#set document(title: "11.3 Graphing Linear Equations", 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")) == 11.3#h(0.6em)Graphing Linear Equations #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Before you get started, take this readiness quiz. Evaluate: #math.equation(block: false, alt: "3 x plus 2")[$3 x + 2$] when #math.equation(block: false, alt: "x equals −1 .")[$x = −1 .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "minus 1")[$− 1$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the formula: #math.equation(block: false, alt: "5 x plus 2 y equals 20")[$5 x + 2 y = 20$] for #math.equation(block: false, alt: "y .")[$y .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 20 minus 5 x over 2")[$y = frac(20 − 5 x, 2)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 3 over 8 open parenthesis −24 close parenthesis .")[$frac(3, 8) ( −24 ) "."$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "minus 9")[$− 9$] ] ] === Recognize the Relation Between the Solutions of an Equation and its Graph In Use the Rectangular Coordinate System, we found a few solutions to the equation #math.equation(block: false, alt: "3 x plus 2 y equals 6")[$3 x + 2 y = 6$]. They are listed in the table below. So, the ordered pairs #math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$( 0 , 3 )$], #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 , minus 3 close parenthesis")[$( 4 , − 3 )$], are some solutions to the equation#math.equation(block: false, alt: "3 x plus 2 y equals 6")[$3 x + 2 y = 6$]. We can plot these solutions in the rectangular coordinate system as shown on the graph at right. #figure(figph[The image shows a table breaking up points by their x components and y components. These points are then mapped onto a graph. Points include (0, 3), (2, 0), (1, 3/2), and (4, –3).], alt: "The image shows a table breaking up points by their x components and y components. These points are then mapped onto a graph. Points include (0, 3), (2, 0), (1, 3/2), and (4, –3).", caption: none) Notice how the points line up perfectly? We connect the points with a straight line to get the graph of the equation #math.equation(block: false, alt: "3 x plus 2 y equals 6")[$3 x + 2 y = 6$]. Notice the arrows on the ends of each side of the line. These arrows indicate the line continues. #figure(figph[A graph displays the linear equation 3x + 2y = 6. The line passes through points such as (0, 3), (2, 0), and (3, -1.5), demonstrating its downward slope on a Cartesian coordinate system.], alt: "A graph displays the linear equation 3x + 2y = 6. The line passes through points such as (0, 3), (2, 0), and (3, -1.5), demonstrating its downward slope on a Cartesian coordinate system.", caption: none) Every point on the line is a solution of the equation. Also, every solution of this equation is a point on this line. Points not on the line are #emph[not] solutions! Notice that the point whose coordinates are #math.equation(block: false, alt: "open parenthesis minus 2 , 6 close parenthesis")[$( − 2 , 6 )$] is on the line shown. If you substitute #math.equation(block: false, alt: "x equals minus 2")[$x = − 2$] and #math.equation(block: false, alt: "y equals 6")[$y = 6$] into the equation, you find that it is a solution to the equation. #figure(figph[The image shows the process of testing point (–2, 6) as a solution to the given equation. Graphing it out, the line properly intercepts this point but does not intercept point (4, 1) which was also tested but failed with the equation.], alt: "The image shows the process of testing point (–2, 6) as a solution to the given equation. Graphing it out, the line properly intercepts this point but does not intercept point (4, 1) which was also tested but failed with the equation.", caption: none) So #math.equation(block: false, alt: "open parenthesis 4 , 1 close parenthesis")[$( 4 , 1 )$] is not a solution to the equation #math.equation(block: false, alt: "3 x plus 2 y equals 6")[$3 x + 2 y = 6$] . Therefore the point #math.equation(block: false, alt: "open parenthesis 4 , 1 close parenthesis")[$( 4 , 1 )$] is not on the line. This is an example of the saying,” A picture is worth a thousand words.” The line shows you all the solutions to the equation. Every point on the line is a solution of the equation. And, every solution of this equation is on this line. This line is called the #emph[graph] of the equation #math.equation(block: false, alt: "3 x plus 2 y equals 6")[$3 x + 2 y = 6$]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Graph of a Linear Equation] #math.equation(block: true, alt: "A x plus B y equals C")[$A x + B y = C$] - Every point on the line is a solution of the equation. - Every solution of this equation is a point on this line. ] #examplebox("Example 1")[][ The graph of #math.equation(block: false, alt: "y equals 2 x minus 3")[$y = 2 x − 3$] is shown below. #figure(figph[Graph of the linear function y = 2x - 3, illustrating its positive slope and y-intercept at -3.], alt: "Graph of the linear function y = 2x - 3, illustrating its positive slope and y-intercept at -3.", caption: none) For each ordered pair decide + ⓐ Is the ordered pair a solution to the equation? + ⓑ Is the point on the line? + (a) #math.equation(block: false, alt: "open parenthesis 0 , –3 close parenthesis")[$( 0 , –3 )$] + (b) #math.equation(block: false, alt: "open parenthesis 3 , 3 close parenthesis")[$( 3 , 3 )$] + (c) #math.equation(block: false, alt: "open parenthesis 2 , minus 3 close parenthesis")[$( 2 , − 3 )$] + (d) #math.equation(block: false, alt: "open parenthesis minus 1 , minus 5 close parenthesis")[$( − 1 , − 5 )$] #solutionbox[ Substitute the #math.equation(block: false, alt: "x")[$x$]- and #math.equation(block: false, alt: "y")[$y$]-values into the equation to check if the ordered pair is a solution to the equation. ⓐ #linebreak() #figure(figph[This image shows a series of ordered pairs tested with the equation to determine if said pair is a solution to the equation.], alt: "This image shows a series of ordered pairs tested with the equation to determine if said pair is a solution to the equation.", caption: none) ⓑ Plot the points A: #math.equation(block: false, alt: "open parenthesis 0 , minus 3 close parenthesis")[$( 0 , − 3 )$] B: #math.equation(block: false, alt: "open parenthesis 3 , 3 close parenthesis")[$( 3 , 3 )$] C: #math.equation(block: false, alt: "open parenthesis 2 , minus 3 close parenthesis")[$( 2 , − 3 )$] and D: #math.equation(block: false, alt: "open parenthesis minus 1 , minus 5 close parenthesis")[$( − 1 , − 5 )$]. #linebreak() The points #math.equation(block: false, alt: "open parenthesis 0 , minus 3 close parenthesis")[$( 0 , − 3 )$], #math.equation(block: false, alt: "open parenthesis 3 , 3 close parenthesis")[$( 3 , 3 )$], and #math.equation(block: false, alt: "open parenthesis minus 1 , minus 5 close parenthesis")[$( − 1 , − 5 )$] are on the line #math.equation(block: false, alt: "y equals 2 x minus 3")[$y = 2 x − 3$], and the point #math.equation(block: false, alt: "open parenthesis 2 , minus 3 close parenthesis")[$( 2 , − 3 )$] is not on the line. #figure(figph[A graph in which a line runs through points (–1, –5), (0, –3), and (3, 3). The point (2, –3) is also plotted on the graph but does not intercept the line.], alt: "A graph in which a line runs through points (–1, –5), (0, –3), and (3, 3). The point (2, –3) is also plotted on the graph but does not intercept the line.", caption: none) The points which are solutions to #math.equation(block: false, alt: "y equals 2 x minus 3")[$y = 2 x − 3$] are on the line, but the point which is not a solution is not on the line. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The graph of #math.equation(block: false, alt: "y equals 3 x minus 1")[$y = 3 x − 1$] is shown. For each ordered pair, decide + ⓐ is the ordered pair a solution to the equation? + ⓑ is the point on the line? #figure(figph[A two-dimensional graph displays the linear function y = 3x - 1. The line ascends from left to right, intersecting the y-axis at -1.], alt: "A two-dimensional graph displays the linear function y = 3x - 1. The line ascends from left to right, intersecting the y-axis at -1.", caption: none) + #math.equation(block: false, alt: "open parenthesis 0 , minus 1 close parenthesis")[$( 0 , − 1 )$] + #math.equation(block: false, alt: "open parenthesis 2 , 2 close parenthesis")[$( 2 , 2 )$] + #math.equation(block: false, alt: "open parenthesis 3 , minus 1 close parenthesis")[$( 3 , − 1 )$] + #math.equation(block: false, alt: "open parenthesis minus 1 , minus 4 close parenthesis")[$( − 1 , − 4 )$] #solutionbox[ + ⓐ yes ⓑ yes + ⓐ no ⓑ no + ⓐ no ⓑ no + ⓐ yes ⓑ yes ] ] === Graph a Linear Equation by Plotting Points There are several methods that can be used to graph a linear equation. The method we used at the start of this section to graph is called plotting points, or the #strong[Point-Plotting Method]. Let’s graph the equation #math.equation(block: false, alt: "y equals 2 x plus 1")[$y = 2 x + 1$] by plotting points. We start by finding three points that are solutions to the equation. We can choose any value for #math.equation(block: false, alt: "x")[$x$] or #math.equation(block: false, alt: "y ,")[$y ,$] and then solve for the other variable. Since #math.equation(block: false, alt: "y")[$y$] is isolated on the left side of the equation, it is easier to choose values for #math.equation(block: false, alt: "x .")[$x .$] We will use #math.equation(block: false, alt: "0 , 1 ,")[$0 , 1 ,$] and #math.equation(block: false, alt: "-2")[$-2$] for #math.equation(block: false, alt: "x")[$x$] for this example. We substitute each value of #math.equation(block: false, alt: "x")[$x$] into the equation and solve for #math.equation(block: false, alt: "y .")[$y .$] #figure(figph[The figure shows three algebraic substitutions into an equation. The first substitution is for x = -2, with -2 shown in blue. The next line is y = 2 x + 1. The next line is y = 2 open parentheses -2, shown in blue, closed parentheses, + 1. The next line is y = - 4 + 1. The next line is y = -3. The last line is “ordered pair -2, -3”. The second substitution is for x = 0, with 0 shown in blue. The next line is y = 2 x + 1. The next line is y = 2 open parentheses 0, shown in blue, closed parentheses, + 1. The next line is y = 0 + 1. The next line is y = 1. The last line is “ordered pair 0, 2”. The third substitution is for x = 1, with 1 shown in blue. The next line is y = 2 x + 1. The next line is y = 2 open parentheses 1, shown in blue, closed parentheses, + 1. The next line is y = 2 + 1. The next line is y = 3. The last line is “ordered pair -1, 3”.], alt: "The figure shows three algebraic substitutions into an equation. The first substitution is for x = -2, with -2 shown in blue. The next line is y = 2 x + 1. The next line is y = 2 open parentheses -2, shown in blue, closed parentheses, + 1. The next line is y = - 4 + 1. The next line is y = -3. The last line is “ordered pair -2, -3”. The second substitution is for x = 0, with 0 shown in blue. The next line is y = 2 x + 1. The next line is y = 2 open parentheses 0, shown in blue, closed parentheses, + 1. The next line is y = 0 + 1. The next line is y = 1. The last line is “ordered pair 0, 2”. The third substitution is for x = 1, with 1 shown in blue. The next line is y = 2 x + 1. The next line is y = 2 open parentheses 1, shown in blue, closed parentheses, + 1. The next line is y = 2 + 1. The next line is y = 3. The last line is “ordered pair -1, 3”.", caption: none) We can organize the solutions in a table. #figure(table( columns: 3, align: left, inset: 6pt, [#math.equation(block: false, alt: "y equals 2 x plus 1")[$y = 2 x + 1$]], [], [], [#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: "3")[$3$]], [#math.equation(block: false, alt: "open parenthesis 1 , 3 close parenthesis")[$( 1 , 3 )$]], [#math.equation(block: false, alt: "−2")[$−2$]], [#math.equation(block: false, alt: "−3")[$−3$]], [#math.equation(block: false, alt: "open parenthesis −2 , −3 close parenthesis")[$( −2 , −3 )$]], )) Now we plot the points on a rectangular coordinate system. Check that the points line up. If they did not line up, it would mean we made a mistake and should double-check all our work. #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. Three labeled points are shown, “ordered pair -2, -3”, “ordered pair 0, 1”, and ordered pair 1, 3”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. Three labeled points are shown, “ordered pair -2, -3”, “ordered pair 0, 1”, and ordered pair 1, 3”.", caption: none) Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line. The line is the graph of #math.equation(block: false, alt: "y equals 2 x plus 1 .")[$y = 2 x + 1 .$] #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through three labeled points, “ordered pair -2, -3”, “ordered pair 0, 1”, and ordered pair 1, 3”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through three labeled points, “ordered pair -2, -3”, “ordered pair 0, 1”, and ordered pair 1, 3”.", caption: none) #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Graph a linear equation by plotting points.] + Find three points whose coordinates are solutions to the equation. Organize them in a table. + Plot the points on a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work. + Draw the line through the points. Extend the line to fill the grid and put arrows on both ends of the line. ] It is true that it only takes two points to determine a line, but it is a good habit to use three points. If you plot only two points and one of them is incorrect, you can still draw a line but it will not represent the solutions to the equation. It will be the wrong line. If you use three points, and one is incorrect, the points will not line up. This tells you something is wrong and you need to check your work. #figure(figph[There are two figures. Figure a shows three points that are all contained on a straight line. There is a line with arrows that passed through the three points. Figure b shows 3 points that are not all arranged in a straight line. ], alt: "There are two figures. Figure a shows three points that are all contained on a straight line. There is a line with arrows that passed through the three points. Figure b shows 3 points that are not all arranged in a straight line.", caption: [Look at the difference between (a) and (b). All three points in (a) line up so we can draw one line through them. The three points in (b) do not line up. We cannot draw a single straight line through all three points.]) #examplebox("Example 2")[][ Graph the equation #math.equation(block: false, alt: "y equals −3 x .")[$y = −3 x .$] #solutionbox[ Find three points that are solutions to the equation. It’s easier to choose values for #math.equation(block: false, alt: "x ,")[$x ,$] and solve for #math.equation(block: false, alt: "y .")[$y .$] Do you see why? #figure(figph[The figure shows three algebraic substitutions into an equation. The first substitution is for x = 0, with 0 shown in blue. The next line is y = -3 x. The next line is y = -3 open parentheses 0, shown in blue, closed parentheses. The next line is y = 0. The last line is “ordered pair 0, 0 “. The second substitution is for x = 1, with 0 shown in blue. The next line is y = -3 x. The next line is y = -3 open parentheses 1, shown in blue, closed parentheses. The next line is y = -3. The last line is “ordered pair 1, -3”. The third substitution is for x = -2, with -2 shown in blue. The next line is y = -3 x. The next line is y = -3 open parentheses -2, shown in blue, closed parentheses. The next line is y = 6. The last line is “ordered pair -2, 6 “.], alt: "The figure shows three algebraic substitutions into an equation. The first substitution is for x = 0, with 0 shown in blue. The next line is y = -3 x. The next line is y = -3 open parentheses 0, shown in blue, closed parentheses. The next line is y = 0. The last line is “ordered pair 0, 0 “. The second substitution is for x = 1, with 0 shown in blue. The next line is y = -3 x. The next line is y = -3 open parentheses 1, shown in blue, closed parentheses. The next line is y = -3. The last line is “ordered pair 1, -3”. The third substitution is for x = -2, with -2 shown in blue. The next line is y = -3 x. The next line is y = -3 open parentheses -2, shown in blue, closed parentheses. The next line is y = 6. The last line is “ordered pair -2, 6 “.", caption: none) List the points in a table. #figure(table( columns: 3, align: left, inset: 6pt, [#math.equation(block: false, alt: "y equals −3 x")[$y = −3 x$]], [], [], [#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: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "open parenthesis 1 , −3 close parenthesis")[$( 1 , −3 )$]], [#math.equation(block: false, alt: "−2")[$−2$]], [#math.equation(block: false, alt: "6")[$6$]], [#math.equation(block: false, alt: "open parenthesis −2 , 6 close parenthesis")[$( −2 , 6 )$]], )) Plot the points, check that they line up, and draw the line as shown. #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through three labeled points, “ordered pair -2, 6”, “ordered pair 0, 0”, and ordered pair 1, -3”. The line is labeled y = -3 x.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through three labeled points, “ordered pair -2, 6”, “ordered pair 0, 0”, and ordered pair 1, -3”. The line is labeled y = -3 x.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the equation by plotting points: #math.equation(block: false, alt: "y equals −4 x .")[$y = −4 x .$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 0” and “ordered pair 4, -4”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 0” and “ordered pair 4, -4”.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the equation by plotting points: #math.equation(block: false, alt: "y equals x .")[$y = x .$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 0” and “ordered pair 1, -4”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 0” and “ordered pair 1, -4”.", caption: none) ] ] When an equation includes a fraction as the coefficient of #math.equation(block: false, alt: "x ,")[$x ,$] we can substitute any numbers for #math.equation(block: false, alt: "x .")[$x .$] But the math is easier if we make ‘good’ choices for the values of #math.equation(block: false, alt: "x .")[$x .$] This way we will avoid fraction answers, which are hard to graph precisely. #examplebox("Example 3")[][ Graph the equation #math.equation(block: false, alt: "y equals the fraction 1 over 2 x plus 3 .")[$y = frac(1, 2) x + 3 .$] #solutionbox[ Find three points that are solutions to the equation. Since this equation has the fraction #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] as a coefficient of #math.equation(block: false, alt: "x ,")[$x ,$] we will choose values of #math.equation(block: false, alt: "x")[$x$] carefully. We will use zero as one choice and multiples of #math.equation(block: false, alt: "2")[$2$] for the other choices. #figure(figph[The figure shows three algebraic substitutions into an equation. The first substitution is for x = 0, with 0 shown in blue. The next line is y = 1 over 2 x + 3. The next line is y = 1 over 2 open parentheses 0, shown in blue, closed parentheses, + 3. The next line is y = 3. The last line is “ordered pair 0, 3”. The second substitution is for x = 2, with 2 shown in blue. The next line is y = 1 over 2 x + 3. The next line is y = 1 over 2 open parentheses 2, shown in blue, closed parentheses, + 3. The next line is y = 4. The last line is “ordered pair 2, 4”. The third substitution is for x = 4, with 4 shown in blue. The next line is y = 1 over 2 x + 3. The next line is y = 1 over 2 open parentheses 4, shown in blue, closed parentheses, + 3. The next line is y = 5. The last line is “ordered pair 4, 5”.], alt: "The figure shows three algebraic substitutions into an equation. The first substitution is for x = 0, with 0 shown in blue. The next line is y = 1 over 2 x + 3. The next line is y = 1 over 2 open parentheses 0, shown in blue, closed parentheses, + 3. The next line is y = 3. The last line is “ordered pair 0, 3”. The second substitution is for x = 2, with 2 shown in blue. The next line is y = 1 over 2 x + 3. The next line is y = 1 over 2 open parentheses 2, shown in blue, closed parentheses, + 3. The next line is y = 4. The last line is “ordered pair 2, 4”. The third substitution is for x = 4, with 4 shown in blue. The next line is y = 1 over 2 x + 3. The next line is y = 1 over 2 open parentheses 4, shown in blue, closed parentheses, + 3. The next line is y = 5. The last line is “ordered pair 4, 5”.", caption: none) The points are shown in the table. #figure(table( columns: 3, align: left, inset: 6pt, [#math.equation(block: false, alt: "y equals the fraction 1 over 2 x plus 3")[$y = frac(1, 2) x + 3$]], [], [], [#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: "4")[$4$]], [#math.equation(block: false, alt: "open parenthesis 2 , 4 close parenthesis")[$( 2 , 4 )$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "open parenthesis 4 , 5 close parenthesis")[$( 4 , 5 )$]], )) Plot the points, check that they line up, and draw the line as shown. #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through three labeled points, “ordered pair 0, 3”, “ordered pair 2, 4”, and ordered pair 4, 5”. The line is labeled y = 1 over 2 x + 3.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through three labeled points, “ordered pair 0, 3”, “ordered pair 2, 4”, and ordered pair 4, 5”. The line is labeled y = 1 over 2 x + 3.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the equation: #math.equation(block: false, alt: "y equals the fraction 1 over 3 x minus 1 .")[$y = frac(1, 3) x − 1 .$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, -1” and “ordered pair 3, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, -1” and “ordered pair 3, 0”.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the equation: #math.equation(block: false, alt: "y equals the fraction 1 over 4 x plus 2 .")[$y = frac(1, 4) x + 2 .$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 2” and “ordered pair -12, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 2” and “ordered pair -12, 0”.", caption: none) ] ] So far, all the equations we graphed had #math.equation(block: false, alt: "y")[$y$] given in terms of #math.equation(block: false, alt: "x .")[$x .$] Now we’ll graph an equation with #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$] on the same side. #examplebox("Example 4")[][ Graph the equation #math.equation(block: false, alt: "x plus y equals 5 .")[$x + y = 5 .$] #solutionbox[ Find three points that are solutions to the equation. Remember, you can start with #emph[any] value of #math.equation(block: false, alt: "x")[$x$] or #math.equation(block: false, alt: "y .")[$y .$] #figure(figph[The figure shows three algebraic substitutions into an equation. The first substitution is for x = 0, with 0 shown in blue. The next line is x + y = 5. The next line is 0, shown in blue + y = 5. The next line is y = 5. The last line is “ordered pair 0, 5”. The second substitution is for x = 1, with 1 shown in blue. The next line is x + y = 5. The next line is 1, shown in blue + y = 5. The next line is y = 4. The last line is “ordered pair 1, 4”. The third substitution is for x = 4, with 4 shown in blue. The next line is x + y = 5. The next line is 4, shown in blue + y = 5. The next line is y = 1. The last line is “ordered pair 4, 1”.], alt: "The figure shows three algebraic substitutions into an equation. The first substitution is for x = 0, with 0 shown in blue. The next line is x + y = 5. The next line is 0, shown in blue + y = 5. The next line is y = 5. The last line is “ordered pair 0, 5”. The second substitution is for x = 1, with 1 shown in blue. The next line is x + y = 5. The next line is 1, shown in blue + y = 5. The next line is y = 4. The last line is “ordered pair 1, 4”. The third substitution is for x = 4, with 4 shown in blue. The next line is x + y = 5. The next line is 4, shown in blue + y = 5. The next line is y = 1. The last line is “ordered pair 4, 1”.", caption: none) We list the points in a table. #figure(table( columns: 3, align: left, inset: 6pt, [#math.equation(block: false, alt: "x plus y equals 5")[$x + y = 5$]], [], [], [#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: "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: "4")[$4$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "open parenthesis 4 , 1 close parenthesis")[$( 4 , 1 )$]], )) Then plot the points, check that they line up, and draw the line. #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through three labeled points, “ordered pair 0, 5”, “ordered pair 1, 4”, and ordered pair 4, 1”. The line is labeled x + y = 5.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through three labeled points, “ordered pair 0, 5”, “ordered pair 1, 4”, and ordered pair 4, 1”. The line is labeled x + y = 5.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the equation: #math.equation(block: false, alt: "x plus y equals −2 .")[$x + y = −2 .$] #solutionbox[ #figure(figph[This answer graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. The equation x plus y equals -2 is shown. A line passes through the intercepts with coordinates 0, –2 and –2, 0.], alt: "This answer graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. The equation x plus y equals -2 is shown. A line passes through the intercepts with coordinates 0, –2 and –2, 0.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the equation: #math.equation(block: false, alt: "x minus y equals 6 .")[$x − y = 6 .$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points “ordered pair 6, 0” and “ordered pair 0, -6”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points “ordered pair 6, 0” and “ordered pair 0, -6”.", caption: none) ] ] In the previous example, the three points we found were easy to graph. But this is not always the case. Let’s see what happens in the equation #math.equation(block: false, alt: "2 x plus y equals 3 .")[$2 x + y = 3 .$] If #math.equation(block: false, alt: "y")[$y$] is #math.equation(block: false, alt: "0 ,")[$0 ,$] what is the value of #math.equation(block: false, alt: "x ?")[$x ?$] #figure(figph[This figure shows an algebraic substitution. The first line is 2 x + y = 3. The second line is 2 x + 0, with 0 shown in red. The third line is 2 x = 3. The last line is x = 3 over 2.], alt: "This figure shows an algebraic substitution. The first line is 2 x + y = 3. The second line is 2 x + 0, with 0 shown in red. The third line is 2 x = 3. The last line is x = 3 over 2.", caption: none) The solution is the point #math.equation(block: false, alt: "open parenthesis the fraction 3 over 2 , 0 close parenthesis .")[$( frac(3, 2) , 0 ) .$] This point has a fraction for the #math.equation(block: false, alt: "x")[$x$]-coordinate. While we could graph this point, it is hard to be precise graphing fractions. Remember in the example #math.equation(block: false, alt: "y equals the fraction 1 over 2 x plus 3 ,")[$y = frac(1, 2) x + 3 ,$] we carefully chose values for #math.equation(block: false, alt: "x")[$x$] so as not to graph fractions at all. If we solve the equation #math.equation(block: false, alt: "2 x plus y equals 3")[$2 x + y = 3$] for #math.equation(block: false, alt: "y ,")[$y ,$] it will be easier to find three solutions to the equation. #math.equation(block: true, alt: "2 x plus y equals 3")[$2 x + y = 3$]#math.equation(block: true, alt: "y equals −2 x plus 3")[$y = −2 x + 3$]Now we can choose values for #math.equation(block: false, alt: "x")[$x$] that will give coordinates that are integers. The solutions for #math.equation(block: false, alt: "x equals 0 , x equals 1 ,")[$x = 0 , x = 1 ,$] and #math.equation(block: false, alt: "x equals −1")[$x = −1$] are shown. #figure(table( columns: 3, align: left, inset: 6pt, [#math.equation(block: false, alt: "y equals −2 x plus 3")[$y = −2 x + 3$]], [], [], [#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: "1")[$1$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "open parenthesis 1 , 1 close parenthesis")[$( 1 , 1 )$]], [#math.equation(block: false, alt: "−1")[$−1$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "open parenthesis minus 1 , 5 close parenthesis")[$( − 1 , 5 )$]], )) #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through three labeled points, “ordered pair -1, 5”, “ordered pair 0, 3”, and ordered pair 1, 1”. The line is labeled 2 x + y = 3.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through three labeled points, “ordered pair -1, 5”, “ordered pair 0, 3”, and ordered pair 1, 1”. The line is labeled 2 x + y = 3.", caption: none) #examplebox("Example 5")[][ Graph the equation #math.equation(block: false, alt: "3 x plus y equals −1 .")[$3 x + y = −1 .$] #solutionbox[ Find three points that are solutions to the equation. First, solve the equation for #math.equation(block: false, alt: "y .")[$y .$] #math.equation(block: true, alt: "3 x plus y, equals, −1; y, equals, −3 x minus 1")[$3 x + y & = & −1 \ y & = & −3 x − 1$] We’ll let #math.equation(block: false, alt: "x")[$x$] be #math.equation(block: false, alt: "0 , 1 ,")[$0 , 1 ,$] and #math.equation(block: false, alt: "−1")[$−1$] to find three points. The ordered pairs are shown in the table. Plot the points, check that they line up, and draw the line. #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: "−1")[$−1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "open parenthesis −1 , 2 close parenthesis")[$( −1 , 2 )$]], )) #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through three labeled points, “ordered pair -1, 2”, “ordered pair 0, -1”, and ordered pair 1, -4”. The line is labeled 3 x + y = -1.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through three labeled points, “ordered pair -1, 2”, “ordered pair 0, -1”, and ordered pair 1, -4”. The line is labeled 3 x + y = -1.", caption: none) If you can choose any three points to graph a line, how will you know if your graph matches the one shown in the answers in the book? If the points where the graphs cross the #math.equation(block: false, alt: "x -")[$x "-"$] and #math.equation(block: false, alt: "y")[$y$]-axes are the same, the graphs match. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph each equation: #math.equation(block: false, alt: "2 x plus y equals 2 .")[$2 x + y = 2 .$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through three labeled points, “ordered pair -1, 2”, “ordered pair 0, -1”, and ordered pair 1, -4”. The line is labeled 3 x + y = -1.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through three labeled points, “ordered pair -1, 2”, “ordered pair 0, -1”, and ordered pair 1, -4”. The line is labeled 3 x + y = -1.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph each equation: #math.equation(block: false, alt: "4 x plus y equals −3 .")[$4 x + y = −3 .$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair -1, 4” and “ordered pair 0, -3”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair -1, 4” and “ordered pair 0, -3”.", caption: none) ] ] === Graph Vertical and Horizontal Lines Can we graph an equation with only one variable? Just #math.equation(block: false, alt: "x")[$x$] and no #math.equation(block: false, alt: "y ,")[$y ,$] or just #math.equation(block: false, alt: "y")[$y$] without an #math.equation(block: false, alt: "x ?")[$x ?$] How will we make a table of values to get the points to plot? Let’s consider the equation #math.equation(block: false, alt: "x equals −3 .")[$x = −3 .$] The equation says that #math.equation(block: false, alt: "x")[$x$] is always equal to #math.equation(block: false, alt: "−3 ,")[$−3 ,$] so its value does not depend on #math.equation(block: false, alt: "y .")[$y .$] No matter what #math.equation(block: false, alt: "y")[$y$] is, the value of #math.equation(block: false, alt: "x")[$x$] is always #math.equation(block: false, alt: "−3 .")[$−3 .$] To make a table of solutions, we write #math.equation(block: false, alt: "−3")[$−3$] for all the #math.equation(block: false, alt: "x")[$x$] values. Then choose any values for #math.equation(block: false, alt: "y .")[$y .$] Since #math.equation(block: false, alt: "x")[$x$] does not depend on #math.equation(block: false, alt: "y ,")[$y ,$] you can choose any numbers you like. But to fit the size of our coordinate graph, we’ll use #math.equation(block: false, alt: "1 , 2 ,")[$1 , 2 ,$] and #math.equation(block: false, alt: "3")[$3$] for the #math.equation(block: false, alt: "y")[$y$]-coordinates as shown in the table. #figure(table( columns: 3, align: left, inset: 6pt, [#math.equation(block: false, alt: "x equals −3")[$x = −3$]], [], [], [#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: "−3")[$−3$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "open parenthesis −3 , 1 close parenthesis")[$( −3 , 1 )$]], [#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: false, alt: "−3")[$−3$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "open parenthesis −3 , 3 close parenthesis")[$( −3 , 3 )$]], )) Then plot the points and connect them with a straight line. Notice in that the graph is a #strong[vertical line]. #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A vertical line passes through three labeled points, “ordered pair -3, 3”, “ordered pair -3, 2”, and ordered pair -3, 1”. The line is labeled x = -3.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A vertical line passes through three labeled points, “ordered pair -3, 3”, “ordered pair -3, 2”, and ordered pair -3, 1”. The line is labeled x = -3.", caption: none) #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Vertical Line] A vertical line is the graph of an equation that can be written in the form #math.equation(block: false, alt: "x equals a .")[$x = a .$] The line passes through the #math.equation(block: false, alt: "x")[$x$]-axis at #math.equation(block: false, alt: "open parenthesis a , 0 close parenthesis")[$( a , 0 )$]. ] #examplebox("Example 6")[][ Graph the equation #math.equation(block: false, alt: "x equals 2 .")[$x = 2 .$] What type of line does it form? #solutionbox[ The equation has only variable, #math.equation(block: false, alt: "x ,")[$x ,$] and #math.equation(block: false, alt: "x")[$x$] is always equal to #math.equation(block: false, alt: "2 .")[$2 .$] We make a table where #math.equation(block: false, alt: "x")[$x$] is always #math.equation(block: false, alt: "2")[$2$] and we put in any values for #math.equation(block: false, alt: "y .")[$y .$] #figure(table( columns: 3, align: left, inset: 6pt, [#math.equation(block: false, alt: "x equals 2")[$x = 2$]], [], [], [#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: "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: "2")[$2$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "open parenthesis 2 , 2 close parenthesis")[$( 2 , 2 )$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "open parenthesis 2 , 3 close parenthesis")[$( 2 , 3 )$]], )) Plot the points and connect them as shown. #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A vertical line passes through three labeled points, “ordered pair 2, 3”, “ordered pair 2, 2”, and ordered pair 2, 1”. The line is labeled x = 2.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A vertical line passes through three labeled points, “ordered pair 2, 3”, “ordered pair 2, 2”, and ordered pair 2, 1”. The line is labeled x = 2.", caption: none) The graph is a vertical line passing through the #math.equation(block: false, alt: "x")[$x$]-axis at #math.equation(block: false, alt: "2 .")[$2 .$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the equation: #math.equation(block: false, alt: "x equals 5 .")[$x = 5 .$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A vertical line passes through the points “ordered pair 5, 0” and “ordered pair 5, 1”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A vertical line passes through the points “ordered pair 5, 0” and “ordered pair 5, 1”.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the equation: #math.equation(block: false, alt: "x equals −2 .")[$x = −2 .$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A vertical line passes through the points “ordered pair 5, 0” and “ordered pair 5, 1”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A vertical line passes through the points “ordered pair 5, 0” and “ordered pair 5, 1”.", caption: none) ] ] What if the equation has #math.equation(block: false, alt: "y")[$y$] but no #math.equation(block: false, alt: "x")[$x$]? Let’s graph the equation #math.equation(block: false, alt: "y equals 4 .")[$y = 4 .$] This time the #math.equation(block: false, alt: "y")[$y$]-value is a constant, so in this equation #math.equation(block: false, alt: "y")[$y$] does not depend on #math.equation(block: false, alt: "x .")[$x .$] To make a table of solutions, write #math.equation(block: false, alt: "4")[$4$] for all the #math.equation(block: false, alt: "y")[$y$] values and then choose any values for #math.equation(block: false, alt: "x .")[$x .$] We’ll use #math.equation(block: false, alt: "0 , 2 ,")[$0 , 2 ,$] and #math.equation(block: false, alt: "4")[$4$] for the #math.equation(block: false, alt: "x")[$x$]-values. #figure(table( columns: 3, align: left, inset: 6pt, [#math.equation(block: false, alt: "y equals 4")[$y = 4$]], [], [], [#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: "2")[$2$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "open parenthesis 2 , 4 close parenthesis")[$( 2 , 4 )$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "open parenthesis 4 , 4 close parenthesis")[$( 4 , 4 )$]], )) Plot the points and connect them, as shown. This graph is a #strong[horizontal line] passing through the #math.equation(block: false, alt: "y -axis")[$y "-axis"$] at #math.equation(block: false, alt: "4 .")[$4 .$] #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A horizontal line passes through three labeled points, “ordered pair 0, 4”, “ordered pair 2, 4”, and ordered pair 4, 4”. The line is labeled y = 4.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A horizontal line passes through three labeled points, “ordered pair 0, 4”, “ordered pair 2, 4”, and ordered pair 4, 4”. The line is labeled y = 4.", caption: none) #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Horizontal Line] A horizontal line is the graph of an equation that can be written in the form #math.equation(block: false, alt: "y equals b .")[$y = b .$] The line passes through the #math.equation(block: false, alt: "y -axis")[$y "-axis"$] at #math.equation(block: false, alt: "open parenthesis 0 , b close parenthesis .")[$( 0 , b ) .$] ] #examplebox("Example 7")[][ Graph the equation #math.equation(block: false, alt: "y equals −1 .")[$y = −1 .$] #solutionbox[ The equation #math.equation(block: false, alt: "y equals −1")[$y = −1$] has only variable, #math.equation(block: false, alt: "y .")[$y .$] The value of #math.equation(block: false, alt: "y")[$y$] is constant. All the ordered pairs in the table have the same #math.equation(block: false, alt: "y")[$y$]-coordinate, #math.equation(block: false, alt: "−1")[$−1$]. We choose #math.equation(block: false, alt: "0 , 3 ,")[$0 , 3 ,$] and #math.equation(block: false, alt: "−3")[$−3$] as values for #math.equation(block: false, alt: "x .")[$x .$] #figure(table( columns: 3, align: left, inset: 6pt, [#math.equation(block: false, alt: "y equals −1")[$y = −1$]], [], [], [#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: "−3")[$−3$]], [#math.equation(block: false, alt: "−1")[$−1$]], [#math.equation(block: false, alt: "open parenthesis −3 , −1 close parenthesis")[$( −3 , −1 )$]], [#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: "3")[$3$]], [#math.equation(block: false, alt: "−1")[$−1$]], [#math.equation(block: false, alt: "open parenthesis 3 , −1 close parenthesis")[$( 3 , −1 )$]], )) The graph is a horizontal line passing through the #math.equation(block: false, alt: "y")[$y$]-axis at #math.equation(block: false, alt: "–1")[$–1$] as shown. #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A horizontal line passes through three labeled points, “ordered pair -3, -1”, “ordered pair 0, -1”, and ordered pair 3, -1”. The line is labeled y = -1.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A horizontal line passes through three labeled points, “ordered pair -3, -1”, “ordered pair 0, -1”, and ordered pair 3, -1”. The line is labeled y = -1.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the equation: #math.equation(block: false, alt: "y equals −4 .")[$y = −4 .$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A horizontal line passes through the points “ordered pair 0, -4” and “ordered pair 1, -4”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A horizontal line passes through the points “ordered pair 0, -4” and “ordered pair 1, -4”.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the equation: #math.equation(block: false, alt: "y equals 3 .")[$y = 3 .$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A horizontal line passes through the points “ordered pair 0, 3” and “ordered pair 1, 3”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A horizontal line passes through the points “ordered pair 0, 3” and “ordered pair 1, 3”.", caption: none) ] ] The equations for vertical and horizontal lines look very similar to equations like #math.equation(block: false, alt: "y equals 4 x .")[$y = 4 x .$] What is the difference between the equations #math.equation(block: false, alt: "y equals 4 x")[$y = 4 x$] and #math.equation(block: false, alt: "y equals 4 ?")[$y = 4 ?$] The equation #math.equation(block: false, alt: "y equals 4 x")[$y = 4 x$] has both #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y .")[$y .$] The value of #math.equation(block: false, alt: "y")[$y$] depends on the value of #math.equation(block: false, alt: "x .")[$x .$] The #math.equation(block: false, alt: "y -coordinate")[$y "-coordinate"$] changes according to the value of #math.equation(block: false, alt: "x .")[$x .$] The equation #math.equation(block: false, alt: "y equals 4")[$y = 4$] has only one variable. The value of #math.equation(block: false, alt: "y")[$y$] is constant. The #math.equation(block: false, alt: "y -coordinate")[$y "-coordinate"$] is always #math.equation(block: false, alt: "4 .")[$4 .$] It does not depend on the value of #math.equation(block: false, alt: "x .")[$x .$] #figure(figph[There are two tables. This first table is titled y = 4 x, which is shown in blue. It has 4 rows and 3 columns. The first row is a header row and it labels each column “x”, “y”, and “ordered pair x, y”. Under the column “x” are the values 0, 1, and 2. Under the column “y” are the values 0, 4, and 8. Under the column “ordered pair x, y” are the values “ordered pair 0, 0”, “ordered pair 1, 4”, and “ordered pair 2, 8”. This second table is titled y = 4 , which is shown in red. It has 4 rows and 3 columns. The first row is a header row and it labels each column “x”, “y”, and “ordered pair x, y”. Under the column “x” are the values 0, 1, and 2. Under the column “y” are the values 4, 4, and 4. Under the column “ordered pair x, y” are the values “ordered pair 0, 4”, “ordered pair 1, 4”, and “ordered pair 2, 4”.], alt: "There are two tables. This first table is titled y = 4 x, which is shown in blue. It has 4 rows and 3 columns. The first row is a header row and it labels each column “x”, “y”, and “ordered pair x, y”. Under the column “x” are the values 0, 1, and 2. Under the column “y” are the values 0, 4, and 8. Under the column “ordered pair x, y” are the values “ordered pair 0, 0”, “ordered pair 1, 4”, and “ordered pair 2, 8”. This second table is titled y = 4 , which is shown in red. It has 4 rows and 3 columns. The first row is a header row and it labels each column “x”, “y”, and “ordered pair x, y”. Under the column “x” are the values 0, 1, and 2. Under the column “y” are the values 4, 4, and 4. Under the column “ordered pair x, y” are the values “ordered pair 0, 4”, “ordered pair 1, 4”, and “ordered pair 2, 4”.", caption: none) The graph shows both equations. #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A horizontal line passes through “ordered pair 0, 4” and “ordered pair 1, 4” and is labeled y = 4. A second line passes through “ordered pair 0, 0” and “ordered pair 1, 4” and is labeled y = 4 x. The two lines intersect at “ordered pair 1, 4”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A horizontal line passes through “ordered pair 0, 4” and “ordered pair 1, 4” and is labeled y = 4. A second line passes through “ordered pair 0, 0” and “ordered pair 1, 4” and is labeled y = 4 x. The two lines intersect at “ordered pair 1, 4”.", caption: none) Notice that the equation #math.equation(block: false, alt: "y equals 4 x")[$y = 4 x$] gives a slanted line whereas #math.equation(block: false, alt: "y equals 4")[$y = 4$] gives a horizontal line. #examplebox("Example 8")[][ Graph #math.equation(block: false, alt: "y equals −3 x")[$y = −3 x$] and #math.equation(block: false, alt: "y equals −3")[$y = −3$] in the same rectangular coordinate system. #solutionbox[ Find three solutions for each equation. Notice that the first equation has the variable #math.equation(block: false, alt: "x ,")[$x ,$] while the second does not. Solutions for both equations are listed. #linebreak() #figure(figph[There are two tables. This first table is titled y = -3 x, which is shown in red. It has 4 rows and 3 columns. The first row is a header row and it labels each column “x”, “y”, and “ordered pair x, y”. Under the column “x” are the values 0, 1, and 2. Under the column “y” are the values 0, -3, and -6. Under the column “ordered pair x, y” are the values “ordered pair 0, 0”, “ordered pair 1, -3”, and “ordered pair 2, -6”. This second table is titled y = -3 , which is shown in red. It has 4 rows and 3 columns. The first row is a header row and it labels each column “x”, “y”, and “ordered pair x, y”. Under the column “x” are the values 0, 1, and 2. Under the column “y” are the values -3, -3, and -3. Under the column “ordered pair x, y” are the values “ordered pair 0, -3”, “ordered pair 1, -3”, and “ordered pair 2, -3”.], alt: "There are two tables. This first table is titled y = -3 x, which is shown in red. It has 4 rows and 3 columns. The first row is a header row and it labels each column “x”, “y”, and “ordered pair x, y”. Under the column “x” are the values 0, 1, and 2. Under the column “y” are the values 0, -3, and -6. Under the column “ordered pair x, y” are the values “ordered pair 0, 0”, “ordered pair 1, -3”, and “ordered pair 2, -6”. This second table is titled y = -3 , which is shown in red. It has 4 rows and 3 columns. The first row is a header row and it labels each column “x”, “y”, and “ordered pair x, y”. Under the column “x” are the values 0, 1, and 2. Under the column “y” are the values -3, -3, and -3. Under the column “ordered pair x, y” are the values “ordered pair 0, -3”, “ordered pair 1, -3”, and “ordered pair 2, -3”.", caption: none) The graph shows both equations. #linebreak() #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A horizontal line passes through “ordered pair 0, -3” and “ordered pair 1, -3” and is labeled y = -3. A second line passes through “ordered pair 0, 0” and “ordered pair 1, -3” and is labeled y = -3 x. The two lines intersect at “ordered pair 1, -3”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A horizontal line passes through “ordered pair 0, -3” and “ordered pair 1, -3” and is labeled y = -3. A second line passes through “ordered pair 0, 0” and “ordered pair 1, -3” and is labeled y = -3 x. The two lines intersect at “ordered pair 1, -3”.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the equations in the same rectangular coordinate system: #math.equation(block: false, alt: "y equals −4 x")[$y = −4 x$] and #math.equation(block: false, alt: "y equals −4 .")[$y = −4 .$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A horizontal line passes through “ordered pair 0, -4” and “ordered pair 1, -4” . A second line passes through “ordered pair 0, 0” and “ordered pair 1, -4” . The two lines intersect at “ordered pair 1, -4”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A horizontal line passes through “ordered pair 0, -4” and “ordered pair 1, -4” . A second line passes through “ordered pair 0, 0” and “ordered pair 1, -4” . The two lines intersect at “ordered pair 1, -4”.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the equations in the same rectangular coordinate system: #math.equation(block: false, alt: "y equals 3")[$y = 3$] and #math.equation(block: false, alt: "y equals 3 x .")[$y = 3 x .$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A horizontal line passes through “ordered pair 0, 3” and “ordered pair 1, 3” . A second line passes through “ordered pair 0, 0” and “ordered pair 1, 3” . The two lines intersect at “ordered pair 1, 3”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A horizontal line passes through “ordered pair 0, 3” and “ordered pair 1, 3” . A second line passes through “ordered pair 0, 0” and “ordered pair 1, 3” . The two lines intersect at “ordered pair 1, 3”.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[ACCESS ADDITIONAL ONLINE RESOURCES] - #link("https://www.openstax.org/l/24tabofval")[Use a Table of Values] - #link("https://www.openstax.org/l/24graphlineq")[Graph a Linear Equation Involving Fractions] - #link("https://www.openstax.org/l/24graphhorvert")[Graph Horizontal and Vertical Lines] ] === Key Concepts - #emph[Graph a linear equation by plotting points.]+ Find three points whose coordinates are solutions to the equation. Organize them in a table. + Plot the points on a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work. + Draw the line through the points. Extend the line to fill the grid and put arrows on both ends of the line. - #emph[Graph of a Linear Equation:]The graph of a linear equation #math.equation(block: false, alt: "a x plus b y equals c")[$a x + b y = c$] is a straight line.- Every point on the line is a solution of the equation. - Every solution of this equation is a point on this line. ==== Practice Makes Perfect #strong[Recognize the Relation Between the Solutions of an Equation and its Graph] In each of the following exercises, an equation and its graph is shown. For each ordered pair, decide + ⓐ is the ordered pair a solution to the equation? + ⓑ is the point on the line? #math.equation(block: true, alt: "y equals x plus 2")[$y = x + 2$] #linebreak() #figure(figph[A graph with both x- and y-axis ranging from –8 to 8. A line runs upwards, intercepting the x-axis at (–2, 0) and the y-axis at (0, 2).], alt: "A graph with both x- and y-axis ranging from –8 to 8. A line runs upwards, intercepting the x-axis at (–2, 0) and the y-axis at (0, 2).", caption: none) + #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$] + #math.equation(block: false, alt: "open parenthesis 1 , 2 close parenthesis")[$( 1 , 2 )$] + #math.equation(block: false, alt: "open parenthesis minus 1 , 1 close parenthesis")[$( − 1 , 1 )$] + #math.equation(block: false, alt: "open parenthesis minus 3 , 1 close parenthesis")[$( − 3 , 1 )$] #solutionbox[ + ⓐ yes ⓑ yes + ⓐ no ⓑ no + ⓐ yes ⓑ yes + ⓐ no ⓑ no ] #math.equation(block: true, alt: "y equals x minus 4")[$y = x − 4$] #linebreak() #figure(figph[A graph with both x- and y-axis running from –8 to 8. A line is drawn that intercepts the y-axis at (0, –4) and the x-axis at (4, 0)], alt: "A graph with both x- and y-axis running from –8 to 8. A line is drawn that intercepts the y-axis at (0, –4) and the x-axis at (4, 0)", caption: none) + #math.equation(block: false, alt: "open parenthesis 0 , minus 4 close parenthesis")[$( 0 , − 4 )$] + #math.equation(block: false, alt: "open parenthesis 3 , minus 1 close parenthesis")[$( 3 , − 1 )$] + #math.equation(block: false, alt: "open parenthesis 2 , 2 close parenthesis")[$( 2 , 2 )$] + #math.equation(block: false, alt: "open parenthesis 1 , minus 5 close parenthesis")[$( 1 , − 5 )$] #math.equation(block: true, alt: "y equals the fraction 1 over 2 x minus 3")[$y = frac(1, 2) x − 3$] #linebreak() #figure(figph[A graph with both x- and y-axis running from –8 to 8. A line is drawn that intercepts the y-axis at (0, –3) and the x-axis at (6, 0)], alt: "A graph with both x- and y-axis running from –8 to 8. A line is drawn that intercepts the y-axis at (0, –3) and the x-axis at (6, 0)", caption: none) + #math.equation(block: false, alt: "open parenthesis 0 , minus 3 close parenthesis")[$( 0 , − 3 )$] + #math.equation(block: false, alt: "open parenthesis 2 , minus 2 close parenthesis")[$( 2 , − 2 )$] + #math.equation(block: false, alt: "open parenthesis minus 2 , minus 4 close parenthesis")[$( − 2 , − 4 )$] + #math.equation(block: false, alt: "open parenthesis 4 , 1 close parenthesis")[$( 4 , 1 )$] #solutionbox[ + ⓐ yes ⓑ yes + ⓐ yes ⓑ yes + ⓐ yes ⓑ yes + ⓐ no ⓑ no ] #math.equation(block: true, alt: "y equals the fraction 1 over 3 x plus 2")[$y = frac(1, 3) x + 2$] #linebreak() #figure(figph[A linear graph on a grid, showing a line with a positive slope, passing through y-intercept (0, 2) and x-intercept (-7, 0). The x and y axes range from -8 to 8.], alt: "A linear graph on a grid, showing a line with a positive slope, passing through y-intercept (0, 2) and x-intercept (-7, 0). The x and y axes range from -8 to 8.", caption: none) + #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$] + #math.equation(block: false, alt: "open parenthesis 3 , 3 close parenthesis")[$( 3 , 3 )$] + #math.equation(block: false, alt: "open parenthesis minus 3 , 2 close parenthesis")[$( − 3 , 2 )$] + #math.equation(block: false, alt: "open parenthesis minus 6 , 0 close parenthesis")[$( − 6 , 0 )$] #strong[Graph a Linear Equation by Plotting Points] In the following exercises, graph by plotting points. #math.equation(block: true, alt: "y equals 3 x minus 1")[$y = 3 x − 1$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, -1” and “ordered pair 1/3, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, -1” and “ordered pair 1/3, 0”.", caption: none) ] #math.equation(block: true, alt: "y equals 2 x plus 3")[$y = 2 x + 3$] #math.equation(block: true, alt: "y equals −2 x plus 2")[$y = −2 x + 2$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 4” and “ordered pair 1, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 4” and “ordered pair 1, 0”.", caption: none) ] #math.equation(block: true, alt: "y equals −3 x plus 1")[$y = −3 x + 1$] #math.equation(block: true, alt: "y equals x plus 2")[$y = x + 2$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 2” and “ordered pair 2, 4”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 2” and “ordered pair 2, 4”.", caption: none) ] #math.equation(block: true, alt: "y equals x minus 3")[$y = x − 3$] #math.equation(block: true, alt: "y equals minus x minus 3")[$y = − x − 3$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, -3” and “ordered pair 1, -4”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, -3” and “ordered pair 1, -4”.", caption: none) ] #math.equation(block: true, alt: "y equals minus x minus 2")[$y = − x − 2$] #math.equation(block: true, alt: "y equals 2 x")[$y = 2 x$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 0” and “ordered pair 2, 4”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 0” and “ordered pair 2, 4”.", caption: none) ] #math.equation(block: true, alt: "y equals 3 x")[$y = 3 x$] #math.equation(block: true, alt: "y equals −4 x")[$y = −4 x$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 0” and “ordered pair 1, -4”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 0” and “ordered pair 1, -4”.", caption: none) ] #math.equation(block: true, alt: "y equals −2 x")[$y = −2 x$] #math.equation(block: true, alt: "y equals the fraction 1 over 2 x plus 2")[$y = frac(1, 2) x + 2$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 2” and “ordered pair 4, 4”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 2” and “ordered pair 4, 4”.", caption: none) ] #math.equation(block: true, alt: "y equals the fraction 1 over 3 x minus 1")[$y = frac(1, 3) x − 1$] #math.equation(block: true, alt: "y equals the fraction 4 over 3 x minus 5")[$y = frac(4, 3) x − 5$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 5” and “ordered pair 4, 2”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 5” and “ordered pair 4, 2”.", caption: none) ] #math.equation(block: true, alt: "y equals the fraction 3 over 2 x minus 3")[$y = frac(3, 2) x − 3$] #math.equation(block: true, alt: "y equals minus the fraction 2 over 5 x plus 1")[$y = − frac(2, 5) x + 1$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 1” and “ordered pair 8, -2”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 1” and “ordered pair 8, -2”.", caption: none) ] #math.equation(block: true, alt: "y equals minus the fraction 4 over 5 x minus 1")[$y = − frac(4, 5) x − 1$] #math.equation(block: true, alt: "y equals minus the fraction 3 over 2 x plus 2")[$y = − frac(3, 2) x + 2$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 2” and “ordered pair 4, -4”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 2” and “ordered pair 4, -4”.", caption: none) ] #math.equation(block: true, alt: "y equals minus the fraction 5 over 3 x plus 4")[$y = − frac(5, 3) x + 4$] #math.equation(block: true, alt: "x plus y equals 6")[$x + y = 6$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 6” and “ordered pair 6, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 6” and “ordered pair 6, 0”.", caption: none) ] #math.equation(block: true, alt: "x plus y equals 4")[$x + y = 4$] #math.equation(block: true, alt: "x plus y equals −3")[$x + y = −3$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, -3” and “ordered pair -3, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, -3” and “ordered pair -3, 0”.", caption: none) ] #math.equation(block: true, alt: "x plus y equals −2")[$x + y = −2$] #math.equation(block: true, alt: "x minus y equals 2")[$x − y = 2$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, -2” and “ordered pair 2, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, -2” and “ordered pair 2, 0”.", caption: none) ] #math.equation(block: true, alt: "x minus y equals 1")[$x − y = 1$] #math.equation(block: true, alt: "x minus y equals −1")[$x − y = −1$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 1” and “ordered pair 8, 8”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 1” and “ordered pair 8, 8”.", caption: none) ] #math.equation(block: true, alt: "x minus y equals −3")[$x − y = −3$] #math.equation(block: true, alt: "minus x plus y equals 4")[$− x + y = 4$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair -4, 0” and “ordered pair 0, 4”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair -4, 0” and “ordered pair 0, 4”.", caption: none) ] #math.equation(block: true, alt: "minus x plus y equals 3")[$− x + y = 3$] #math.equation(block: true, alt: "minus x minus y equals 5")[$− x − y = 5$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair -5, 0” and “ordered pair 0, -5”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair -5, 0” and “ordered pair 0, -5”.", caption: none) ] #math.equation(block: true, alt: "minus x minus y equals 1")[$− x − y = 1$] #math.equation(block: true, alt: "3 x plus y equals 7")[$3 x + y = 7$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 4, -4” and “ordered pair 0, 8”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 4, -4” and “ordered pair 0, 8”.", caption: none) ] #math.equation(block: true, alt: "5 x plus y equals 6")[$5 x + y = 6$] #math.equation(block: true, alt: "2 x plus y equals −3")[$2 x + y = −3$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, -3” and “ordered pair 2, -7”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, -3” and “ordered pair 2, -7”.", caption: none) ] #math.equation(block: true, alt: "4 x plus y equals −5")[$4 x + y = −5$] #math.equation(block: true, alt: "2 x plus 3 y equals 12")[$2 x + 3 y = 12$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 4” and “ordered pair 6, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 4” and “ordered pair 6, 0”.", caption: none) ] #math.equation(block: true, alt: "3 x minus 4 y equals 12")[$3 x − 4 y = 12$] #math.equation(block: true, alt: "the fraction 1 over 3 x plus y equals 2")[$frac(1, 3) x + y = 2$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 2” and “ordered pair 6, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 2” and “ordered pair 6, 0”.", caption: none) ] #math.equation(block: true, alt: "the fraction 1 over 2 x plus y equals 3")[$frac(1, 2) x + y = 3$] #strong[Graph Vertical and Horizontal lines] In the following exercises, graph the vertical and horizontal lines. #math.equation(block: true, alt: "x equals 4")[$x = 4$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A vertical line passes through the points “ordered pair 4, 0” and “ordered pair 4, 1”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A vertical line passes through the points “ordered pair 4, 0” and “ordered pair 4, 1”.", caption: none) ] #math.equation(block: true, alt: "x equals 3")[$x = 3$] #math.equation(block: true, alt: "x equals −2")[$x = −2$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A vertical line passes through the points “ordered pair -2, 0” and “ordered pair -2, 1”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A vertical line passes through the points “ordered pair -2, 0” and “ordered pair -2, 1”.", caption: none) ] #math.equation(block: true, alt: "x equals −5")[$x = −5$] #math.equation(block: true, alt: "y equals 3")[$y = 3$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A horizontal line passes through the points “ordered pair 0, 3” and “ordered pair 1, 3”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A horizontal line passes through the points “ordered pair 0, 3” and “ordered pair 1, 3”.", caption: none) ] #math.equation(block: true, alt: "y equals 1")[$y = 1$] #math.equation(block: true, alt: "y equals −5")[$y = −5$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A horizontal line passes through the points “ordered pair 0, -5” and “ordered pair 1, -5”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A horizontal line passes through the points “ordered pair 0, -5” and “ordered pair 1, -5”.", caption: none) ] #math.equation(block: true, alt: "y equals −2")[$y = −2$] #math.equation(block: true, alt: "x equals the fraction 7 over 3")[$x = frac(7, 3)$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A vertical line passes through the points “ordered pair 7 over 3, 0” and “ordered pair 7 over 3 , 1”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A vertical line passes through the points “ordered pair 7 over 3, 0” and “ordered pair 7 over 3 , 1”.", caption: none) ] #math.equation(block: true, alt: "x equals the fraction 5 over 4")[$x = frac(5, 4)$] In the following exercises, graph each pair of equations in the same rectangular coordinate system. #math.equation(block: false, alt: "y equals minus the fraction 1 over 2 x")[$y = − frac(1, 2) x$] and #math.equation(block: false, alt: "y equals minus the fraction 1 over 2")[$y = − frac(1, 2)$] #solutionbox[ #figure(figph[A Cartesian coordinate system displays a dark blue line that passes through the origin (0,0). The line has a negative slope, intersecting points like (-4, 2) and (4, -2).], alt: "A Cartesian coordinate system displays a dark blue line that passes through the origin (0,0). The line has a negative slope, intersecting points like (-4, 2) and (4, -2).", caption: none) ] #math.equation(block: false, alt: "y equals minus the fraction 1 over 3 x")[$y = − frac(1, 3) x$] and #math.equation(block: false, alt: "y equals minus the fraction 1 over 3")[$y = − frac(1, 3)$] #math.equation(block: false, alt: "y equals 2 x")[$y = 2 x$] and #math.equation(block: false, alt: "y equals 2")[$y = 2$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A horizontal line passes through “ordered pair 0, 2” and “ordered pair 1, 2” . A second line passes through “ordered pair 0, 0” and “ordered pair 1, 2” . The two lines intersect at “ordered pair 1, 2”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A horizontal line passes through “ordered pair 0, 2” and “ordered pair 1, 2” . A second line passes through “ordered pair 0, 0” and “ordered pair 1, 2” . The two lines intersect at “ordered pair 1, 2”.", caption: none) ] #math.equation(block: false, alt: "y equals 5 x")[$y = 5 x$] and #math.equation(block: false, alt: "y equals 5")[$y = 5$] #strong[Mixed Practice] In the following exercises, graph each equation. #math.equation(block: true, alt: "y equals 4 x")[$y = 4 x$] #solutionbox[ #figure(figph[A coordinate plane shows a straight line passing through the origin (0,0) and the points (1,4) and (-1,-4), indicating a positive slope of 4.], alt: "A coordinate plane shows a straight line passing through the origin (0,0) and the points (1,4) and (-1,-4), indicating a positive slope of 4.", caption: none) ] #math.equation(block: true, alt: "y equals 2 x")[$y = 2 x$] #math.equation(block: true, alt: "y equals minus the fraction 1 over 2 x plus 3")[$y = − frac(1, 2) x + 3$] #solutionbox[ #figure(figph[A graph displays a blue line with a negative slope on a Cartesian coordinate plane. The line intersects the y-axis at approximately 3.5 and the x-axis at approximately 7.], alt: "A graph displays a blue line with a negative slope on a Cartesian coordinate plane. The line intersects the y-axis at approximately 3.5 and the x-axis at approximately 7.", caption: none) ] #math.equation(block: true, alt: "y equals the fraction 1 over 4 x minus 2")[$y = frac(1, 4) x − 2$] #math.equation(block: true, alt: "y equals minus x")[$y = − x$] #solutionbox[ #figure(figph[A line graph on a Cartesian plane showing a straight line passing through the origin (0,0) with a negative slope. The x and y axes range from -8 to 8.], alt: "A line graph on a Cartesian plane showing a straight line passing through the origin (0,0) with a negative slope. The x and y axes range from -8 to 8.", caption: none) ] #math.equation(block: true, alt: "y equals x")[$y = x$] #math.equation(block: true, alt: "x minus y equals 3")[$x − y = 3$] #solutionbox[ #figure(figph[A graph showing a straight line on a Cartesian coordinate plane. The line has a positive slope, intersecting the y-axis at -3 and the x-axis at 4.5. It represents a linear function.], alt: "A graph showing a straight line on a Cartesian coordinate plane. The line has a positive slope, intersecting the y-axis at -3 and the x-axis at 4.5. It represents a linear function.", caption: none) ] #math.equation(block: true, alt: "x plus y equals minus 5")[$x + y = − 5$] #math.equation(block: true, alt: "4 x plus y equals 2")[$4 x + y = 2$] #solutionbox[ #figure(figph[A graph shows a linear function, represented by a solid downward-sloping line, plotted on a Cartesian coordinate system. The line intersects the y-axis at (0, 2) and the x-axis at (1, 0), extending infinitely in both directions as indicated by arrows.], alt: "A graph shows a linear function, represented by a solid downward-sloping line, plotted on a Cartesian coordinate system. The line intersects the y-axis at (0, 2) and the x-axis at (1, 0), extending infinitely in both directions as indicated by arrows.", caption: none) ] #math.equation(block: true, alt: "2 x plus y equals 6")[$2 x + y = 6$] #math.equation(block: true, alt: "y equals −1")[$y = −1$] #solutionbox[ #figure(figph[A Cartesian coordinate system with a grid. The x and y axes are labeled from -8 to 8. A horizontal double-headed arrow is drawn along the x-axis, indicating the entire range.], alt: "A Cartesian coordinate system with a grid. The x and y axes are labeled from -8 to 8. A horizontal double-headed arrow is drawn along the x-axis, indicating the entire range.", caption: none) ] #math.equation(block: true, alt: "y equals 5")[$y = 5$] #math.equation(block: true, alt: "2 x plus 6 y equals 12")[$2 x + 6 y = 12$] #solutionbox[ #figure(figph[A graph on a coordinate plane shows a downward-sloping line. The line intersects the y-axis at (0, 2) and the x-axis at (6, 0), extending infinitely in both directions.], alt: "A graph on a coordinate plane shows a downward-sloping line. The line intersects the y-axis at (0, 2) and the x-axis at (6, 0), extending infinitely in both directions.", caption: none) ] #math.equation(block: true, alt: "5 x plus 2 y equals 10")[$5 x + 2 y = 10$] #math.equation(block: true, alt: "x equals 3")[$x = 3$] #solutionbox[ #figure(figph[A graph with x- and y- axis both ranging from –8 to 8. A line is drawn to demonstrate x = 3.], alt: "A graph with x- and y- axis both ranging from –8 to 8. A line is drawn to demonstrate x = 3.", caption: none) ] #math.equation(block: true, alt: "x equals −4")[$x = −4$] ==== Everyday Math #strong[Motor home cost] The Robinsons rented a motor home for one week to go on vacation. It cost them #math.equation(block: false, alt: "$594")[$"$594"$] plus #math.equation(block: false, alt: "$0.32")[$"$0.32"$] per mile to rent the motor home, so the linear equation #math.equation(block: false, alt: "y equals 594 plus 0.32 x")[$y = 594 + 0.32 x$] gives the cost, #math.equation(block: false, alt: "y ,")[$y ,$] for driving #math.equation(block: false, alt: "x")[$x$] miles. Calculate the rental cost for driving #math.equation(block: false, alt: "400 , 800 , and 1,200")[$400 , 800 , #h(0.2em) "and" #h(0.2em) 1,200$] miles, and then graph the line. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x-axis runs from 0 to 1000. The y-axis runs from 0 to 1200. A line passes through the points “ordered pair 0, 594” and “ordered pair 800, 850”.], alt: "The graph shows the x y-coordinate plane. The x-axis runs from 0 to 1000. The y-axis runs from 0 to 1200. A line passes through the points “ordered pair 0, 594” and “ordered pair 800, 850”.", caption: none) #linebreak() \$722, \$850, \$978 ] #strong[Weekly earning] At the art gallery where he works, Salvador gets paid #math.equation(block: false, alt: "$200")[$"$200"$] per week plus #math.equation(block: false, alt: "15%")[$"15%"$] of the sales he makes, so the equation #math.equation(block: false, alt: "y equals 200 plus 0.15 x")[$y = 200 + 0.15 x$] gives the amount #math.equation(block: false, alt: "y")[$y$] he earns for selling #math.equation(block: false, alt: "x")[$x$] dollars of artwork. Calculate the amount Salvador earns for selling #math.equation(block: false, alt: "$900, $1,600 , and $2,000 ,")[$"$900, $1,600" , #h(0.2em) "and" #h(0.2em) "$2,000" ,$] and then graph the line. ==== Writing Exercises Explain how you would choose three #math.equation(block: false, alt: "x -values")[$x "-values"$] to make a table to graph the line #math.equation(block: false, alt: "y equals the fraction 1 over 5 x minus 2 .")[$y = frac(1, 5) x − 2 .$] #solutionbox[ Answers will vary. ] What is the difference between the equations of a vertical and a horizontal line? ==== Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. #figure(figph[A self-assessment table for math skills, asking students to rate their ability to graph linear equations by plotting points, and graph vertical and horizontal lines.], alt: "A self-assessment table for math skills, asking students to rate their ability to graph linear equations by plotting points, and graph vertical and horizontal lines.", caption: none) ⓑ After reviewing this checklist, what will you do to become confident for all objectives?