#set document(title: "4.2 Graph Linear Equations in Two Variables", 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")) == 4.2#h(0.6em)Graph Linear Equations in Two Variables #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[ −1 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve #math.equation(block: false, alt: "3 x plus 2 y equals 12")[$3 x + 2 y = 12$] for #math.equation(block: false, alt: "y")[$y$] in general. #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 12 minus 3 x over 2")[$y = frac(12 − 3 x, 2)$] ] ] === Recognize the Relationship Between the Solutions of an Equation and its Graph In the previous section, we found several 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 . 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")[$( 2 , 0 )$], and #math.equation(block: false, alt: "open parenthesis 1 , the fraction 3 over 2 close parenthesis")[$( 1 , frac(3, 2) )$] 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. #figure(table( columns: 3, align: left, inset: 6pt, [#strong[#math.equation(block: false, alt: "3 x plus 2 y equals 6")[$3 x + 2 y = 6$]]], [], [], [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]], [0], [3], [#math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$( 0 , 3 )$]], [2], [0], [#math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$]], [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")[$( 1 , frac(3, 2) )$]], )) #figure(figph[The figure shows four points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the four points at (0, 3), (1, three halves), (2, 0), and (4, negative 3). The four points appear to line up along a straight line.], alt: "The figure shows four points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the four points at (0, 3), (1, three halves), (2, 0), and (4, negative 3). The four points appear to line up along a straight line.", caption: none) Notice how the points line up perfectly? We connect the points with a 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[The figure shows a straight line drawn through four points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the four points at (0, 3), (1, three halves), (2, 0), and (4, negative 3). A straight line with a negative slope goes through all four points. The line has arrows on both ends pointing to the edge of the figure. The line is labeled with the equation 3x plus 2y equals 6.], alt: "The figure shows a straight line drawn through four points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the four points at (0, 3), (1, three halves), (2, 0), and (4, negative 3). A straight line with a negative slope goes through all four points. The line has arrows on both ends pointing to the edge of the figure. The line is labeled with the equation 3x plus 2y equals 6.", 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 #emph[not] on the line are not solutions. Notice that the point whose coordinates are #math.equation(block: false, alt: "open parenthesis −2 , 6 close parenthesis")[$( −2 , 6 )$] is on the line shown. If you substitute #math.equation(block: false, alt: "x equals −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 figure shows a straight line and two points and on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the two points and are labeled by the coordinates “(negative 2, 6)” and “(4, 1)”. The straight line goes through the point (negative 2, 6) but does not go through the point (4, 1).], alt: "The figure shows a straight line and two points and on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the two points and are labeled by the coordinates “(negative 2, 6)” and “(4, 1)”. The straight line goes through the point (negative 2, 6) but does not go through the point (4, 1).", caption: none) #figure(figph[Checking if ordered pair negative 2, 6 solves 3x + 2y = 6. Substituting gives 3 times negative 2 plus 2 times 6 equals 6. Simplifying yields negative 6 plus 12 equals 6, then 6 equals 6. A checkmark indicates the solution is correct.], alt: "Checking if ordered pair negative 2, 6 solves 3x + 2y = 6. Substituting gives 3 times negative 2 plus 2 times 6 equals 6. Simplifying yields negative 6 plus 12 equals 6, then 6 equals 6. A checkmark indicates the solution is correct.", caption: none) So the point #math.equation(block: false, alt: "open parenthesis −2 , 6 close parenthesis")[$( −2 , 6 )$] is a solution to the equation #math.equation(block: false, alt: "3 x plus 2 y equals 6")[$3 x + 2 y = 6$]. (The phrase “the point whose coordinates are #math.equation(block: false, alt: "open parenthesis −2 , 6 close parenthesis")[$( −2 , 6 )$]” is often shortened to “the point #math.equation(block: false, alt: "open parenthesis −2 , 6 close parenthesis")[$( −2 , 6 )$].”) #figure(figph[Checking if (4, 1) solves 3x + 2y = 6. Substituting gives 3(4) + 2(1) = 6, which simplifies to 12 + 2 = 6. This further simplifies to 14 does not equal 6, so the ordered pair is not a solution.], alt: "Checking if (4, 1) solves 3x + 2y = 6. Substituting gives 3(4) + 2(1) = 6, which simplifies to 12 + 2 = 6. This further simplifies to 14 does not equal 6, so the ordered pair is not a solution.", 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 #emph[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] The #strong[graph of a linear equation] #math.equation(block: false, alt: "A x plus B y equals C")[$A x + B y = C$] is a line. - 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. #figure(figph[The figure shows a straight line on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line has a positive slope and goes through the y-axis at the (0, negative 3). The line is labeled with the equation y equals 2x negative 3.], alt: "The figure shows a straight line on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line has a positive slope and goes through the y-axis at the (0, negative 3). The line is labeled with the equation y equals 2x negative 3.", caption: none) For each ordered pair, decide: ⓐ Is the ordered pair a solution to the equation? #linebreak() ⓑ 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 , −3 close parenthesis")[$( 2 , −3 )$] D #math.equation(block: false, alt: "open parenthesis −1 , −5 close parenthesis")[$( −1 , −5 )$] #solutionbox[ Substitute the #emph[x]- and #emph[y]- values into the equation to check if the ordered pair is a solution to the equation. + ⓐ #linebreak() #figure(figph[The figure shows ordered pairs and the equation y equals 2x minus 3. Four ordered pairs are tested: (0, negative 3), (3, 3), (2, negative 3), and (negative 1, negative 5). Each pair is substituted into the equation to check for solutions. The results show (0, negative 3) and (3, 3) are solutions, (2, negative 3) is not a solution, and (negative 1, negative 5) is a solution.], alt: "The figure shows ordered pairs and the equation y equals 2x minus 3. Four ordered pairs are tested: (0, negative 3), (3, 3), (2, negative 3), and (negative 1, negative 5). Each pair is substituted into the equation to check for solutions. The results show (0, negative 3) and (3, 3) are solutions, (2, negative 3) is not a solution, and (negative 1, negative 5) is a solution.", caption: none) + ⓑ Plot the points 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 , −3 close parenthesis")[$( 2 , −3 )$], and D #math.equation(block: false, alt: "open parenthesis −1 , −5 close parenthesis")[$( −1 , −5 )$]. #linebreak() #figure(figph[A graph shows a line and four points on an x-y coordinate plane. The points are located at coordinates: negative 1, negative 5; 0, negative 3; 2, negative 3; and 3, 3. The line y equals 2x minus 3 passes through the points negative 1, negative 5 and 3, 3.], alt: "A graph shows a line and four points on an x-y coordinate plane. The points are located at coordinates: negative 1, negative 5; 0, negative 3; 2, negative 3; and 3, 3. The line y equals 2x minus 3 passes through the points negative 1, negative 5 and 3, 3.", caption: none) The points #math.equation(block: false, alt: "open parenthesis 0 , 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 −1 , −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 , −3 close parenthesis")[$( 2 , −3 )$] is not on the line. The points that 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 that is not a solution is not on the line. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use the graph of #math.equation(block: false, alt: "y equals 3 x minus 1")[$y = 3 x − 1$] to decide whether each ordered pair is: - a solution to the equation. - on the line. ⓐ #math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$( 0 , −1 )$] ⓑ #math.equation(block: false, alt: "open parenthesis 2 , 5 close parenthesis")[$( 2 , 5 )$] #figure(figph[The figure shows a straight line on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the point (negative 2, negative 7) and for every 3 units it goes up, it goes one unit to the right. The line is labeled with the equation y equals 3x minus 1.], alt: "The figure shows a straight line on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the point (negative 2, negative 7) and for every 3 units it goes up, it goes one unit to the right. The line is labeled with the equation y equals 3x minus 1.", caption: none) #solutionbox[ ⓐ yes, yes ⓑ yes, yes ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use graph of #math.equation(block: false, alt: "y equals 3 x minus 1")[$y = 3 x − 1$] to decide whether each ordered pair is: - a solution to the equation - on the line ⓐ #math.equation(block: false, alt: "open parenthesis 3 , −1 close parenthesis")[$( 3 , −1 )$] ⓑ #math.equation(block: false, alt: "open parenthesis −1 , −4 close parenthesis")[$( −1 , −4 )$] #figure(figph[The figure shows a straight line on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the point (negative 2, negative 7) and for every 3 units it goes up, it goes one unit to the right. The line is labeled with the equation y equals 3x minus 1.], alt: "The figure shows a straight line on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the point (negative 2, negative 7) and for every 3 units it goes up, it goes one unit to the right. The line is labeled with the equation y equals 3x minus 1.", caption: none) #solutionbox[ ⓐ 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 to graph #math.equation(block: false, alt: "3 x plus 2 y equals 6")[$3 x + 2 y = 6$] is called plotting points, or the Point–Plotting Method. #examplebox("Example 2")[How To Graph an Equation By Plotting Points][ Graph the equation #math.equation(block: false, alt: "y equals 2 x plus 1")[$y = 2 x + 1$] by plotting points. #solutionbox[ #figure(figph[The figure shows the three step procedure for graphing a line from the equation using the example equation y equals 2x minus 1. The first step is to “Find three points whose coordinates are solutions to the equation. Organize the solutions in a table”. The remark is made that “You can choose any values for x or y. In this case, since y is isolated on the left side of the equation, it is easier to choose values for x”. The work for the first step of the example is shown through a series of equations aligned vertically. From the top down, the equations are y equals 2x plus 1, x equals 0 (where the 0 is blue), y equals 2x plus 1, y equals 2(0) plus 1 (where the 0 is blue), y equals 0 plus 1, y equals 1, x equals 1 (where the 1 is blue), y equals 2x plus 1, y equals 2(1) plus 1 (where the 1 is blue), y equals 2 plus 1, y equals 3, x equals negative 2 (where the negative 2 is blue), y equals 2x plus 1, y equals 2(negative 2) plus 1 (where the negative 2 is blue), y equals negative 4 plus 1, y equals negative 3. The work is then organized in a table. The table has 5 rows and 3 columns. The first row is a title row with the equation y equals 2x plus 1. The second row is a header row and it labels each column. The first column header is “x”, the second is “y” and the third is “(x, y)”. Under the first column are the numbers 0, 1, and negative 2. Under the second column are the numbers 1, 3, and negative 3. Under the third column are the ordered pairs (0, 1), (1, 3), and (negative 2, negative 3).], alt: "The figure shows the three step procedure for graphing a line from the equation using the example equation y equals 2x minus 1. The first step is to “Find three points whose coordinates are solutions to the equation. Organize the solutions in a table”. The remark is made that “You can choose any values for x or y. In this case, since y is isolated on the left side of the equation, it is easier to choose values for x”. The work for the first step of the example is shown through a series of equations aligned vertically. From the top down, the equations are y equals 2x plus 1, x equals 0 (where the 0 is blue), y equals 2x plus 1, y equals 2(0) plus 1 (where the 0 is blue), y equals 0 plus 1, y equals 1, x equals 1 (where the 1 is blue), y equals 2x plus 1, y equals 2(1) plus 1 (where the 1 is blue), y equals 2 plus 1, y equals 3, x equals negative 2 (where the negative 2 is blue), y equals 2x plus 1, y equals 2(negative 2) plus 1 (where the negative 2 is blue), y equals negative 4 plus 1, y equals negative 3. The work is then organized in a table. The table has 5 rows and 3 columns. The first row is a title row with the equation y equals 2x plus 1. The second row is a header row and it labels each column. The first column header is “x”, the second is “y” and the third is “(x, y)”. Under the first column are the numbers 0, 1, and negative 2. Under the second column are the numbers 1, 3, and negative 3. Under the third column are the ordered pairs (0, 1), (1, 3), and (negative 2, negative 3).", caption: none) #figure(figph[The second step is to “Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work!” For the example the points are (0, 1), (1, 3), and (negative 2, negative 3). A graph shows the three points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the three points at (0, 1), (1, 3), and (negative 2, negative 3). The question “Do the points line up?” is stated and followed with the answer “Yes, the points line up.”], alt: "The second step is to “Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work!” For the example the points are (0, 1), (1, 3), and (negative 2, negative 3). A graph shows the three points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the three points at (0, 1), (1, 3), and (negative 2, negative 3). The question “Do the points line up?” is stated and followed with the answer “Yes, the points line up.”", caption: none) #figure(figph[The third step of the procedure is “Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.” A graph shows a straight line drawn through three points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the three points at (0, 1), (1, 3), and (negative 2, negative 3). A straight line goes through all three points. The line has arrows on both ends pointing to the edge of the figure. The line is labeled with the equation y equals 2x plus 1. The statement “This line is the graph of y equals 2x plus 1” is included next to the graph.], alt: "The third step of the procedure is “Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.” A graph shows a straight line drawn through three points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the three points at (0, 1), (1, 3), and (negative 2, negative 3). A straight line goes through all three points. The line has arrows on both ends pointing to the edge of the figure. The line is labeled with the equation y equals 2x plus 1. The statement “This line is the graph of y equals 2x plus 1” is included next to the graph.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the equation by plotting points: #math.equation(block: false, alt: "y equals 2 x minus 3")[$y = 2 x − 3$]. #solutionbox[ #figure(figph[The figure shows a straight line on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 2, negative 7), (negative 1, negative 5), (0, negative 3), (1, negative 1), (2, 1), (3, 3), (4, 5), and (5, 7). There are arrows at the ends of the line pointing to the outside of the figure.], alt: "The figure shows a straight line on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 2, negative 7), (negative 1, negative 5), (0, negative 3), (1, negative 1), (2, 1), (3, 3), (4, 5), and (5, 7). There are arrows at the ends of the line pointing to the outside of the figure.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the equation by plotting points: #math.equation(block: false, alt: "y equals −2 x plus 4")[$y = −2 x + 4$]. #solutionbox[ #figure(figph[The figure shows a straight line on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 1, 6), (0, 4), (1, 2), (2, 0), (3, negative 2), (4, negative 4), and (5, negative 6). There are arrows at the ends of the line pointing to the outside of the figure.], alt: "The figure shows a straight line on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 1, 6), (0, 4), (1, 2), (2, 0), (3, negative 2), (4, negative 4), and (5, negative 6). There are arrows at the ends of the line pointing to the outside of the figure.", caption: none) ] ] The steps to take when graphing a linear equation by plotting points are summarized below. #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 in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work. + Draw the line through the three 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 only plot 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. Look at the difference between part (a) and part (b) in . #figure(figph[Figure a shows three points with a straight line going through them. Figure b shows three points that do not lie on the same line.], alt: "Figure a shows three points with a straight line going through them. Figure b shows three points that do not lie on the same line.", caption: none) Let’s do another example. This time, we’ll show the last two steps all on one grid. #examplebox("Example 3")[][ 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. Here, again, it’s easier to choose values for #math.equation(block: false, alt: "x")[$x$]. Do you see why? #linebreak() #figure(figph[Three sets of equations demonstrate finding ordered pairs for y equals negative three times x. The first set uses x equals zero, resulting in y equals zero. The second uses x equals one, yielding y equals negative three. The third uses x equals negative two, giving y equals six.], alt: "Three sets of equations demonstrate finding ordered pairs for y equals negative three times x. The first set uses x equals zero, resulting in y equals zero. The second uses x equals one, yielding y equals negative three. The third uses x equals negative two, giving y equals six.", caption: none) We list the points in . #linebreak() #figure(table( columns: 3, align: left, inset: 6pt, [#strong[#math.equation(block: false, alt: "y equals −3 x")[$y = −3 x$]]], [], [], [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]], [0], [0], [#math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]], [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")[$−2$]], [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. #linebreak() #figure(figph[The figure shows a straight line drawn through three points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the three points which are labeled by their ordered pairs (negative 2, 6), (0, 0), and (1, negative 3). A straight line goes through all three points. The line has arrows on both ends pointing to the outside of the figure. The line is labeled with the equation y equals negative 3x.], alt: "The figure shows a straight line drawn through three points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the three points which are labeled by their ordered pairs (negative 2, 6), (0, 0), and (1, negative 3). A straight line goes through all three points. The line has arrows on both ends pointing to the outside of the figure. The line is labeled with the equation y equals negative 3x.", 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 figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 2, 8), (0, 0), and (2, negative 8). The line has arrows on both ends pointing to the outside of the figure.], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 2, 8), (0, 0), and (2, negative 8). The line has arrows on both ends pointing to the outside of the figure.", 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 figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 8, negative 8), (negative 6, negative 6), (negative 4, negative 4), (negative 2, negative 2), (0, 0), (2, 2), (4, 4), (6, 6), and (8, 8). The line has arrows on both ends pointing to the outside of the figure.], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 8, negative 8), (negative 6, negative 6), (negative 4, negative 4), (negative 2, negative 2), (0, 0), (2, 2), (4, 4), (6, 6), and (8, 8). The line has arrows on both ends pointing to the outside of the figure.", caption: none) ] ] When an equation includes a fraction as the coefficient of #math.equation(block: false, alt: "x")[$x$], we can still 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 4")[][ 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 2 for the other choices. Why are multiples of 2 a good choice for values of #math.equation(block: false, alt: "x")[$x$]? #linebreak() #figure(figph[Three sets of equations demonstrate finding ordered pairs for the equation y equals one-half x plus three. The first set uses x equals zero, resulting in y equals three. The second uses x equals two, yielding y equals four. The third uses x equals four, giving y equals five.], alt: "Three sets of equations demonstrate finding ordered pairs for the equation y equals one-half x plus three. The first set uses x equals zero, resulting in y equals three. The second uses x equals two, yielding y equals four. The third uses x equals four, giving y equals five.", caption: none) The points are shown. #figure(table( columns: 3, align: left, inset: 6pt, [#strong[#math.equation(block: false, alt: "y equals the fraction 1 over 2 x plus 3")[$y = frac(1, 2) x + 3$]]], [], [], [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]], [0], [3], [#math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$( 0 , 3 )$]], [2], [4], [#math.equation(block: false, alt: "open parenthesis 2 , 4 close parenthesis")[$( 2 , 4 )$]], [4], [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. #linebreak() #figure(figph[The figure shows a straight line drawn through three points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the three points which are labeled by their ordered pairs (0, 3), (2, 4), and (4, 5). A straight line goes through all three points. The line has arrows on both ends pointing to the outside of the figure. The line is labeled with the equation y equals (one half)x plus 3.], alt: "The figure shows a straight line drawn through three points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the three points which are labeled by their ordered pairs (0, 3), (2, 4), and (4, 5). A straight line goes through all three points. The line has arrows on both ends pointing to the outside of the figure. The line is labeled with the equation y equals (one half)x plus 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 figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 9, negative 4), (negative 6, negative 3), (negative 3, negative 2), (0, negative 1), (3, 0), (6, 1), and (9, 2). The line has arrows on both ends pointing to the outside of the figure.], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 9, negative 4), (negative 6, negative 3), (negative 3, negative 2), (0, negative 1), (3, 0), (6, 1), and (9, 2). The line has arrows on both ends pointing to the outside of the figure.", 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 figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 12, negative 1), (negative 8, 0), (negative 4, 1), (0, 2), (4, 3), (8, 4), and (12, 5). The line has arrows on both ends pointing to the outside of the figure.], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 12, negative 1), (negative 8, 0), (negative 4, 1), (0, 2), (4, 3), (8, 4), and (12, 5). The line has arrows on both ends pointing to the outside of the figure.", 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. 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 equals 0")[$y = 0$] what is the value of #math.equation(block: false, alt: "x")[$x$]? #figure(figph[The figure shows a set of equations used to determine an ordered pair from the equation 2x plus y equals 3. The first equation is y equals 0 (where the 0 is red). The second equation is the two- variable equation 2x plus y equals 3. The third equation is the onenegative variable equation 2x plus 0 equals 3 (where the 0 is red). The fourth equation is 2x equals 3. The fifth equation is x equals three halves. The last line is the ordered pair (three halves, 0).], alt: "The figure shows a set of equations used to determine an ordered pair from the equation 2x plus y equals 3. The first equation is y equals 0 (where the 0 is red). The second equation is the two- variable equation 2x plus y equals 3. The third equation is the onenegative variable equation 2x plus 0 equals 3 (where the 0 is red). The fourth equation is 2x equals 3. The fifth equation is x equals three halves. The last line is the ordered pair (three halves, 0).", caption: none) This point has a fraction for the #emph[x]- coordinate and, 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; y, equals, −2 x plus 3")[$2 x + y & = & 3 \ y & = & −2 x + 3$]The solutions for #math.equation(block: false, alt: "x equals 0")[$x = 0$], #math.equation(block: false, alt: "x equals 1")[$x = 1$], and #math.equation(block: false, alt: "x equals −1")[$x = −1$] are shown in the . The graph is shown. #figure(table( columns: 3, align: left, inset: 6pt, [#strong[#math.equation(block: false, alt: "2 x plus y equals 3")[$2 x + y = 3$]]], [], [], [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]], [0], [3], [#math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$( 0 , 3 )$]], [1], [1], [#math.equation(block: false, alt: "open parenthesis 1 , 1 close parenthesis")[$( 1 , 1 )$]], [#math.equation(block: false, alt: "−1")[$−1$]], [5], [#math.equation(block: false, alt: "open parenthesis −1 , 5 close parenthesis")[$( −1 , 5 )$]], )) #figure(figph[The figure shows a straight line drawn through three points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the three points which are labeled by their ordered pairs (negative 1, 5), (0, 3), and (1, 1). A straight line goes through all three points. The line has arrows on both ends pointing to the outside of the figure. The line is labeled with the equation 2x plus y equals 3.], alt: "The figure shows a straight line drawn through three points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the three points which are labeled by their ordered pairs (negative 1, 5), (0, 3), and (1, 1). A straight line goes through all three points. The line has arrows on both ends pointing to the outside of the figure. The line is labeled with the equation 2x plus y equals 3.", caption: none) Can you locate the point #math.equation(block: false, alt: "open parenthesis the fraction 3 over 2 , 0 close parenthesis")[$( frac(3, 2) , 0 )$], which we found by letting #math.equation(block: false, alt: "y equals 0")[$y = 0$], on the line? #examplebox("Example 5")[][ Graph the equation #math.equation(block: false, alt: "3 x plus y equals −1")[$3 x + y = −1$]. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Find three points that are solutions to the equation.], [#math.equation(block: false, alt: "3 x plus y equals −1")[$3 x + y #h(0.5em) = #h(0.5em) −1$]]), [First, solve the equation for #math.equation(block: false, alt: "y")[$y$].], [#math.equation(block: false, alt: "y equals −3 x minus 1")[$y #h(0.5em) = #h(0.5em) −3 x − 1$]], )) We’ll let #math.equation(block: false, alt: "x")[$x$] be 0, 1, and #math.equation(block: false, alt: "−1")[$−1$] to find 3 points. The ordered pairs are shown. Plot the points, check that they line up, and draw the line. #figure(table( columns: 3, align: left, inset: 6pt, [#strong[#math.equation(block: false, alt: "3 x plus y equals −1")[$3 x + y = −1$]]], [], [], [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]], [0], [#math.equation(block: false, alt: "−1")[$−1$]], [#math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$( 0 , −1 )$]], [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")[$−1$]], [2], [#math.equation(block: false, alt: "open parenthesis −1 , 2 close parenthesis")[$( −1 , 2 )$]], )) #figure(figph[A straight line passes through the points negative 1 comma 2, 0 comma negative 1, and 1 comma negative 4 on a coordinate plane. The line's equation is three x plus y equals negative one, indicating it continues infinitely in both directions.], alt: "A straight line passes through the points negative 1 comma 2, 0 comma negative 1, and 1 comma negative 4 on a coordinate plane. The line's equation is three x plus y equals negative one, indicating it continues infinitely in both directions.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the equation #math.equation(block: false, alt: "2 x plus y equals 2")[$2 x + y = 2$]. #solutionbox[ #figure(figph[The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 4, 10), (negative 2, 6), (0, 2), (2, negative 2), (4, negative 6), and (6, negative 10). The line has arrows on both ends pointing to the outside of the figure.], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 4, 10), (negative 2, 6), (0, 2), (2, negative 2), (4, negative 6), and (6, negative 10). The line has arrows on both ends pointing to the outside of the figure.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the equation #math.equation(block: false, alt: "4 x plus y equals −3")[$4 x + y = −3$]. #solutionbox[ #figure(figph[The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 3, 9), (negative 2, 5), (negative 1, 1), (0, negative 3), (1, negative 7), and (2, negative 10). The line has arrows on both ends pointing to the outside of the figure.], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 3, 9), (negative 2, 5), (negative 1, 1), (0, negative 3), (1, negative 7), and (2, negative 10). The line has arrows on both ends pointing to the outside of the figure.", 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 #emph[x]- and #emph[y]-axes are the same, the graphs match! The equation in was written in standard form, with both #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$] on the same side. We solved that equation for #math.equation(block: false, alt: "y")[$y$] in just one step. But for other equations in standard form it is not that easy to solve for #math.equation(block: false, alt: "y")[$y$], so we will leave them in standard form. We can still find a first point to plot by letting #math.equation(block: false, alt: "x equals 0")[$x = 0$] and solving for #math.equation(block: false, alt: "y")[$y$]. We can plot a second point by letting #math.equation(block: false, alt: "y equals 0")[$y = 0$] and then solving for #math.equation(block: false, alt: "x")[$x$]. Then we will plot a third point by using some other value for #math.equation(block: false, alt: "x")[$x$] or #math.equation(block: false, alt: "y")[$y$]. #examplebox("Example 6")[][ Graph the equation #math.equation(block: false, alt: "2 x minus 3 y equals 6")[$2 x − 3 y = 6$]. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Find three points that are solutions to the equation.], [#math.equation(block: false, alt: "2 x minus 3 y, equals, 6")[$2 x − 3 y & = & 6$]]), [First, let #math.equation(block: false, alt: "x equals 0")[$x = 0$].], [#math.equation(block: false, alt: "2 open parenthesis 0 close parenthesis minus 3 y, equals, 6")[$2 ( 0 ) − 3 y & = & 6$]], [Solve for #math.equation(block: false, alt: "y")[$y$].], [#math.equation(block: false, alt: "−3 y, equals, 6; y, equals, −2")[$−3 y & = & 6 \ y & = & −2$]], [Now let #math.equation(block: false, alt: "y equals 0")[$y = 0$].], [#math.equation(block: false, alt: "2 x minus 3 open parenthesis 0 close parenthesis, equals, 6")[$2 x − 3 ( 0 ) & = & 6$]], [Solve for #math.equation(block: false, alt: "x")[$x$].], [#math.equation(block: false, alt: "2 x, equals, 6; x, equals, 3")[$2 x & = & 6 \ x & = & 3$]], [We need a third point. Remember, we can choose any value for #math.equation(block: false, alt: "x")[$x$] or #math.equation(block: false, alt: "y")[$y$]. We'll let #math.equation(block: false, alt: "x equals 6")[$x = 6$].], [#math.equation(block: false, alt: "2 open parenthesis 6 close parenthesis minus 3 y, equals, 6")[$2 ( 6 ) − 3 y & = & 6$]], [Solve for #math.equation(block: false, alt: "y")[$y$].], [#math.equation(block: false, alt: "12 minus 3 y, equals, 6; −3 y, equals, −6; y, equals, 2")[$12 − 3 y & = & 6 \ −3 y & = & −6 \ y & = & 2$]], )) We list the ordered pairs in . Plot the points, check that they line up, and draw the line. #figure(table( columns: 3, align: left, inset: 6pt, [#strong[#math.equation(block: false, alt: "2 x minus 3 y equals 6")[$2 x − 3 y = 6$]]], [], [], [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]], [0], [#math.equation(block: false, alt: "−2")[$−2$]], [#math.equation(block: false, alt: "open parenthesis 0 , −2 close parenthesis")[$( 0 , −2 )$]], [3], [0], [#math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis")[$( 3 , 0 )$]], [6], [2], [#math.equation(block: false, alt: "open parenthesis 6 , 2 close parenthesis")[$( 6 , 2 )$]], )) #figure(figph[The figure shows a straight line drawn through three points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the three points which are labeled by their ordered pairs (0, negative 2), (3, 0), and (6, 2). A straight line goes through all three points. The line has arrows on both ends pointing to the outside of the figure. The line is labeled with the equation 2x minus 3y equals 6.], alt: "The figure shows a straight line drawn through three points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the three points which are labeled by their ordered pairs (0, negative 2), (3, 0), and (6, 2). A straight line goes through all three points. The line has arrows on both ends pointing to the outside of the figure. The line is labeled with the equation 2x minus 3y equals 6.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the equation #math.equation(block: false, alt: "4 x plus 2 y equals 8")[$4 x + 2 y = 8$]. #solutionbox[ #figure(figph[The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 1, 6), (0, 4), (1, 2), (2, 0), (3, negative 2), and (4, negative 4). The line has arrows on both ends pointing to the outside of the figure.], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 1, 6), (0, 4), (1, 2), (2, 0), (3, negative 2), and (4, negative 4). The line has arrows on both ends pointing to the outside of the figure.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the equation #math.equation(block: false, alt: "2 x minus 4 y equals 8")[$2 x − 4 y = 8$]. #solutionbox[ #figure(figph[The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 6, negative 5), (negative 4, negative 4), (negative 2, negative 3), (0, negative 2), (2, negative 1), (4, 0), and (6, 1). The line has arrows on both ends pointing to the outside of the figure.], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 6, negative 5), (negative 4, negative 4), (negative 2, negative 3), (0, negative 2), (2, negative 1), (4, 0), and (6, 1). The line has arrows on both ends pointing to the outside of the figure.", 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")[$x = −3$]. This equation has only one variable, #math.equation(block: false, alt: "x")[$x$]. The equation says that #math.equation(block: false, alt: "x")[$x$] is #emph[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")[$−3$]. So to make a table of values, write #math.equation(block: false, alt: "−3")[$−3$] in 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 points on our coordinate graph, we’ll use 1, 2, and 3 for the #emph[y]-coordinates. #figure(table( columns: 3, align: left, inset: 6pt, [#strong[#math.equation(block: false, alt: "x equals −3")[$x = −3$]]], [], [], [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]], [#math.equation(block: false, alt: "−3")[$−3$]], [1], [#math.equation(block: false, alt: "open parenthesis −3 , 1 close parenthesis")[$( −3 , 1 )$]], [#math.equation(block: false, alt: "−3")[$−3$]], [2], [#math.equation(block: false, alt: "open parenthesis −3 , 2 close parenthesis")[$( −3 , 2 )$]], [#math.equation(block: false, alt: "−3")[$−3$]], [3], [#math.equation(block: false, alt: "open parenthesis −3 , 3 close parenthesis")[$( −3 , 3 )$]], )) Plot the points from and connect them with a straight line. Notice in that we have graphed a #emph[vertical line]. #figure(figph[A vertical line passes through the points negative 3 comma 1, negative 3 comma 2, and negative 3 comma 3 on a coordinate plane. The line represents x equals negative 3.], alt: "A vertical line passes through the points negative 3 comma 1, negative 3 comma 2, and negative 3 comma 3 on a coordinate plane. The line represents x equals negative 3.", caption: none) #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Vertical Line] A #strong[vertical line] is the graph of an equation of the form #math.equation(block: false, alt: "x equals a")[$x = a$]. The line passes through the #emph[x]-axis at #math.equation(block: false, alt: "open parenthesis a , 0 close parenthesis")[$( a , 0 )$]. ] #examplebox("Example 7")[][ Graph the equation #math.equation(block: false, alt: "x equals 2")[$x = 2$]. #solutionbox[ The equation has only one variable, #math.equation(block: false, alt: "x")[$x$], and #math.equation(block: false, alt: "x")[$x$] is always equal to 2. We create where #math.equation(block: false, alt: "x")[$x$] is always 2 and then put in any values for #math.equation(block: false, alt: "y")[$y$]. The graph is a vertical line passing through the #emph[x]-axis at 2. #figure(table( columns: 3, align: left, inset: 6pt, [#strong[#math.equation(block: false, alt: "x equals 2")[$x = 2$]]], [], [], [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]], [2], [1], [#math.equation(block: false, alt: "open parenthesis 2 , 1 close parenthesis")[$( 2 , 1 )$]], [2], [2], [#math.equation(block: false, alt: "open parenthesis 2 , 2 close parenthesis")[$( 2 , 2 )$]], [2], [3], [#math.equation(block: false, alt: "open parenthesis 2 , 3 close parenthesis")[$( 2 , 3 )$]], )) #figure(figph[The figure shows a straight vertical line drawn through three points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the three points which are labeled by their ordered pairs (2, 1), (2, 2), and (2, 3). A vertical straight line goes through all three points. The line has arrows on both ends pointing to the outside of the figure. The line is labeled with the equation x equals 2.], alt: "The figure shows a straight vertical line drawn through three points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the three points which are labeled by their ordered pairs (2, 1), (2, 2), and (2, 3). A vertical straight line goes through all three points. The line has arrows on both ends pointing to the outside of the figure. The line is labeled with the equation x equals 2.", caption: none) ] ] #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 figure shows a straight vertical line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (5, 1), (5, 2), (5, 3), and all other points with first coordinate 5. The line has arrows on both ends pointing to the outside of the figure.], alt: "The figure shows a straight vertical line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (5, 1), (5, 2), (5, 3), and all other points with first coordinate 5. The line has arrows on both ends pointing to the outside of the figure.", 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 figure shows a straight vertical line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 2, 1), (negative 2, 2), (negative 2, 3), and all other points with first coordinate negative 2. The line has arrows on both ends pointing to the outside of the figure.], alt: "The figure shows a straight vertical line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 2, 1), (negative 2, 2), (negative 2, 3), and all other points with first coordinate negative 2. The line has arrows on both ends pointing to the outside of the figure.", 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 #emph[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")[$x$]. Fill in 4 for all the #math.equation(block: false, alt: "y")[$y$]’s in and then choose any values for #math.equation(block: false, alt: "x")[$x$]. We’ll use 0, 2, and 4 for the #emph[x]-coordinates. #figure(table( columns: 3, align: left, inset: 6pt, [#strong[#math.equation(block: false, alt: "y equals 4")[$y = 4$]]], [], [], [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]], [0], [4], [#math.equation(block: false, alt: "open parenthesis 0 , 4 close parenthesis")[$( 0 , 4 )$]], [2], [4], [#math.equation(block: false, alt: "open parenthesis 2 , 4 close parenthesis")[$( 2 , 4 )$]], [4], [4], [#math.equation(block: false, alt: "open parenthesis 4 , 4 close parenthesis")[$( 4 , 4 )$]], )) The graph is a horizontal line passing through the #emph[y]-axis at 4. #figure(figph[The figure shows a straight horizontal line drawn through three points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the three points which are labeled by their ordered pairs (0, 4), (2, 4), and (4, 4). A straight horizontal line goes through all three points. The line has arrows on both ends pointing to the outside of the figure. The line is labeled with the equation y equals 4.], alt: "The figure shows a straight horizontal line drawn through three points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the three points which are labeled by their ordered pairs (0, 4), (2, 4), and (4, 4). A straight horizontal line goes through all three points. The line has arrows on both ends pointing to the outside of the figure. The line is labeled with the equation y equals 4.", caption: none) #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Horizontal Line] A #strong[horizontal line] is the graph of an equation of the form #math.equation(block: false, alt: "y equals b")[$y = b$]. The line passes through the #emph[y]-axis at #math.equation(block: false, alt: "open parenthesis 0 , b close parenthesis")[$( 0 , b )$]. ] #examplebox("Example 8")[][ 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 one 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 have the same #emph[y]-coordinate. The graph is a horizontal line passing through the #emph[y]-axis at #math.equation(block: false, alt: "−1")[$−1$], as shown. #figure(table( columns: 3, align: left, inset: 6pt, [#strong[#math.equation(block: false, alt: "y equals −1")[$y = −1$]]], [], [], [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]], [0], [#math.equation(block: false, alt: "−1")[$−1$]], [#math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$( 0 , −1 )$]], [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: "−1")[$−1$]], [#math.equation(block: false, alt: "open parenthesis −3 , −1 close parenthesis")[$( −3 , −1 )$]], )) #figure(figph[A graph shows a horizontal line at y equals negative 1. Three points are marked on the line: negative 3, negative 1; 0, negative 1; and 3, negative 1. The x and y axes range from negative 7 to positive 7.], alt: "A graph shows a horizontal line at y equals negative 1. Three points are marked on the line: negative 3, negative 1; 0, negative 1; and 3, negative 1. The x and y axes range from negative 7 to positive 7.", 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 figure shows a straight horizontal line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 4, negative 4), (0, negative 4), (4, negative 4), and all other points with second coordinate negative 4. The line has arrows on both ends pointing to the outside of the figure.], alt: "The figure shows a straight horizontal line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 4, negative 4), (0, negative 4), (4, negative 4), and all other points with second coordinate negative 4. The line has arrows on both ends pointing to the outside of the figure.", 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 figure shows a straight horizontal line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 4, 3), (0, 3), (4, 3), and all other points with second coordinate 3. The line has arrows on both ends pointing to the outside of the figure.], alt: "The figure shows a straight horizontal line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 4, 3), (0, 3), (4, 3), and all other points with second coordinate 3. The line has arrows on both ends pointing to the outside of the figure.", 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 #emph[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 #emph[y]-coordinate is always 4. It does not depend on the value of #math.equation(block: false, alt: "x")[$x$]. #figure(table( columns: 6, align: left, inset: 6pt, [#strong[#math.equation(block: false, alt: "y equals 4 x")[$y = 4 x$]]], [], [#strong[#math.equation(block: false, alt: "y equals 4")[$y = 4$]]], [], [], [], [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]], [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]], [0], [0], [#math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]], [0], [4], [#math.equation(block: false, alt: "open parenthesis 0 , 4 close parenthesis")[$( 0 , 4 )$]], [1], [4], [#math.equation(block: false, alt: "open parenthesis 1 , 4 close parenthesis")[$( 1 , 4 )$]], [1], [4], [#math.equation(block: false, alt: "open parenthesis 1 , 4 close parenthesis")[$( 1 , 4 )$]], [2], [8], [#math.equation(block: false, alt: "open parenthesis 2 , 8 close parenthesis")[$( 2 , 8 )$]], [2], [4], [#math.equation(block: false, alt: "open parenthesis 2 , 4 close parenthesis")[$( 2 , 4 )$]], )) #figure(figph[The figure shows a two straight lines drawn on the same x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. One line is a straight horizontal line labeled with the equation y equals 4. The other line is a slanted line labeled with the equation y equals 4x.], alt: "The figure shows a two straight lines drawn on the same x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. One line is a straight horizontal line labeled with the equation y equals 4. The other line is a slanted line labeled with the equation y equals 4x.", caption: none) Notice, in , the equation #math.equation(block: false, alt: "y equals 4 x")[$y = 4 x$] gives a slanted line, while #math.equation(block: false, alt: "y equals 4")[$y = 4$] gives a horizontal line. #examplebox("Example 9")[][ 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[ Notice that the first equation has the variable #math.equation(block: false, alt: "x")[$x$], while the second does not. The two graphs are shown. #figure(table( columns: 6, align: left, inset: 6pt, [#strong[#math.equation(block: false, alt: "y equals −3 x")[$y = −3 x$]]], [], [#strong[#math.equation(block: false, alt: "y equals −3")[$y = −3$]]], [], [], [], [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]], [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]], [0], [0], [#math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]], [0], [#math.equation(block: false, alt: "−3")[$−3$]], [#math.equation(block: false, alt: "open parenthesis 0 , −3 close parenthesis")[$( 0 , −3 )$]], [1], [#math.equation(block: false, alt: "−3")[$−3$]], [#math.equation(block: false, alt: "open parenthesis 1 , −3 close parenthesis")[$( 1 , −3 )$]], [1], [#math.equation(block: false, alt: "−3")[$−3$]], [#math.equation(block: false, alt: "open parenthesis 1 , −3 close parenthesis")[$( 1 , −3 )$]], [2], [#math.equation(block: false, alt: "−6")[$−6$]], [#math.equation(block: false, alt: "open parenthesis 2 , −6 close parenthesis")[$( 2 , −6 )$]], [2], [#math.equation(block: false, alt: "−3")[$−3$]], [#math.equation(block: false, alt: "open parenthesis 2 , −3 close parenthesis")[$( 2 , −3 )$]], )) #figure(figph[The figure shows a two straight lines drawn on the same x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. One line is a straight horizontal line labeled with the equation y equals negative 3. The other line is a slanted line labeled with the equation y equals negative 3x.], alt: "The figure shows a two straight lines drawn on the same x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. One line is a straight horizontal line labeled with the equation y equals negative 3. The other line is a slanted line labeled with the equation y equals negative 3x.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph #math.equation(block: false, alt: "y equals −4 x")[$y = −4 x$] and #math.equation(block: false, alt: "y equals −4")[$y = −4$] in the same rectangular coordinate system. #solutionbox[ #figure(figph[The figure shows a two straight lines drawn on the same x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. One line is a straight horizontal line going through the points (negative 4, negative 4), (0, negative 4), (4, negative 4), and all other points with second coordinate negative 4. The other line is a slanted line going through the points (negative 2, 8), (negative 1, 4), (0, 0), (1, negative 4), and (2, negative 8).], alt: "The figure shows a two straight lines drawn on the same x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. One line is a straight horizontal line going through the points (negative 4, negative 4), (0, negative 4), (4, negative 4), and all other points with second coordinate negative 4. The other line is a slanted line going through the points (negative 2, 8), (negative 1, 4), (0, 0), (1, negative 4), and (2, negative 8).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph #math.equation(block: false, alt: "y equals 3")[$y = 3$] and #math.equation(block: false, alt: "y equals 3 x")[$y = 3 x$] in the same rectangular coordinate system. #solutionbox[ #figure(figph[The figure shows a two straight lines drawn on the same x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. One line is a straight horizontal line going through the points (negative 4, 3) (0, 3), (4, 3), and all other points with second coordinate 3. The other line is a slanted line going through the points (negative 2, negative 6), (negative 1, negative 3), (0, 0), (1, 3), and (2, 6).], alt: "The figure shows a two straight lines drawn on the same x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. One line is a straight horizontal line going through the points (negative 4, 3) (0, 3), (4, 3), and all other points with second coordinate 3. The other line is a slanted line going through the points (negative 2, negative 6), (negative 1, negative 3), (0, 0), (1, 3), and (2, 6).", caption: none) ] ] === Key Concepts - #strong[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 in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work! + Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line. ==== Practice Makes Perfect #strong[Recognize the Relationship Between the Solutions of an Equation and its Graph] In the following exercises, 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")[$y = x + 2$] + ⓐ #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 −1 , 1 close parenthesis")[$( −1 , 1 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis −3 , −1 close parenthesis")[$( −3 , −1 )$] #figure(figph[The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 6, negative 4), (negative 5, negative 3), (negative 4, negative 2), (negative 3, negative 1), (negative 2, 0), (negative 1, 1), (0, 2), (1, 3), (2, 4), (3, 5), (4, 6), and (5, 7).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 6, negative 4), (negative 5, negative 3), (negative 4, negative 2), (negative 3, negative 1), (negative 2, 0), (negative 1, 1), (0, 2), (1, 3), (2, 4), (3, 5), (4, 6), and (5, 7).", caption: none) #solutionbox[ ⓐ yes; yes ⓑ no; no ⓒ yes; yes ⓓ yes; yes ] #math.equation(block: true, alt: "y equals x minus 4")[$y = x − 4$] + ⓐ #math.equation(block: false, alt: "open parenthesis 0 , −4 close parenthesis")[$( 0 , −4 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 3 , −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 , −5 close parenthesis")[$( 1 , −5 )$] #figure(figph[The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 3, negative 7), (negative 2, negative 6), (negative 1, negative 5), (0, negative 4), (1, negative 3), (2, negative 2), (3, negative 1), (4, 0), (5, 1), (6, 2), and (7, 3).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 3, negative 7), (negative 2, negative 6), (negative 1, negative 5), (0, negative 4), (1, negative 3), (2, negative 2), (3, negative 1), (4, 0), (5, 1), (6, 2), and (7, 3).", caption: none) #math.equation(block: true, alt: "y equals the fraction 1 over 2 x minus 3")[$y = frac(1, 2) x − 3$] + ⓐ #math.equation(block: false, alt: "open parenthesis 0 , −3 close parenthesis")[$( 0 , −3 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 2 , −2 close parenthesis")[$( 2 , −2 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis −2 , −4 close parenthesis")[$( −2 , −4 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis 4 , 1 close parenthesis")[$( 4 , 1 )$] #figure(figph[The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 6, negative 6), (negative 4, negative 5), (negative 2, negative 4), (0, negative 3), (2, negative 2), (4, negative 1), and (6, 0).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 6, negative 6), (negative 4, negative 5), (negative 2, negative 4), (0, negative 3), (2, negative 2), (4, negative 1), and (6, 0).", caption: none) #solutionbox[ ⓐ yes; yes ⓑ yes; yes ⓒ yes; yes ⓓ no; no ] #math.equation(block: true, alt: "y equals the fraction 1 over 3 x plus 2")[$y = frac(1, 3) x + 2$] + ⓐ #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 −3 , 2 close parenthesis")[$( −3 , 2 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis −6 , 0 close parenthesis")[$( −6 , 0 )$] #figure(figph[The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 6, 0), (negative 3, 1), (0, 2), (3, 3), and (6, 4).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 6, 0), (negative 3, 1), (0, 2), (3, 3), and (6, 4).", caption: none) #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 figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 3, negative 10), (negative 2, negative 7), (negative 1, negative 4), (0, negative 1), (1, 2), (2, 5), and (3, 8).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 3, negative 10), (negative 2, negative 7), (negative 1, negative 4), (0, negative 1), (1, 2), (2, 5), and (3, 8).", 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 figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 4, 10), (negative 3, 8), (negative 2, 6), (negative 1, 4), (0, 2), (1, 0), (2, negative 2), (3, negative 4), (4, negative 6), and (5, negative 8).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 4, 10), (negative 3, 8), (negative 2, 6), (negative 1, 4), (0, 2), (1, 0), (2, negative 2), (3, negative 4), (4, negative 6), and (5, negative 8).", 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 figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 10, negative 8), (negative 9, negative 7), (negative 8, negative 6), (negative 7, negative 5), (negative 6, negative 4), (negative 5, negative 3), (negative 4, negative 2), (negative 3, negative 1), (negative 2, 0), (negative 1, 1), (0, 2), (1, 3), (2, 4), (3, 5), (4, 6), (5, 7), (6, 8), (7, 9), and (8, 10).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 10, negative 8), (negative 9, negative 7), (negative 8, negative 6), (negative 7, negative 5), (negative 6, negative 4), (negative 5, negative 3), (negative 4, negative 2), (negative 3, negative 1), (negative 2, 0), (negative 1, 1), (0, 2), (1, 3), (2, 4), (3, 5), (4, 6), (5, 7), (6, 8), (7, 9), and (8, 10).", caption: none) ] #math.equation(block: true, alt: "y equals x minus 3")[$y = x − 3$] #math.equation(block: true, alt: "y equals − x minus 3")[$y = "−" x − 3$] #solutionbox[ #figure(figph[The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 10, 7), (negative 9, 6), (negative 8, 5), (negative 7, 4), (negative 6, 3), (negative 5, 2), (negative 4, 1), (negative 3, 0), (negative 2, negative 1), (negative 1, negative 2), (0, negative 3), (1, negative 4), (2, negative 5), (3, negative 6), (4, negative 7), (5, negative 8), (6, negative 9), and (7, negative 10).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 10, 7), (negative 9, 6), (negative 8, 5), (negative 7, 4), (negative 6, 3), (negative 5, 2), (negative 4, 1), (negative 3, 0), (negative 2, negative 1), (negative 1, negative 2), (0, negative 3), (1, negative 4), (2, negative 5), (3, negative 6), (4, negative 7), (5, negative 8), (6, negative 9), and (7, negative 10).", caption: none) ] #math.equation(block: true, alt: "y equals − x minus 2")[$y = "−" x − 2$] #math.equation(block: true, alt: "y equals 2 x")[$y = 2 x$] #solutionbox[ #figure(figph[The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 5, negative 10), (negative 4, negative 8), (negative 3, negative 6), (negative 2, negative 4), (negative 1, negative 2), (0, 0), (1, 2), (2, 4), (3, 6), (4, 8), and (5, 10).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 5, negative 10), (negative 4, negative 8), (negative 3, negative 6), (negative 2, negative 4), (negative 1, negative 2), (0, 0), (1, 2), (2, 4), (3, 6), (4, 8), and (5, 10).", 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 figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 3, 12), (negative 2, 8), (negative 1, 4), (0, 0), (1, negative 4), (2, negative 8), and (3, negative 12).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 3, 12), (negative 2, 8), (negative 1, 4), (0, 0), (1, negative 4), (2, negative 8), and (3, negative 12).", 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 figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 12, negative 4), (negative 10, negative 3), (negative 8, negative 2), (negative 6, negative 1), (negative 4, 0), (negative 2, 1), (0, 2), (2, 3), (4, 4), (6, 5), (8, 6), and (10, 7).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 12, negative 4), (negative 10, negative 3), (negative 8, negative 2), (negative 6, negative 1), (negative 4, 0), (negative 2, 1), (0, 2), (2, 3), (4, 4), (6, 5), (8, 6), and (10, 7).", 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 figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 3, negative 9), (0, negative 5), (3, negative 1), (6, 3), and (9, 7).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 3, negative 9), (0, negative 5), (3, negative 1), (6, 3), and (9, 7).", 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 figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 10, 5), (negative 5, 3), (0, 1), (5, negative 1), and (10, negative 3).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 10, 5), (negative 5, 3), (0, 1), (5, negative 1), and (10, negative 3).", 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 figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 6, 11), (negative 4, 8), (negative 2, 5), (0, 2), (2, negative 1), (4, negative 4), (6, negative 7), and (8, negative 10).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 6, 11), (negative 4, 8), (negative 2, 5), (0, 2), (2, negative 1), (4, negative 4), (6, negative 7), and (8, negative 10).", 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 figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 4, 10), (negative 3, 9), (negative 2, 8), (negative 1, 7), (0, 6), (1, 5), (2, 4), (3, 3), (4, 2), (5, 1), (6, 0), (7, negative 1), (8, negative 2), (9, negative 3), and (10, negative 4).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 4, 10), (negative 3, 9), (negative 2, 8), (negative 1, 7), (0, 6), (1, 5), (2, 4), (3, 3), (4, 2), (5, 1), (6, 0), (7, negative 1), (8, negative 2), (9, negative 3), and (10, negative 4).", 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 figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 10, 7), (negative 9, 6), (negative 8, 5), (negative 7, 4), (negative 6, 3), (negative 5, 2), (negative 4, 1), (negative 3, 0), (negative 2, negative 1), (negative 1, negative 2), (0, negative 3), (1, negative 4), (2, negative 5), (3, negative 6), (4, negative 7), (5, negative 8), (6, negative 9), and (7, negative 10).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 10, 7), (negative 9, 6), (negative 8, 5), (negative 7, 4), (negative 6, 3), (negative 5, 2), (negative 4, 1), (negative 3, 0), (negative 2, negative 1), (negative 1, negative 2), (0, negative 3), (1, negative 4), (2, negative 5), (3, negative 6), (4, negative 7), (5, negative 8), (6, negative 9), and (7, negative 10).", 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 figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 8, negative 10), (negative 7, negative 9), (negative 6, negative 8), (negative 5, negative 7), (negative 4, negative 6), (negative 3, negative 5), (negative 2, negative 4), (negative 1, negative 3), (0, negative 2), (1, negative 1), (2, 0), (3, 1), (4, 2), (5, 3), (6, 4), (7, 5), (8, 6), (9, 7), and (10, 8).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 8, negative 10), (negative 7, negative 9), (negative 6, negative 8), (negative 5, negative 7), (negative 4, negative 6), (negative 3, negative 5), (negative 2, negative 4), (negative 1, negative 3), (0, negative 2), (1, negative 1), (2, 0), (3, 1), (4, 2), (5, 3), (6, 4), (7, 5), (8, 6), (9, 7), and (10, 8).", 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 figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 9, negative 8), (negative 8, negative 7), (negative 7, negative 6), (negative 6, negative 5), (negative 5, negative 4), (negative 4, negative 3), (negative 3, negative 2), (negative 2, negative 1), (negative 1, 0), (0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 8), (8, 9), and (9, 10).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 9, negative 8), (negative 8, negative 7), (negative 7, negative 6), (negative 6, negative 5), (negative 5, negative 4), (negative 4, negative 3), (negative 3, negative 2), (negative 2, negative 1), (negative 1, 0), (0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 8), (8, 9), and (9, 10).", caption: none) ] #math.equation(block: true, alt: "x minus y equals −3")[$x − y = −3$] #math.equation(block: true, alt: "3 x plus y equals 7")[$3 x + y = 7$] #solutionbox[ #figure(figph[The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to -7. The equation 3 x plus y equals 7 is graphed.], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to -7. The equation 3 x plus y equals 7 is graphed.", 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 figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 5, 7), (negative 4, 5), (negative 3, 3), (negative 2, 1), (negative 1, negative 1), (0, negative 3), (1, negative 5), and (2, negative 7).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 5, 7), (negative 4, 5), (negative 3, 3), (negative 2, 1), (negative 1, negative 1), (0, negative 3), (1, negative 5), and (2, negative 7).", caption: none) ] #math.equation(block: true, alt: "4 x plus y equals −5")[$4 x + y = −5$] #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 figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 6, 4), (negative 3, 3), (0, 2), (3, 1), and (6, 0).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 6, 4), (negative 3, 3), (0, 2), (3, 1), and (6, 0).", caption: none) ] #math.equation(block: true, alt: "the fraction 1 over 2 x plus y equals 3")[$frac(1, 2) x + y = 3$] #math.equation(block: true, alt: "the fraction 2 over 5 x minus y equals 4")[$frac(2, 5) x − y = 4$] #solutionbox[ #figure(figph[The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 5, negative 2), (0, negative 4), and (5, negative 6).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 5, negative 2), (0, negative 4), and (5, negative 6).", caption: none) ] #math.equation(block: true, alt: "the fraction 3 over 4 x minus y equals 6")[$frac(3, 4) x − y = 6$] #math.equation(block: true, alt: "2 x plus 3 y equals 12")[$2 x + 3 y = 12$] #solutionbox[ #figure(figph[The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 3, 6), (0, 4), (3, 2), and (6, 0).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 3, 6), (0, 4), (3, 2), and (6, 0).", caption: none) ] #math.equation(block: true, alt: "4 x plus 2 y equals 12")[$4 x + 2 y = 12$] #math.equation(block: true, alt: "3 x minus 4 y equals 12")[$3 x − 4 y = 12$] #solutionbox[ #figure(figph[The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 4, negative 6), (0, negative 3), (4, 0), and (8, 3).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 4, negative 6), (0, negative 3), (4, 0), and (8, 3).", caption: none) ] #math.equation(block: true, alt: "2 x minus 5 y equals 10")[$2 x − 5 y = 10$] #math.equation(block: true, alt: "x minus 6 y equals 3")[$x − 6 y = 3$] #solutionbox[ #figure(figph[The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 6, negative three halves), (negative 3, negative 1), (0, negative one half), (3, 0), and (6, one half).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 6, negative three halves), (negative 3, negative 1), (0, negative one half), (3, 0), and (6, one half).", caption: none) ] #math.equation(block: true, alt: "x minus 4 y equals 2")[$x − 4 y = 2$] #math.equation(block: true, alt: "5 x plus 2 y equals 4")[$5 x + 2 y = 4$] #solutionbox[ #figure(figph[The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 2, 7), (0, 2), (2, negative 3), and (4, negative 8).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 2, 7), (0, 2), (2, negative 3), and (4, negative 8).", caption: none) ] #math.equation(block: true, alt: "3 x plus 5 y equals 5")[$3 x + 5 y = 5$] #strong[Graph Vertical and Horizontal Lines] In the following exercises, graph each equation. #math.equation(block: true, alt: "x equals 4")[$x = 4$] #solutionbox[ #figure(figph[The figure shows a straight vertical line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The vertical line goes through the points (4, 0), (4, 1), (4, 2) and all points with first coordinate 4.], alt: "The figure shows a straight vertical line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The vertical line goes through the points (4, 0), (4, 1), (4, 2) and all points with first coordinate 4.", 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 figure shows a straight vertical line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The vertical line goes through the points (negative 2, 0), (negative 2, 1), (negative 2, 2) and all points with first coordinate negative 2.], alt: "The figure shows a straight vertical line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The vertical line goes through the points (negative 2, 0), (negative 2, 1), (negative 2, 2) and all points with first coordinate negative 2.", 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 figure shows a straight horizontal line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The horizontal line goes through the points (0, 3), (1, 3), (2, 3) and all points with second coordinate 3.], alt: "The figure shows a straight horizontal line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The horizontal line goes through the points (0, 3), (1, 3), (2, 3) and all points with second coordinate 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 figure shows a straight horizontal line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The horizontal line goes through the points (0, negative 5), (1, negative 5), (2, negative 5) and all points with second coordinate negative 5.], alt: "The figure shows a straight horizontal line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The horizontal line goes through the points (0, negative 5), (1, negative 5), (2, negative 5) and all points with second coordinate negative 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 figure shows a straight vertical line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The vertical line goes through the points (7/3, 0), (7/3, 1), (7/3, 2) and all points with first coordinate 7/3.], alt: "The figure shows a straight vertical line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The vertical line goes through the points (7/3, 0), (7/3, 1), (7/3, 2) and all points with first coordinate 7/3.", caption: none) ] #math.equation(block: true, alt: "x equals the fraction 5 over 4")[$x = frac(5, 4)$] #math.equation(block: true, alt: "y equals minus the fraction 15 over 4")[$y = − frac(15, 4)$] #solutionbox[ #figure(figph[The figure shows a straight horizontal line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The horizontal line goes through the points (0, negative 15/4), (1, negative 15/4), (2, negative 15/4) and all points with second coordinate negative 15/4.], alt: "The figure shows a straight horizontal line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The horizontal line goes through the points (0, negative 15/4), (1, negative 15/4), (2, negative 15/4) and all points with second coordinate negative 15/4.", caption: none) ] #math.equation(block: true, alt: "y equals minus the fraction 5 over 3")[$y = − frac(5, 3)$] In the following exercises, graph each pair of equations in the same rectangular coordinate system. #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 figure shows a two straight lines drawn on the same x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. One line is a straight horizontal line going through the points (negative 4, 2) (0, 2), (4, 2), and all other points with second coordinate 2. The other line is a slanted line going through the points (negative 5, negative 10), (negative 4, negative 8), (negative 3, negative 6), (negative 2, negative 4), (negative 1, negative 2), (0, 0), (1, 2), (2, 4), (3, 6), (4, 8), and (5, 10).], alt: "The figure shows a two straight lines drawn on the same x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. One line is a straight horizontal line going through the points (negative 4, 2) (0, 2), (4, 2), and all other points with second coordinate 2. The other line is a slanted line going through the points (negative 5, negative 10), (negative 4, negative 8), (negative 3, negative 6), (negative 2, negative 4), (negative 1, negative 2), (0, 0), (1, 2), (2, 4), (3, 6), (4, 8), and (5, 10).", caption: none) ] #math.equation(block: false, alt: "y equals 5 x")[$y = 5 x$] and #math.equation(block: false, alt: "y equals 5")[$y = 5$] #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[The figure shows a two straight lines drawn on the same x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. One line is a straight horizontal line going through the points (negative 4, negative one half) (0, negative one half), (4, negative one half), and all other points with second coordinate negative one half. The other line is a slanted line going through the points (negative 10, 5), (negative 8, 4), (negative 6, 3), (negative 4, 2), (negative 2, 1), (0, 0), (1, negative 2), (2, negative 4), (3, negative 6), (4, negative 8), and (5, negative 10).], alt: "The figure shows a two straight lines drawn on the same x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. One line is a straight horizontal line going through the points (negative 4, negative one half) (0, negative one half), (4, negative one half), and all other points with second coordinate negative one half. The other line is a slanted line going through the points (negative 10, 5), (negative 8, 4), (negative 6, 3), (negative 4, 2), (negative 2, 1), (0, 0), (1, negative 2), (2, negative 4), (3, negative 6), (4, negative 8), and (5, negative 10).", 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")[$y = − frac(1, 3)$] ==== Mixed Practice In the following exercises, graph each equation. #math.equation(block: true, alt: "y equals 4 x")[$y = 4 x$] #solutionbox[ #figure(figph[The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 2, negative 8), (negative 1, negative 4), (0, 0), (1, 4), and (2, 8).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 2, negative 8), (negative 1, negative 4), (0, 0), (1, 4), and (2, 8).", 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[The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 6, 6), (negative 4, 5), (negative 2, 4), (0, 3), (2, 2), (4, 1), and (6, 0).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 6, 6), (negative 4, 5), (negative 2, 4), (0, 3), (2, 2), (4, 1), and (6, 0).", 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 − x")[$y = "−" x$] #solutionbox[ #figure(figph[The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 6, 6), (negative 5, 5), (negative 4, 4), (negative 3, 3), (negative 2, 2), (negative 1, 1), (0, 0), (1, negative 1), (2, negative 2), (3, negative 3), (4, negative 4), (5, negative 5), and (6, negative 6).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 6, 6), (negative 5, 5), (negative 4, 4), (negative 3, 3), (negative 2, 2), (negative 1, 1), (0, 0), (1, negative 1), (2, negative 2), (3, negative 3), (4, negative 4), (5, negative 5), and (6, negative 6).", 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[The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 3, negative 7), (negative 2, negative 6), (negative 1, negative 4), (0, negative 3), (1, negative 2), (2, negative 1), (3, 0), (4, 1), (5, 2), and (6, 3).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 3, negative 7), (negative 2, negative 6), (negative 1, negative 4), (0, negative 3), (1, negative 2), (2, negative 1), (3, 0), (4, 1), (5, 2), and (6, 3).", caption: none) ] #math.equation(block: true, alt: "x plus y equals −5")[$x + y = −5$] #math.equation(block: true, alt: "4 x plus y equals 2")[$4 x + y = 2$] #solutionbox[ #figure(figph[The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 2, 6), (negative 1, 4), (0, 2), (1, negative 2), and (2, negative 6).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 2, 6), (negative 1, 4), (0, 2), (1, negative 2), and (2, negative 6).", 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[The figure shows a straight horizontal line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The horizontal line goes through the points (0, negative 1), (1, negative 1), (2, negative 1) and all points with second coordinate negative 1.], alt: "The figure shows a straight horizontal line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The horizontal line goes through the points (0, negative 1), (1, negative 1), (2, negative 1) and all points with second coordinate negative 1.", 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[The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 6, 4), (negative 3, 3), (0, 2), (3, 1), and (6, 0).], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 6, 4), (negative 3, 3), (0, 2), (3, 1), and (6, 0).", 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[The figure shows a straight vertical line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The vertical line goes through the points (3, 0), (3, 1), (3, 2) and all points with first coordinate 3.], alt: "The figure shows a straight vertical line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The vertical line goes through the points (3, 0), (3, 1), (3, 2) and all points with first coordinate 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 \$594 plus \$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 400, 800, and 1200 miles, and then graph the line. #solutionbox[ \$722, \$850, \$978 #linebreak() #figure(figph[The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from 0 to 1200 in increments of 100. The y-axis of the plane runs from 0 to 1000 in increments of 100. The straight line starts at the point (0, 594) and goes through the points (400, 722), (800, 850), and (1200, 978). The right end of the line has an arrow pointing up and to the right.], alt: "The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from 0 to 1200 in increments of 100. The y-axis of the plane runs from 0 to 1000 in increments of 100. The straight line starts at the point (0, 594) and goes through the points (400, 722), (800, 850), and (1200, 978). The right end of the line has an arrow pointing up and to the right.", caption: none) ] #strong[Weekly earnings.] At the art gallery where he works, Salvador gets paid \$200 per week plus 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 \$900, \$1600, and \$2000, and then graph the line. ==== Writing Exercises Explain how you would choose three #emph[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[This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “…recognize the relation between the solutions of an equation and its graph.”, “…graph a linear equation by plotting points.”, and “…graph vertical and horizontal lines.”. The other columns are left blank so that the learner may indicate their mastery level for each topic.], alt: "This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “…recognize the relation between the solutions of an equation and its graph.”, “…graph a linear equation by plotting points.”, and “…graph vertical and horizontal lines.”. The other columns are left blank so that the learner may indicate their mastery level for each topic.", caption: none) ⓑ After reviewing this checklist, what will you do to become confident for all goals?