 0” is labeled “T”. The point “ordered pair -4, 0” is labeled “S”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point “ordered pair 3, 0” is labeled “T”. The point “ordered pair -4, 0” is labeled “S”.", caption: none) #strong[Verify Solutions to an Equation in Two Variables] In the following exercises, determine which ordered pairs are solutions to the given equation. #math.equation(block: true, alt: "2 x plus y equals 6")[$2 x + y = 6$] + ⓐ #math.equation(block: false, alt: "open parenthesis 1 , 4 close parenthesis")[$#h(0.2em) ( 1 , 4 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis")[$#h(0.2em) ( 3 , 0 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 2 , 3 close parenthesis")[$#h(0.2em) ( 2 , 3 )$] #solutionbox[ ⓐ , ⓑ ] #math.equation(block: true, alt: "x plus 3 y equals 9")[$x + 3 y = 9$] + ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$#h(0.2em) ( 0 , 3 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 6 , 1 close parenthesis")[$#h(0.2em) ( 6 , 1 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis −3 , −3 close parenthesis")[$#h(0.2em) ( −3 , −3 )$] #math.equation(block: true, alt: "4 x minus 2 y equals 8")[$4 x − 2 y = 8$] + ⓐ #math.equation(block: false, alt: "open parenthesis 3 , 2 close parenthesis")[$#h(0.2em) ( 3 , 2 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 1 , 4 close parenthesis")[$#h(0.2em) ( 1 , 4 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 0 , −4 close parenthesis")[$#h(0.2em) ( 0 , −4 )$] #solutionbox[ ⓐ , ⓒ ] #math.equation(block: true, alt: "3 x minus 2 y equals 12")[$3 x − 2 y = 12$] + ⓐ #math.equation(block: false, alt: "open parenthesis 4 , 0 close parenthesis")[$#h(0.2em) ( 4 , 0 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 2 , −3 close parenthesis")[$#h(0.2em) ( 2 , −3 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 1 , 6 close parenthesis")[$#h(0.2em) ( 1 , 6 )$] #math.equation(block: true, alt: "y equals 4 x plus 3")[$y = 4 x + 3$] + ⓐ #math.equation(block: false, alt: "open parenthesis 4 , 3 close parenthesis")[$#h(0.2em) ( 4 , 3 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis −1 , −1 close parenthesis")[$#h(0.2em) ( −1 , −1 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis the fraction 1 over 2 , 5 close parenthesis")[$#h(0.2em) ( frac(1, 2) , 5 )$] #solutionbox[ ⓑ , ⓒ ] #math.equation(block: true, alt: "y equals 2 x minus 5")[$y = 2 x − 5$] + ⓐ #math.equation(block: false, alt: "open parenthesis 0 , −5 close parenthesis")[$#h(0.2em) ( 0 , −5 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 2 , 1 close parenthesis")[$#h(0.2em) ( 2 , 1 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis the fraction 1 over 2 , −4 close parenthesis")[$#h(0.2em) ( frac(1, 2) , −4 )$] #math.equation(block: true, alt: "y equals the fraction 1 over 2 x minus 1")[$y = frac(1, 2) x − 1$] + ⓐ #math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$#h(0.2em) ( 2 , 0 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis −6 , −4 close parenthesis")[$#h(0.2em) ( −6 , −4 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis −4 , −1 close parenthesis")[$#h(0.2em) ( −4 , −1 )$] #solutionbox[ ⓐ , ⓑ ] #math.equation(block: true, alt: "y equals the fraction 1 over 3 x plus 1")[$y = frac(1, 3) x + 1$] + ⓐ #math.equation(block: false, alt: "open parenthesis −3 , 0 close parenthesis")[$#h(0.2em) ( −3 , 0 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 9 , 4 close parenthesis")[$#h(0.2em) ( 9 , 4 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis −6 , −1 close parenthesis")[$#h(0.2em) ( −6 , −1 )$] #strong[Find Solutions to Linear Equations in Two Variables] In the following exercises, complete the table to find solutions to each linear equation. #math.equation(block: true, alt: "y equals 2 x minus 4")[$y = 2 x − 4$] #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]), [#math.equation(block: false, alt: "−1")[$−1$]], [], [], [#math.equation(block: false, alt: "0")[$0$]], [], [], [#math.equation(block: false, alt: "2")[$2$]], [], [], )) #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]), [#math.equation(block: false, alt: "−1")[$−1$]], [#math.equation(block: false, alt: "−6")[$−6$]], [#math.equation(block: false, alt: "open parenthesis −1 , −6 close parenthesis")[$( −1 , −6 )$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "−4")[$−4$]], [#math.equation(block: false, alt: "open parenthesis 0 , −4 close parenthesis")[$( 0 , −4 )$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$]], )) ] #math.equation(block: true, alt: "y equals 3 x minus 1")[$y = 3 x − 1$] #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]), [#math.equation(block: false, alt: "−1")[$−1$]], [], [], [#math.equation(block: false, alt: "0")[$0$]], [], [], [#math.equation(block: false, alt: "2")[$2$]], [], [], )) #math.equation(block: true, alt: "y equals minus x plus 5")[$y = − x + 5$] #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]), [#math.equation(block: false, alt: "−2")[$−2$]], [], [], [#math.equation(block: false, alt: "0")[$0$]], [], [], [#math.equation(block: false, alt: "3")[$3$]], [], [], )) #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]), [#math.equation(block: false, alt: "−2")[$−2$]], [#math.equation(block: false, alt: "7")[$7$]], [#math.equation(block: false, alt: "open parenthesis −2 , 7 close parenthesis")[$( −2 , 7 )$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis")[$( 0 , 5 )$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "open parenthesis 3 , 2 close parenthesis")[$( 3 , 2 )$]], )) ] #math.equation(block: true, alt: "y equals the fraction 1 over 3 x plus 1")[$y = frac(1, 3) x + 1$] #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]), [#math.equation(block: false, alt: "0")[$0$]], [], [], [#math.equation(block: false, alt: "3")[$3$]], [], [], [#math.equation(block: false, alt: "6")[$6$]], [], [], )) #math.equation(block: true, alt: "y equals minus the fraction 3 over 2 x minus 2")[$y = − frac(3, 2) x − 2$] #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]), [#math.equation(block: false, alt: "−2")[$−2$]], [], [], [#math.equation(block: false, alt: "0")[$0$]], [], [], [#math.equation(block: false, alt: "2")[$2$]], [], [], )) #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]), [#math.equation(block: false, alt: "−2")[$−2$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis")[$( −2 , 1 )$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "−2")[$−2$]], [#math.equation(block: false, alt: "open parenthesis 0 , −2 close parenthesis")[$( 0 , −2 )$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "−5")[$−5$]], [#math.equation(block: false, alt: "open parenthesis 2 , −5 close parenthesis")[$( 2 , −5 )$]], )) ] #math.equation(block: true, alt: "x plus 2 y equals 8")[$x + 2 y = 8$] #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]), [#math.equation(block: false, alt: "0")[$0$]], [], [], [#math.equation(block: false, alt: "4")[$4$]], [], [], [], [#math.equation(block: false, alt: "0")[$0$]], [], )) ==== Everyday Math #strong[Weight of a baby] Mackenzie recorded her baby’s weight every two months. The baby’s age, in months, and weight, in pounds, are listed in the table, and shown as an ordered pair in the third column. ⓐ Plot the points on a coordinate grid. #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "Age")[$"Age"$]], [#math.equation(block: false, alt: "Weight")[$"Weight"$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]), [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "7")[$7$]], [#math.equation(block: false, alt: "open parenthesis 0 , 7 close parenthesis")[$( 0 , 7 )$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "11")[$11$]], [#math.equation(block: false, alt: "open parenthesis 2 , 11 close parenthesis")[$( 2 , 11 )$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "15")[$15$]], [#math.equation(block: false, alt: "open parenthesis 4 , 15 close parenthesis")[$( 4 , 15 )$]], [#math.equation(block: false, alt: "6")[$6$]], [#math.equation(block: false, alt: "16")[$16$]], [#math.equation(block: false, alt: "open parenthesis 6 , 16 close parenthesis")[$( 6 , 16 )$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "19")[$19$]], [#math.equation(block: false, alt: "open parenthesis 8 , 19 close parenthesis")[$( 8 , 19 )$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "open parenthesis 10 , 20 close parenthesis")[$( 10 , 20 )$]], [#math.equation(block: false, alt: "12")[$12$]], [#math.equation(block: false, alt: "21")[$21$]], [#math.equation(block: false, alt: "open parenthesis 12 , 21 close parenthesis")[$( 12 , 21 )$]], )) ⓑ Why is only Quadrant I needed? #solutionbox[ + ⓐ #figure(figph[This figure shows points plot on the x y coordinate plane. There are 7 points graphed without labeled at approximately the points “ordered pair 0, 7”, “ordered pair 2, 11”, “ordered pair 4, 15”, “ordered pair 6, 16”, “ordered pair 8, 19”, “ordered pair 10, 20”, “ordered pair 12, 21”.], alt: "This figure shows points plot on the x y coordinate plane. There are 7 points graphed without labeled at approximately the points “ordered pair 0, 7”, “ordered pair 2, 11”, “ordered pair 4, 15”, “ordered pair 6, 16”, “ordered pair 8, 19”, “ordered pair 10, 20”, “ordered pair 12, 21”.", caption: none) + ⓑ Age and weight are only positive. ] #strong[Weight of a child] Latresha recorded her son’s height and weight every year. His height, in inches, and weight, in pounds, are listed in the table, and shown as an ordered pair in the third column. ⓐ Plot the points on a coordinate grid. #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "Height; x")[$"Height" \ x$]], [#math.equation(block: false, alt: "Weight; y")[$"Weight" \ y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]), [#math.equation(block: false, alt: "28")[$28$]], [#math.equation(block: false, alt: "22")[$22$]], [#math.equation(block: false, alt: "open parenthesis 28 , 22 close parenthesis")[$( 28 , 22 )$]], [#math.equation(block: false, alt: "31")[$31$]], [#math.equation(block: false, alt: "27")[$27$]], [#math.equation(block: false, alt: "open parenthesis 31 , 27 close parenthesis")[$( 31 , 27 )$]], [#math.equation(block: false, alt: "33")[$33$]], [#math.equation(block: false, alt: "33")[$33$]], [#math.equation(block: false, alt: "open parenthesis 33 , 33 close parenthesis")[$( 33 , 33 )$]], [#math.equation(block: false, alt: "37")[$37$]], [#math.equation(block: false, alt: "35")[$35$]], [#math.equation(block: false, alt: "open parenthesis 37 , 35 close parenthesis")[$( 37 , 35 )$]], [#math.equation(block: false, alt: "40")[$40$]], [#math.equation(block: false, alt: "41")[$41$]], [#math.equation(block: false, alt: "open parenthesis 40 , 41 close parenthesis")[$( 40 , 41 )$]], [#math.equation(block: false, alt: "42")[$42$]], [#math.equation(block: false, alt: "45")[$45$]], [#math.equation(block: false, alt: "open parenthesis 42 , 45 close parenthesis")[$( 42 , 45 )$]], )) ⓑ Why is only Quadrant I needed? ==== Writing Exercises Have you ever used a map with a rectangular coordinate system? Describe the map and how you used it. #solutionbox[ Answers may vary. ] How do you determine if an ordered pair is a solution to a given equation? ==== Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. #figure(figph[An empty self-assessment checklist for students to rate their confidence in math skills like plotting points, identifying points on graphs, and solving linear equations with categories: Confidently, With some help, No-I don't get it!], alt: "An empty self-assessment checklist for students to rate their confidence in math skills like plotting points, identifying points on graphs, and solving linear equations with categories: Confidently, With some help, No-I don't get it!", caption: none) ⓑ If most of your checks were: …confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific. …with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved? …no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.