#set document(title: "4.6 Find the Equation of a Line", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 4.6#h(0.6em)Find the Equation of a Line #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Before you get started, take this readiness quiz. Solve: #math.equation(block: false, alt: "the fraction 2 over 3 equals the fraction x over 5")[$frac(2, 3) = frac(x, 5)$]. #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "x equals the fraction 10 over 3")[$x = frac(10, 3)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "minus the fraction 2 over 5 open parenthesis x minus 15 close parenthesis")[$− frac(2, 5) ( x − 15 )$]. #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "minus the fraction 2 over 5 x plus 6")[$− frac(2, 5) x + 6$] ] ] How do online retailers know that ‘you may also like’ a particular item based on something you just ordered? How can economists know how a rise in the minimum wage will affect the unemployment rate? How do medical researchers create drugs to target cancer cells? How can traffic engineers predict the effect on your commuting time of an increase or decrease in gas prices? It’s all mathematics. You are at an exciting point in your mathematical journey as the mathematics you are studying has interesting applications in the real world. The physical sciences, social sciences, and the business world are full of situations that can be modeled with linear equations relating two variables. Data is collected and graphed. If the data points appear to form a straight line, an equation of that line can be used to predict the value of one variable based on the value of the other variable. To create a mathematical model of a linear relation between two variables, we must be able to find the equation of the line. In this section we will look at several ways to write the equation of a line. The specific method we use will be determined by what information we are given. === Find an Equation of the Line Given the Slope and #emph[y]-Intercept We can easily determine the slope and intercept of a line if the equation was written in slope–intercept form, #math.equation(block: false, alt: "y equals m x plus b .")[$y = m x + b .$] Now, we will do the reverse—we will start with the slope and #emph[y]-intercept and use them to find the equation of the line. #examplebox("Example 1")[][ Find an equation of a line with slope #math.equation(block: false, alt: "−7")[$−7$] and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$( 0 , −1 )$]. #solutionbox[ Since we are given the slope and #emph[y]-intercept of the line, we can substitute the needed values into the slope–intercept form, #math.equation(block: false, alt: "y equals m x plus b")[$y = m x + b$]. #figure(table( columns: 2, align: left, inset: 6pt, table.header([Name the slope.], [#figure(figph[A mathematical equation shows 'm = -7' written in black characters with the number 7 highlighted in red against a plain white background, indicating a variable assigned a negative integer value.], alt: "A mathematical equation shows 'm = -7' written in black characters with the number 7 highlighted in red against a plain white background, indicating a variable assigned a negative integer value.", caption: none)]), [Name the #emph[y]-intercept.], [#figure(figph[The image displays the text 'y-intercept (0, -1)'.], alt: "The image displays the text 'y-intercept (0, -1)'.", caption: none)], [Substitute the values into #math.equation(block: false, alt: "y equals m x plus b .")[$y = m x + b .$]], [#figure(figph[The equation y = mx + b, which represents the slope-intercept form of a linear equation, is displayed. The variable 'm' is colored red, and 'b' is colored blue, highlighting their roles as slope and y-intercept.], alt: "The equation y = mx + b, which represents the slope-intercept form of a linear equation, is displayed. The variable 'm' is colored red, and 'b' is colored blue, highlighting their roles as slope and y-intercept.", caption: none)], [], [#figure(figph[A mathematical equation is displayed on a white background, reading 'y = -7x + (-1)'. The '-7x' is highlighted in red, and the '(-1)' is highlighted in blue.], alt: "A mathematical equation is displayed on a white background, reading 'y = -7x + (-1)'. The '-7x' is highlighted in red, and the '(-1)' is highlighted in blue.", caption: none)], [], [#figure(figph[A mathematical equation is displayed, reading 'y = -7x - 1' in a clear, sans-serif font against a plain white background.], alt: "A mathematical equation is displayed, reading 'y = -7x - 1' in a clear, sans-serif font against a plain white background.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find an equation of a line with slope #math.equation(block: false, alt: "the fraction 2 over 5")[$frac(2, 5)$] and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 4 close parenthesis")[$( 0 , 4 )$]. #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 2 over 5 x plus 4")[$y = frac(2, 5) x + 4$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find an equation of a line with slope #math.equation(block: false, alt: "−1")[$−1$] and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , −3 close parenthesis")[$( 0 , −3 )$]. #solutionbox[ #math.equation(block: true, alt: "y equals − x minus 3")[$y = "−" x − 3$] ] ] Sometimes, the slope and intercept need to be determined from the graph. #examplebox("Example 2")[][ Find the equation of the line shown. #figure(figph[The graph shows the x y-coordinate plane. The x and y-axes each run from negative 7 to 7. A line intercepts the y-axis at (0, negative 4), passes through the plotted point (3, negative 2), and intercepts the x-axis at (4, 0).], alt: "The graph shows the x y-coordinate plane. The x and y-axes each run from negative 7 to 7. A line intercepts the y-axis at (0, negative 4), passes through the plotted point (3, negative 2), and intercepts the x-axis at (4, 0).", caption: none) #solutionbox[ We need to find the slope and #emph[y]-intercept of the line from the graph so we can substitute the needed values into the slope–intercept form, #math.equation(block: false, alt: "y equals m x plus b")[$y = m x + b$]. To find the slope, we choose two points on the graph. The #emph[y]-intercept is #math.equation(block: false, alt: "open parenthesis 0 , −4 close parenthesis")[$( 0 , −4 )$] and the graph passes through #math.equation(block: false, alt: "open parenthesis 3 , −2 close parenthesis")[$( 3 , −2 )$]. #figure(table( columns: 2, align: left, inset: 6pt, table.header([Find the slope by counting the rise and run.], [#figure(figph[The mathematical formula for slope, 'm = rise / run,' is displayed on a white background.], alt: "The mathematical formula for slope, 'm = rise / run,' is displayed on a white background.", caption: none)]), [], [#figure(figph[A mathematical equation is displayed against a white background, showing 'm = 2/3'. The variable 'm' is in black, and the fraction '2/3' is in red, with '2' above a horizontal line and '3' below.], alt: "A mathematical equation is displayed against a white background, showing 'm = 2/3'. The variable 'm' is in black, and the fraction '2/3' is in red, with '2' above a horizontal line and '3' below.", caption: none)], [Find the #emph[y]-intercept.], [#figure(figph[The image displays the text 'y-intercept (0, -4)' in black font, with the number -4 highlighted in light blue, indicating the point where a line or curve crosses the y-axis.], alt: "The image displays the text 'y-intercept (0, -4)' in black font, with the number -4 highlighted in light blue, indicating the point where a line or curve crosses the y-axis.", caption: none)], [Substitute the values into #math.equation(block: false, alt: "y equals m x plus b .")[$y = m x + b .$]], [#figure(figph[A close-up of the algebraic formula y=mx+b, commonly known as the slope-intercept form of a straight line, with 'm' representing the slope and 'b' the y-intercept.], alt: "A close-up of the algebraic formula y=mx+b, commonly known as the slope-intercept form of a straight line, with 'm' representing the slope and 'b' the y-intercept.", caption: none)], [], [#figure(figph[The image displays the linear equation y = (2/3)x - 4. The fraction 2/3 is colored red, indicating the slope, while the number 4, part of the y-intercept, is colored blue, with a minus sign preceding it.], alt: "The image displays the linear equation y = (2/3)x - 4. The fraction 2/3 is colored red, indicating the slope, while the number 4, part of the y-intercept, is colored blue, with a minus sign preceding it.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the equation of the line shown in the graph. #figure(figph[The graph shows the x y-coordinate plane. The x and y-axes each run from negative 7 to 7. A line intercepts the x-axis at (negative 2, 0), intercepts the y-axis at (0, 1) and passes through the plotted point (5, 4).], alt: "The graph shows the x y-coordinate plane. The x and y-axes each run from negative 7 to 7. A line intercepts the x-axis at (negative 2, 0), intercepts the y-axis at (0, 1) and passes through the plotted point (5, 4).", caption: none) #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 3 over 5 x plus 1")[$y = frac(3, 5) x + 1$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the equation of the line shown in the graph. #figure(figph[The graph shows the x y-coordinate plane. The x and y-axes each run from negative 7 to 7. A line intercepts the y-axis at (0, negative 5), passes through the plotted point (3, negative 1), and intercepts the x-axis at (15 fourths, 0).], alt: "The graph shows the x y-coordinate plane. The x and y-axes each run from negative 7 to 7. A line intercepts the y-axis at (0, negative 5), passes through the plotted point (3, negative 1), and intercepts the x-axis at (15 fourths, 0).", caption: none) #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 4 over 3 x minus 5")[$y = frac(4, 3) x − 5$] ] ] === Find an Equation of the Line Given the Slope and a Point Finding an equation of a line using the slope–intercept form of the equation works well when you are given the slope and #emph[y]-intercept or when you read them off a graph. But what happens when you have another point instead of the #emph[y]-intercept? We are going to use the slope formula to derive another form of an equation of the line. Suppose we have a line that has slope #math.equation(block: false, alt: "m")[$m$] and that contains some specific point #math.equation(block: false, alt: "open parenthesis x sub 1 , y sub 1 close parenthesis")[$( x_(1) , y_(1) )$] and some other point, which we will just call #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]. We can write the slope of this line and then change it to a different form. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "m equals the fraction y minus y sub 1 over x minus x sub 1")[$m = frac(y − y_(1), x − x_(1))$]]), [Multiply both sides of the equation by #math.equation(block: false, alt: "x minus x sub 1")[$x − x_(1)$].], [#math.equation(block: false, alt: "m open parenthesis x minus x sub 1 close parenthesis, equals, open parenthesis the fraction y minus y sub 1 over x minus x sub 1 close parenthesis open parenthesis x minus x sub 1 close parenthesis")[$#h(4em) m ( x − x_(1) ) & = & ( frac(y − y_(1), x − x_(1)) ) ( x − x_(1) )$]], [Simplify.], [#math.equation(block: false, alt: "m open parenthesis x minus x sub 1 close parenthesis, equals, y minus y sub 1")[$#h(4em) m ( x − x_(1) ) & = & y − y_(1)$]], [Rewrite the equation with the #math.equation(block: false, alt: "y")[$y$] terms on the left.], [#math.equation(block: false, alt: "y minus y sub 1, equals, m open parenthesis x minus x sub 1 close parenthesis")[$#h(4em) y − y_(1) & = & m ( x − x_(1) )$]], )) This format is called the point–slope form of an equation of a line. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Point–slope Form of an Equation of a Line] The #strong[point–slope form] of an equation of a line with slope #math.equation(block: false, alt: "m")[$m$] and containing the point #math.equation(block: false, alt: "open parenthesis x sub 1 , y sub 1 close parenthesis")[$( x_(1) , y_(1) )$] is #figure(figph[This image shows and equation formatted to find the point-slope equation of the line.], alt: "This image shows and equation formatted to find the point-slope equation of the line.", caption: none) ] We can use the point–slope form of an equation to find an equation of a line when we are given the slope and one point. Then we will rewrite the equation in slope–intercept form. Most applications of linear equations use the the slope–intercept form. #examplebox("Example 3")[Find an Equation of a Line Given the Slope and a Point][ Find an equation of a line with slope #math.equation(block: false, alt: "m equals the fraction 2 over 5")[$m = frac(2, 5)$] that contains the point #math.equation(block: false, alt: "open parenthesis 10 , 3 close parenthesis")[$( 10 , 3 )$]. Write the equation in slope–intercept form. #solutionbox[ #figure(figph[This figure is a table that has three columns and four rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. In the first row of the table, the first cell on the left reads: “Step 1. Identify the slope.” The text in the second cell reads: “The slope is given.” The third cell contains the slope of a line, defined as m equals 2 fifths.], alt: "This figure is a table that has three columns and four rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. In the first row of the table, the first cell on the left reads: “Step 1. Identify the slope.” The text in the second cell reads: “The slope is given.” The third cell contains the slope of a line, defined as m equals 2 fifths.", caption: none) #figure(figph[In the second row, the first cell reads: “Step 2. Identify the point.” The second cell reads: “The point is given.” The third cell contains the ordered pair (10, 3). A superscript x subscript 1 is written over 10, and a superscript y subscript 1 is written over 3.], alt: "In the second row, the first cell reads: “Step 2. Identify the point.” The second cell reads: “The point is given.” The third cell contains the ordered pair (10, 3). A superscript x subscript 1 is written over 10, and a superscript y subscript 1 is written over 3.", caption: none) #figure(figph[Step 3 substitutes values into point-slope form: y minus y sub one equals m times x minus x sub one. The equation becomes y minus 3 equals 2/5 times x minus 10. Simplify to get y minus 3 equals 2/5 x minus 4.], alt: "Step 3 substitutes values into point-slope form: y minus y sub one equals m times x minus x sub one. The equation becomes y minus 3 equals 2/5 times x minus 10. Simplify to get y minus 3 equals 2/5 x minus 4.", caption: none) #figure(figph[In the fourth row, the first cell reads: “Write the equation in slope-intercept form.” The second cell is blank. In the third cell is y equals 2 fifths x minus 1.], alt: "In the fourth row, the first cell reads: “Write the equation in slope-intercept form.” The second cell is blank. In the third cell is y equals 2 fifths x minus 1.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find an equation of a line with slope #math.equation(block: false, alt: "m equals the fraction 5 over 6")[$m = frac(5, 6)$] and containing the point #math.equation(block: false, alt: "open parenthesis 6 , 3 close parenthesis")[$( 6 , 3 )$]. #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 5 over 6 x minus 2")[$y = frac(5, 6) x − 2$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find an equation of a line with slope #math.equation(block: false, alt: "m equals the fraction 2 over 3")[$m = frac(2, 3)$] and containing the point #math.equation(block: false, alt: "open parenthesis 9 , 2 close parenthesis")[$( 9 , 2 )$]. #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 2 over 3 x minus 4")[$y = frac(2, 3) x − 4$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Find an equation of a line given the slope and a point.] + Identify the slope. + Identify the point. + Substitute the values into the point-slope form, #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis")[$y − y_(1) = m ( x − x_(1) )$]. + Write the equation in slope–intercept form. ] #examplebox("Example 4")[][ Find an equation of a line with slope #math.equation(block: false, alt: "m equals minus the fraction 1 over 3")[$m = − frac(1, 3)$] that contains the point #math.equation(block: false, alt: "open parenthesis 6 , −4 close parenthesis")[$( 6 , −4 )$]. Write the equation in slope–intercept form. #solutionbox[ Since we are given a point and the slope of the line, we can substitute the needed values into the point–slope form, #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis")[$y − y_(1) = m ( x − x_(1) )$]. #figure(table( columns: 2, align: left, inset: 6pt, table.header([Identify the slope.], [#figure(figph[The equation m = -1/3, displaying a blue 'm' and an equal sign, followed by a negative sign and the fraction 1/3 in red, all on a white background.], alt: "The equation m = -1/3, displaying a blue 'm' and an equal sign, followed by a negative sign and the fraction 1/3 in red, all on a white background.", caption: none)]), [Identify the point.], [#figure(figph[A mathematical expression showing a point (x1, y1) with values (6, -4) stacked vertically within parentheses. It indicates that x1 = 6 and y1 = -4.], alt: "A mathematical expression showing a point (x1, y1) with values (6, -4) stacked vertically within parentheses. It indicates that x1 = 6 and y1 = -4.", caption: none)], [Substitute the values into #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis .")[$y − y_(1) = m ( x − x_(1) ) .$]], [#figure(figph[The point-slope form of a linear equation, y - y\_1 = m(x - x\_1), where (x\_1, y\_1) is a point on the line and m is the slope, displayed with color-coded variables.], alt: "The point-slope form of a linear equation, y - y_1 = m(x - x_1), where (x_1, y_1) is a point on the line and m is the slope, displayed with color-coded variables.", caption: none)], [], [#figure(figph[A mathematical equation in point-slope form: y - (-4) = -1/3(x - 6). The numbers -4 and -6 are highlighted in red, and the fraction 1/3 is highlighted in blue.], alt: "A mathematical equation in point-slope form: y - (-4) = -1/3(x - 6). The numbers -4 and -6 are highlighted in red, and the fraction 1/3 is highlighted in blue.", caption: none)], [Simplify.], [#figure(figph[The image displays the linear equation y + 4 = -1/3x + 2, written in an algebraic format. It shows variables 'y' and 'x' with coefficients and constants.], alt: "The image displays the linear equation y + 4 = -1/3x + 2, written in an algebraic format. It shows variables 'y' and 'x' with coefficients and constants.", caption: none)], [Write in slope–intercept form.], [#figure(figph[A mathematical equation is displayed, reading 'y = -1/3x - 2'. The equation represents a linear function in slope-intercept form, with a negative slope of -1/3 and a y-intercept of -2.], alt: "A mathematical equation is displayed, reading 'y = -1/3x - 2'. The equation represents a linear function in slope-intercept form, with a negative slope of -1/3 and a y-intercept of -2.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find an equation of a line with slope #math.equation(block: false, alt: "m equals minus the fraction 2 over 5")[$m = − frac(2, 5)$] and containing the point #math.equation(block: false, alt: "open parenthesis 10 , −5 close parenthesis")[$( 10 , −5 )$]. #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 2 over 5 x minus 1")[$y = − frac(2, 5) x − 1$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find an equation of a line with slope #math.equation(block: false, alt: "m equals minus the fraction 3 over 4")[$m = − frac(3, 4)$], and containing the point #math.equation(block: false, alt: "open parenthesis 4 , −7 close parenthesis")[$( 4 , −7 )$]. #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 3 over 4 x minus 4")[$y = − frac(3, 4) x − 4$] ] ] #examplebox("Example 5")[][ Find an equation of a horizontal line that contains the point #math.equation(block: false, alt: "open parenthesis −1 , 2 close parenthesis")[$( −1 , 2 )$]. Write the equation in slope–intercept form. #solutionbox[ Every horizontal line has slope 0. We can substitute the slope and points into the point–slope form, #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis")[$y − y_(1) = m ( x − x_(1) )$]. #figure(table( columns: 2, align: left, inset: 6pt, table.header([Identify the slope.], [#figure(figph[A white background shows the equation 'm = 0'. The 'm' is light blue and the '0' is reddish-orange, both slightly out of focus.], alt: "A white background shows the equation 'm = 0'. The 'm' is light blue and the '0' is reddish-orange, both slightly out of focus.", caption: none)]), [Identify the point.], [#figure(figph[A mathematical notation displaying a coordinate point (x1, y1) directly above the specific coordinates (-1, 2), indicating that x1 = -1 and y1 = 2.], alt: "A mathematical notation displaying a coordinate point (x1, y1) directly above the specific coordinates (-1, 2), indicating that x1 = -1 and y1 = 2.", caption: none)], [Substitute the values into #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis .")[$y − y_(1) = m ( x − x_(1) ) .$]], [#figure(figph[The point-slope form of a linear equation, y - y\_1 = m(x - x\_1), is displayed with y\_1 and x\_1 highlighted in red, and the slope 'm' highlighted in blue.], alt: "The point-slope form of a linear equation, y - y_1 = m(x - x_1), is displayed with y_1 and x_1 highlighted in red, and the slope 'm' highlighted in blue.", caption: none)], [], [#figure(figph[The equation y - 2 = 0(x - (-1)) represents a horizontal line at y=2 with a slope of 0, passing through the point (-1, 2).], alt: "The equation y - 2 = 0(x - (-1)) represents a horizontal line at y=2 with a slope of 0, passing through the point (-1, 2).", caption: none)], [Simplify.], [#figure(figph[The equation y - 2 = 0(x + 1) is displayed on a white background. This simplifies to y=2, representing a horizontal line.], alt: "The equation y - 2 = 0(x + 1) is displayed on a white background. This simplifies to y=2, representing a horizontal line.", caption: none)], [], [#figure(figph[The image displays a mathematical equation written in black text on a white background, which reads 'y - 2 = 0'.], alt: "The image displays a mathematical equation written in black text on a white background, which reads 'y - 2 = 0'.", caption: none)], [], [#figure(figph[The image displays the simple algebraic equation 'y=2' in black text against a plain white background, clearly indicating a constant value for the variable y.], alt: "The image displays the simple algebraic equation 'y=2' in black text against a plain white background, clearly indicating a constant value for the variable y.", caption: none)], [Write in slope–intercept form.], [It is in #emph[y]-form, but could be written #math.equation(block: false, alt: "y equals 0 x plus 2")[$y = 0 x + 2$].], )) Did we end up with the form of a horizontal line, #math.equation(block: false, alt: "y equals a")[$y = a$]? ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find an equation of a horizontal line containing the point #math.equation(block: false, alt: "open parenthesis −3 , 8 close parenthesis")[$( −3 , 8 )$]. #solutionbox[ #math.equation(block: true, alt: "y equals 8")[$y = 8$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find an equation of a horizontal line containing the point #math.equation(block: false, alt: "open parenthesis −1 , 4 close parenthesis")[$( −1 , 4 )$]. #solutionbox[ #math.equation(block: true, alt: "y equals 4")[$y = 4$] ] ] === Find an Equation of the Line Given Two Points When real-world data is collected, a linear model can be created from two data points. In the next example we’ll see how to find an equation of a line when just two points are given. We have two options so far for finding an equation of a line: slope–intercept or point–slope. Since we will know two points, it will make more sense to use the point–slope form. But then we need the slope. Can we find the slope with just two points? Yes. Then, once we have the slope, we can use it and one of the given points to find the equation. #examplebox("Example 6")[Find an Equation of a Line Given Two Points][ Find an equation of a line that contains the points #math.equation(block: false, alt: "open parenthesis 5 , 4 close parenthesis")[$( 5 , 4 )$] and #math.equation(block: false, alt: "open parenthesis 3 , 6 close parenthesis")[$( 3 , 6 )$]. Write the equation in slope–intercept form. #solutionbox[ #figure(figph[Table showing steps to find the slope of a line. Step 1: Find the slope using the formula y two minus y one, divided by x two minus x one. Calculation shows the slope, m, as negative one.], alt: "Table showing steps to find the slope of a line. Step 1: Find the slope using the formula y two minus y one, divided by x two minus x one. Calculation shows the slope, m, as negative one.", caption: none) #figure(figph[In the second row, the first cell reads: “Step 2. Choose one point.” The second cell reads: “Choose either point.” The third cell contains the ordered pair (5, 4) with a superscript x subscript 1 over 5 and a superscript y subscript 1 over 4.], alt: "In the second row, the first cell reads: “Step 2. Choose one point.” The second cell reads: “Choose either point.” The third cell contains the ordered pair (5, 4) with a superscript x subscript 1 over 5 and a superscript y subscript 1 over 4.", caption: none) #figure(figph[In the third row, the first cell reads: “Step 3. Substitute the values into the point-slope form, y minus y subscript 1 equals m times x minus x subscript 1 in parentheses.” The top line of the second cell is left blank. The third cell contains the point-slope form, y minus y subscript 1 equals m times x minus x subscript 1 in parentheses. Below this is the point-slope form with 5 substituted for x subscript 1, 4 substituted for y subscript 1, and negative 1 substituted for m: y minus 4 equals negative 1 times x minus 5 in parentheses. Below this is y minus 4 equals negative x plus 5.], alt: "In the third row, the first cell reads: “Step 3. Substitute the values into the point-slope form, y minus y subscript 1 equals m times x minus x subscript 1 in parentheses.” The top line of the second cell is left blank. The third cell contains the point-slope form, y minus y subscript 1 equals m times x minus x subscript 1 in parentheses. Below this is the point-slope form with 5 substituted for x subscript 1, 4 substituted for y subscript 1, and negative 1 substituted for m: y minus 4 equals negative 1 times x minus 5 in parentheses. Below this is y minus 4 equals negative x plus 5.", caption: none) #figure(figph[In the fourth row, the first cell reads: “Step 4. Write the equation in slope-intercept form.” The second cell is blank. The third cell contains y equals negative x plus 9.], alt: "In the fourth row, the first cell reads: “Step 4. Write the equation in slope-intercept form.” The second cell is blank. The third cell contains y equals negative x plus 9.", caption: none) Use the point #math.equation(block: false, alt: "open parenthesis 3 , 6 close parenthesis")[$( 3 , 6 )$] and see that you get the same equation. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find an equation of a line containing the points #math.equation(block: false, alt: "open parenthesis 3 , 1 close parenthesis")[$( 3 , 1 )$] and #math.equation(block: false, alt: "open parenthesis 5 , 6 close parenthesis")[$( 5 , 6 )$]. #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 5 over 2 x minus the fraction 13 over 2")[$y = frac(5, 2) x − frac(13, 2)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find an equation of a line containing the points #math.equation(block: false, alt: "open parenthesis 1 , 4 close parenthesis")[$( 1 , 4 )$] and #math.equation(block: false, alt: "open parenthesis 6 , 2 close parenthesis")[$( 6 , 2 )$]. #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 2 over 5 x plus the fraction 22 over 5")[$y = − frac(2, 5) x + frac(22, 5)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Find an equation of a line given two points.] + Find the slope using the given points. + Choose one point. + Substitute the values into the point-slope form, #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis")[$y − y_(1) = m ( x − x_(1) )$]. + Write the equation in slope–intercept form. ] #examplebox("Example 7")[][ Find an equation of a line that contains the points #math.equation(block: false, alt: "open parenthesis −3 , −1 close parenthesis")[$( −3 , −1 )$] and #math.equation(block: false, alt: "open parenthesis 2 , −2 close parenthesis")[$( 2 , −2 )$]. Write the equation in slope–intercept form. #solutionbox[ Since we have two points, we will find an equation of the line using the point–slope form. The first step will be to find the slope. #figure(table( columns: 2, align: left, inset: 6pt, table.header([Find the slope of the line through (−3, −1) and (2, −2).], [#figure(figph[The image displays the mathematical formula for calculating the slope (m) of a line, which is the change in y (rise) divided by the change in x (run): m = (y₂ - y₁)/(x₂ - x₁).], alt: "The image displays the mathematical formula for calculating the slope (m) of a line, which is the change in y (rise) divided by the change in x (run): m = (y₂ - y₁)/(x₂ - x₁).", caption: none)]), [], [#figure(figph[A mathematical formula for calculating the slope 'm' is shown, with the expression m = (-2 - (-1)) / (2 - (-3)).], alt: "A mathematical formula for calculating the slope 'm' is shown, with the expression m = (-2 - (-1)) / (2 - (-3)).", caption: none)], [], [#figure(figph[A mathematical equation is displayed on a white background, showing 'm = -1/5'. The letter 'm' is followed by an equals sign, then a fraction where -1 is the numerator and 5 is the denominator.], alt: "A mathematical equation is displayed on a white background, showing 'm = -1/5'. The letter 'm' is followed by an equals sign, then a fraction where -1 is the numerator and 5 is the denominator.", caption: none)], [], [#figure(figph[A mathematical equation displays 'm = -1/5' in a clean, white background. The variable 'm' is shown in light blue, the equals sign and negative sign are black, and the fraction '1/5' is in red.], alt: "A mathematical equation displays 'm = -1/5' in a clean, white background. The variable 'm' is shown in light blue, the equals sign and negative sign are black, and the fraction '1/5' is in red.", caption: none)], [Choose either point.], [#figure(figph[A coordinate pair (2, -2) with 'x1' labeled above 2 and 'y1' labeled above -2, enclosed in parentheses, against a white background.], alt: "A coordinate pair (2, -2) with 'x1' labeled above 2 and 'y1' labeled above -2, enclosed in parentheses, against a white background.", caption: none)], [Substitute the values into #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis .")[$y − y_(1) = m ( x − x_(1) ) .$]], [#figure(figph[The image displays the point-slope form of a linear equation, y - y1 = m(x - x1), commonly used in algebra and geometry to represent a line given a point (x1, y1) and a slope 'm'.], alt: "The image displays the point-slope form of a linear equation, y - y1 = m(x - x1), commonly used in algebra and geometry to represent a line given a point (x1, y1) and a slope 'm'.", caption: none)], [], [#figure(figph[The point-slope form equation y - (-2) = -1/5 (x - 2) is displayed, with the point (2, -2) indicated in red and the slope -1/5 shown in blue.], alt: "The point-slope form equation y - (-2) = -1/5 (x - 2) is displayed, with the point (2, -2) indicated in red and the slope -1/5 shown in blue.", caption: none)], [], [#figure(figph[The image shows the equation y + 2 = -1/5 x + 2/5.], alt: "The image shows the equation y + 2 = -1/5 x + 2/5.", caption: none)], [Write in slope–intercept form.], [#figure(figph[A mathematical equation on a white background, reading y = -1/5x - 8/5.], alt: "A mathematical equation on a white background, reading y = -1/5x - 8/5.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find an equation of a line containing the points #math.equation(block: false, alt: "open parenthesis −2 , −4 close parenthesis")[$( −2 , −4 )$] and #math.equation(block: false, alt: "open parenthesis 1 , −3 close parenthesis")[$( 1 , −3 )$]. #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 1 over 3 x minus the fraction 10 over 3")[$y = frac(1, 3) x − frac(10, 3)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find an equation of a line containing the points #math.equation(block: false, alt: "open parenthesis −4 , −3 close parenthesis")[$( −4 , −3 )$] and #math.equation(block: false, alt: "open parenthesis 1 , −5 close parenthesis")[$( 1 , −5 )$]. #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 2 over 5 x minus the fraction 23 over 5")[$y = − frac(2, 5) x − frac(23, 5)$] ] ] #examplebox("Example 8")[][ Find an equation of a line that contains the points #math.equation(block: false, alt: "open parenthesis −2 , 4 close parenthesis")[$( −2 , 4 )$] and #math.equation(block: false, alt: "open parenthesis −2 , −3 close parenthesis")[$( −2 , −3 )$]. Write the equation in slope–intercept form. #solutionbox[ Again, the first step will be to find the slope. #figure(table( columns: 2, align: left, inset: 6pt, table.header([Find the slope of the line through #math.equation(block: false, alt: "open parenthesis −2 , 4 close parenthesis and open parenthesis −2 , −3 close parenthesis")[$( −2 , 4 ) #h(0.2em) "and" #h(0.2em) ( −2 , −3 )$].], [#math.equation(block: false, alt: "m equals the fraction y sub 2 minus y sub 1 over x sub 2 minus x sub 1")[$m = frac(y_(2) − y_(1), x_(2) − x_(1))$]]), [], [#math.equation(block: false, alt: "m equals the fraction −3 minus 4 over −2 minus open parenthesis −2 close parenthesis")[$m = frac(−3 − 4, −2 − ( −2 ))$]], [], [#math.equation(block: false, alt: "m equals the fraction −7 over 0")[$m = frac(−7, 0)$]], [], [The slope is undefined.], )) This tells us it is a vertical line. Both of our points have an #emph[x]-coordinate of #math.equation(block: false, alt: "−2")[$−2$]. So our equation of the line is #math.equation(block: false, alt: "x equals −2")[$x = −2$]. Since there is no #math.equation(block: false, alt: "y")[$y$], we cannot write it in slope–intercept form. You may want to sketch a graph using the two given points. Does the graph agree with our conclusion that this is a vertical line? ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find an equation of a line containing the points #math.equation(block: false, alt: "open parenthesis 5 , 1 close parenthesis")[$( 5 , 1 )$] and #math.equation(block: false, alt: "open parenthesis 5 , −4 close parenthesis")[$( 5 , −4 )$]. #solutionbox[ #math.equation(block: true, alt: "x equals 5")[$x = 5$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find an equaion of a line containing the points #math.equation(block: false, alt: "open parenthesis −4 , 4 close parenthesis")[$( −4 , 4 )$] and #math.equation(block: false, alt: "open parenthesis −4 , 3 close parenthesis")[$( −4 , 3 )$]. #solutionbox[ #math.equation(block: true, alt: "x equals −4")[$x = −4$] ] ] We have seen that we can use either the slope–intercept form or the point–slope form to find an equation of a line. Which form we use will depend on the information we are given. This is summarized in . #figure(table( columns: 3, align: left, inset: 6pt, [To Write an Equation of a Line], [], [], [If given:], [Use:], [Form:], [Slope and #emph[y]-intercept], [slope–intercept], [#math.equation(block: false, alt: "y equals m x plus b")[$y = m x + b$]], [Slope and a point], [point–slope], [#math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis")[$y − y_(1) = m ( x − x_(1) )$]], [Two points], [point–slope], [#math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis")[$y − y_(1) = m ( x − x_(1) )$]], )) === Find an Equation of a Line Parallel to a Given Line Suppose we need to find an equation of a line that passes through a specific point and is parallel to a given line. We can use the fact that parallel lines have the same slope. So we will have a point and the slope—just what we need to use the point–slope equation. First let’s look at this graphically. The graph shows the graph of #math.equation(block: false, alt: "y equals 2 x minus 3")[$y = 2 x − 3$]. We want to graph a line parallel to this line and passing through the point #math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis")[$( −2 , 1 )$]. #figure(figph[The graph shows the x y-coordinate plane. The x and y-axes each run from negative 7 to 7. The line whose equation is y equals 2x minus 3 intercepts the y-axis at (0, negative 3) and intercepts the x-axis at (3 halves, 0). Elsewhere on the graph, the point (negative 2, 1) is plotted.], alt: "The graph shows the x y-coordinate plane. The x and y-axes each run from negative 7 to 7. The line whose equation is y equals 2x minus 3 intercepts the y-axis at (0, negative 3) and intercepts the x-axis at (3 halves, 0). Elsewhere on the graph, the point (negative 2, 1) is plotted.", caption: none) We know that parallel lines have the same slope. So the second line will have the same slope as#math.equation(block: false, alt: "y equals 2 x minus 3")[$y = 2 x − 3$]. That slope is#math.equation(block: false, alt: "m sub ∥ equals 2")[$m_(∥) = 2$]. We’ll use the notation #math.equation(block: false, alt: "m sub ∥")[$m_(∥)$] to represent the slope of a line parallel to a line with slope #math.equation(block: false, alt: "m")[$m$]. (Notice that the subscript #math.equation(block: false, alt: "∥")[$∥$] looks like two parallel lines.) The second line will pass through #math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis")[$( −2 , 1 )$] and have #math.equation(block: false, alt: "m equals 2")[$m = 2$]. To graph the line, we start at#math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis")[$( −2 , 1 )$] and count out the rise and run. With #math.equation(block: false, alt: "m equals 2")[$m = 2$] (or #math.equation(block: false, alt: "m equals the fraction 2 over 1")[$m = frac(2, 1)$]), we count out the rise 2 and the run 1. We draw the line. #figure(figph[The graph shows the x y-coordinate plane. The x and y-axes each run from negative 7 to 7. The line whose equation is y equals 2x minus 3 intercepts the y-axis at (0, negative 3) and intercepts the x-axis at (3 halves, 0). The points (negative 2, 1) and (negative 1, 3) are plotted. A second line, parallel to the first, intercepts the x-axis at (negative 5 halves, 0), passes through the points (negative 2, 1) and (negative 1, 3), and intercepts the y-axis at (0, 5).], alt: "The graph shows the x y-coordinate plane. The x and y-axes each run from negative 7 to 7. The line whose equation is y equals 2x minus 3 intercepts the y-axis at (0, negative 3) and intercepts the x-axis at (3 halves, 0). The points (negative 2, 1) and (negative 1, 3) are plotted. A second line, parallel to the first, intercepts the x-axis at (negative 5 halves, 0), passes through the points (negative 2, 1) and (negative 1, 3), and intercepts the y-axis at (0, 5).", caption: none) Do the lines appear parallel? Does the second line pass through #math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis")[$( −2 , 1 )$]? Now, let’s see how to do this algebraically. We can use either the slope–intercept form or the point–slope form to find an equation of a line. Here we know one point and can find the slope. So we will use the point–slope form. #examplebox("Example 9")[How to Find an Equation of a Line Parallel to a Given Line][ Find an equation of a line parallel to #math.equation(block: false, alt: "y equals 2 x minus 3")[$y = 2 x − 3$] that contains the point #math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis")[$( −2 , 1 )$]. Write the equation in slope–intercept form. #solutionbox[ #figure(figph[This figure is a table that has three columns and four rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. In the first row of the table, the first cell on the left reads: “Step 1. Find the slope of the given line.” The second cell reads: “The line is in slope-intercept form. y equals 2x minus 3.” The third cell contains the slope of a line, defined as m equals 2.], alt: "This figure is a table that has three columns and four rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. In the first row of the table, the first cell on the left reads: “Step 1. Find the slope of the given line.” The second cell reads: “The line is in slope-intercept form. y equals 2x minus 3.” The third cell contains the slope of a line, defined as m equals 2.", caption: none) #figure(figph[In the second row, the first cell reads: “Step 2. Find the slope of the parallel line.” The second cell reads “Parallel lines have the same slope.” The third cell contains the slope of the parallel line, defined as m parallel equals 2.], alt: "In the second row, the first cell reads: “Step 2. Find the slope of the parallel line.” The second cell reads “Parallel lines have the same slope.” The third cell contains the slope of the parallel line, defined as m parallel equals 2.", caption: none) #figure(figph[In the third row, the first cell reads “Step 3. Identify the point.” The second cell reads “The given point is (negative 2, 1).” The third cell contains the ordered pair (negative 2, 1) with a superscript x subscript 1 above negative 2 and a superscript y subscript 1 above 1.], alt: "In the third row, the first cell reads “Step 3. Identify the point.” The second cell reads “The given point is (negative 2, 1).” The third cell contains the ordered pair (negative 2, 1) with a superscript x subscript 1 above negative 2 and a superscript y subscript 1 above 1.", caption: none) #figure(figph[In the fourth row, the first cell reads “Step 4. Substitute the values into the point-slope form, y minus y subscript 1 equals m times x minus x subscript 1 in parentheses.” The top of the second cell is blank. The third cell contains the point-slope form, y minus y subscript 1 equals m times x minus x subscript 1 in parentheses. Below this is the form with negative 2 substituted for x subscript 1, 1 substituted for y subscript 1, and 2 substituted for m: y minus 1 equals 2 times x minus negative 2 in parentheses. One line down, the text in the second cell says “Simplify.” The right column contains y minus 1 equals 2 times x plus 2. Below this is y minus 1 equals 2x plus 4.], alt: "In the fourth row, the first cell reads “Step 4. Substitute the values into the point-slope form, y minus y subscript 1 equals m times x minus x subscript 1 in parentheses.” The top of the second cell is blank. The third cell contains the point-slope form, y minus y subscript 1 equals m times x minus x subscript 1 in parentheses. Below this is the form with negative 2 substituted for x subscript 1, 1 substituted for y subscript 1, and 2 substituted for m: y minus 1 equals 2 times x minus negative 2 in parentheses. One line down, the text in the second cell says “Simplify.” The right column contains y minus 1 equals 2 times x plus 2. Below this is y minus 1 equals 2x plus 4.", caption: none) #figure(figph[In the fifth row, the first cell says “Step 5. Write the equation in slope-intercept form.” The second cell is blank. The third cell contains y equals 2x plus 5.], alt: "In the fifth row, the first cell says “Step 5. Write the equation in slope-intercept form.” The second cell is blank. The third cell contains y equals 2x plus 5.", caption: none) Does this equation make sense? What is the #emph[y]-intercept of the line? What is the slope? ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find an equation of a line parallel to the line #math.equation(block: false, alt: "y equals 3 x plus 1")[$y = 3 x + 1$] that contains the point #math.equation(block: false, alt: "open parenthesis 4 , 2 close parenthesis")[$( 4 , 2 )$]. Write the equation in slope–intercept form. #solutionbox[ #math.equation(block: true, alt: "y equals 3 x minus 10")[$y = 3 x − 10$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find an equation of a line parallel to the line #math.equation(block: false, alt: "y equals the fraction 1 over 2 x minus 3")[$y = frac(1, 2) x − 3$] that contains the point #math.equation(block: false, alt: "open parenthesis 6 , 4 close parenthesis")[$( 6 , 4 )$]. #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 1 over 2 x plus 1")[$y = frac(1, 2) x + 1$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Find an equation of a line parallel to a given line.] + Find the slope of the given line. + Find the slope of the parallel line. + Identify the point. + Substitute the values into the point–slope form, #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis")[$y − y_(1) = m ( x − x_(1) )$]. + Write the equation in slope–intercept form. ] === Find an Equation of a Line Perpendicular to a Given Line Now, let’s consider perpendicular lines. Suppose we need to find a line passing through a specific point and which is perpendicular to a given line. We can use the fact that perpendicular lines have slopes that are negative reciprocals. We will again use the point–slope equation, like we did with parallel lines. The graph shows the graph of #math.equation(block: false, alt: "y equals 2 x minus 3")[$y = 2 x − 3$]. Now, we want to graph a line perpendicular to this line and passing through #math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis")[$( −2 , 1 )$]. #figure(figph[The graph shows the x y-coordinate plane. The x and y-axes each run from negative 7 to 7. The line whose equation is y equals 2x minus 3 intercepts the y-axis at (0, negative 3) and intercepts the x-axis at (3 halves, 0). Elsewhere on the graph, the point (negative 2, 1) is plotted.], alt: "The graph shows the x y-coordinate plane. The x and y-axes each run from negative 7 to 7. The line whose equation is y equals 2x minus 3 intercepts the y-axis at (0, negative 3) and intercepts the x-axis at (3 halves, 0). Elsewhere on the graph, the point (negative 2, 1) is plotted.", caption: none) We know that perpendicular lines have slopes that are negative reciprocals. We’ll use the notation #math.equation(block: false, alt: "m sub ⊥")[$m_("⊥")$] to represent the slope of a line perpendicular to a line with slope #math.equation(block: false, alt: "m")[$m$]. (Notice that the subscript ⊥ looks like the right angles made by two perpendicular lines.) #math.equation(block: true, alt: "y equals 2 x minus 3, perpendicular line; m equals 2, m sub ⊥ equals minus the fraction 1 over 2")[$y = 2 x − 3 & & "perpendicular line" \ m = 2 & & m_("⊥") = − frac(1, 2)$]We now know the perpendicular line will pass through #math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis")[$( −2 , 1 )$] with #math.equation(block: false, alt: "m sub ⊥ equals minus the fraction 1 over 2")[$m_("⊥") = − frac(1, 2)$]. To graph the line, we will start at #math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis")[$( −2 , 1 )$] and count out the rise #math.equation(block: false, alt: "−1")[$−1$] and the run 2. Then we draw the line. #figure(figph[Graph of two lines in x y-plane. The first line, y equals 2x minus 3, intersects the y-axis at negative 3 and the x-axis at one and a half, zero. A second line passes through negative 2, 1, forming a right triangle with the first line.], alt: "Graph of two lines in x y-plane. The first line, y equals 2x minus 3, intersects the y-axis at negative 3 and the x-axis at one and a half, zero. A second line passes through negative 2, 1, forming a right triangle with the first line.", caption: none) Do the lines appear perpendicular? Does the second line pass through #math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis")[$( −2 , 1 )$]? Now, let’s see how to do this algebraically. We can use either the slope–intercept form or the point–slope form to find an equation of a line. In this example we know one point, and can find the slope, so we will use the point–slope form. #examplebox("Example 10")[How to Find an Equation of a Line Perpendicular to a Given Line][ Find an equation of a line perpendicular to #math.equation(block: false, alt: "y equals 2 x minus 3")[$y = 2 x − 3$] that contains the point #math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis")[$( −2 , 1 )$]. Write the equation in slope–intercept form. #solutionbox[ #figure(figph[This figure is a table that has three columns and four rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. In the first row of the table, the first cell on the left reads: “Step 1. Find the slope of the given line.” The second cell reads: “The line is in slope-intercept form. y equals 2x minus 3.” The third cell contains the slope of a line, defined as m equals 2.], alt: "This figure is a table that has three columns and four rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. In the first row of the table, the first cell on the left reads: “Step 1. Find the slope of the given line.” The second cell reads: “The line is in slope-intercept form. y equals 2x minus 3.” The third cell contains the slope of a line, defined as m equals 2.", caption: none) #figure(figph[In the second row, the first cell reads: “Step 2. Find the slope of the perpendicular line.” The second cell reads “The slopes of perpendicular lines are negative reciprocals.” The third cell contains the slope of the perpendicular line, defined as m perpendicular equals negative 1 half.], alt: "In the second row, the first cell reads: “Step 2. Find the slope of the perpendicular line.” The second cell reads “The slopes of perpendicular lines are negative reciprocals.” The third cell contains the slope of the perpendicular line, defined as m perpendicular equals negative 1 half.", caption: none) #figure(figph[In the third row, the first cell reads “Step 3. Identify the point.” The second cell reads “The given point is (negative 2, 1).” The third cell contains the ordered pair (negative 2, 1) with a superscript x subscript 1 above negative 2 and a superscript y subscript 1 above 1.], alt: "In the third row, the first cell reads “Step 3. Identify the point.” The second cell reads “The given point is (negative 2, 1).” The third cell contains the ordered pair (negative 2, 1) with a superscript x subscript 1 above negative 2 and a superscript y subscript 1 above 1.", caption: none) #figure(figph[In the fourth row, the first cell reads “Step 4. Substitute the values into the point-slope form, y minus y subscript 1 equals m times x minus x subscript 1 in parentheses.” The top of the second cell is blank. The third cell contains the point-slope form, y minus y subscript 1 equals m times x minus x subscript 1 in parentheses. Below this is the form with negative 2 substituted for x subscript 1, 1 substituted for y subscript 1, and negative 1 half substituted for m: y minus 1 equals negative 1 half times x minus negative 2 in parentheses. One line down, the text in the second cell says “Simplify.” The right column contains y minus 1 equals negative 1 half times x plus 2. Below this is y minus 1 equals negative 1 half x plus minus 1.], alt: "In the fourth row, the first cell reads “Step 4. Substitute the values into the point-slope form, y minus y subscript 1 equals m times x minus x subscript 1 in parentheses.” The top of the second cell is blank. The third cell contains the point-slope form, y minus y subscript 1 equals m times x minus x subscript 1 in parentheses. Below this is the form with negative 2 substituted for x subscript 1, 1 substituted for y subscript 1, and negative 1 half substituted for m: y minus 1 equals negative 1 half times x minus negative 2 in parentheses. One line down, the text in the second cell says “Simplify.” The right column contains y minus 1 equals negative 1 half times x plus 2. Below this is y minus 1 equals negative 1 half x plus minus 1.", caption: none) #figure(figph[In the fifth row, the first cell says “Step 5. Write the equation in slope-intercept form.” The second cell is blank. The third cell contains y equals negative 1 half x.], alt: "In the fifth row, the first cell says “Step 5. Write the equation in slope-intercept form.” The second cell is blank. The third cell contains y equals negative 1 half x.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find an equation of a line perpendicular to the line #math.equation(block: false, alt: "y equals 3 x plus 1")[$y = 3 x + 1$] that contains the point #math.equation(block: false, alt: "open parenthesis 4 , 2 close parenthesis")[$( 4 , 2 )$]. Write the equation in slope–intercept form. #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 1 over 3 x plus the fraction 10 over 3")[$y = − frac(1, 3) x + frac(10, 3)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find an equation of a line perpendicular to the line #math.equation(block: false, alt: "y equals the fraction 1 over 2 x minus 3")[$y = frac(1, 2) x − 3$] that contains the point #math.equation(block: false, alt: "open parenthesis 6 , 4 close parenthesis")[$( 6 , 4 )$]. #solutionbox[ #math.equation(block: true, alt: "y equals −2 x plus 16")[$y = −2 x + 16$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Find an equation of a line perpendicular to a given line.] + Find the slope of the given line. + Find the slope of the perpendicular line. + Identify the point. + Substitute the values into the point–slope form, #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis")[$y − y_(1) = m ( x − x_(1) )$]. + Write the equation in slope–intercept form. ] #examplebox("Example 11")[][ Find an equation of a line perpendicular to #math.equation(block: false, alt: "x equals 5")[$x = 5$] that contains the point #math.equation(block: false, alt: "open parenthesis 3 , −2 close parenthesis")[$( 3 , −2 )$]. Write the equation in slope–intercept form. #solutionbox[ Again, since we know one point, the point–slope option seems more promising than the slope–intercept option. We need the slope to use this form, and we know the new line will be perpendicular to #math.equation(block: false, alt: "x equals 5")[$x = 5$]. This line is vertical, so its perpendicular will be horizontal. This tells us the #math.equation(block: false, alt: "m sub ⊥ equals 0")[$m_("⊥") = 0$]. #figure(table( columns: 2, align: left, inset: 6pt, table.header([Identify the point.], [#math.equation(block: false, alt: "open parenthesis 3 , −2 close parenthesis")[$( 3 , −2 )$]]), [Identify the slope of the perpendicular line.], [#math.equation(block: false, alt: "m sub ⊥ equals 0")[$m_("⊥") = 0$]], [Substitute the values into #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis")[$y − y_(1) = m ( x − x_(1) )$].], [#math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis; y minus open parenthesis −2 close parenthesis equals 0 open parenthesis x minus 3 close parenthesis; y plus 2 equals 0")[$y − y_(1) = m ( x − x_(1) ) \ y − ( −2 ) = 0 ( x − 3 ) \ y + 2 = 0$]], [Simplify.], [#math.equation(block: false, alt: "y equals −2")[$y = −2$]], )) Sketch the graph of both lines. Do they appear to be perpendicular? ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find an equation of a line that is perpendicular to the line #math.equation(block: false, alt: "x equals 4")[$x = 4$] that contains the point #math.equation(block: false, alt: "open parenthesis 4 , −5 close parenthesis")[$( 4 , −5 )$]. Write the equation in slope–intercept form. #solutionbox[ #math.equation(block: true, alt: "y equals −5")[$y = −5$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find an equation of a line that is perpendicular to the line #math.equation(block: false, alt: "x equals 2")[$x = 2$] that contains the point #math.equation(block: false, alt: "open parenthesis 2 , −1 close parenthesis")[$( 2 , −1 )$]. Write the equation in slope–intercept form. #solutionbox[ #math.equation(block: true, alt: "y equals −1")[$y = −1$] ] ] In , we used the point–slope form to find the equation. We could have looked at this in a different way. We want to find a line that is perpendicular to #math.equation(block: false, alt: "x equals 5")[$x = 5$] that contains the point #math.equation(block: false, alt: "open parenthesis 3 , −2 close parenthesis")[$( 3 , −2 )$]. The graph shows us the line#math.equation(block: false, alt: "x equals 5")[$x = 5$] and the point #math.equation(block: false, alt: "open parenthesis 3 , −2 close parenthesis")[$( 3 , −2 )$]. #figure(figph[The graph shows the x y-coordinate plane. The x and y-axes each run from negative 7 to 7. The line whose equation is x equals 5 intercepts the x-axis at (5, 0) and runs parallel to the y-axis. Elsewhere on the graph, the point (3, negative 2) is plotted.], alt: "The graph shows the x y-coordinate plane. The x and y-axes each run from negative 7 to 7. The line whose equation is x equals 5 intercepts the x-axis at (5, 0) and runs parallel to the y-axis. Elsewhere on the graph, the point (3, negative 2) is plotted.", caption: none) We know every line perpendicular to a vetical line is horizontal, so we will sketch the horizontal line through #math.equation(block: false, alt: "open parenthesis 3 , −2 close parenthesis")[$( 3 , −2 )$]. #figure(figph[The graph shows the x y-coordinate plane. The x and y-axes each run from negative 7 to 7. The line whose equation is x equals 5 intercepts the x-axis at (5, 0) and runs parallel to the y-axis. Elsewhere on the graph, the points (negative 2, negative 2), (0, negative 2), (3, negative 2), and (6, negative 2) are plotted. A line perpendicular to the previous line passes through those points and runs parallel to the x-axis.], alt: "The graph shows the x y-coordinate plane. The x and y-axes each run from negative 7 to 7. The line whose equation is x equals 5 intercepts the x-axis at (5, 0) and runs parallel to the y-axis. Elsewhere on the graph, the points (negative 2, negative 2), (0, negative 2), (3, negative 2), and (6, negative 2) are plotted. A line perpendicular to the previous line passes through those points and runs parallel to the x-axis.", caption: none) Do the lines appear perpendicular? If we look at a few points on this horizontal line, we notice they all have #emph[y]-coordinates of #math.equation(block: false, alt: "−2")[$−2$]. So, the equation of the line perpendicular to the vertical line #math.equation(block: false, alt: "x equals 5")[$x = 5$] is #math.equation(block: false, alt: "y equals −2")[$y = −2$]. #examplebox("Example 12")[][ Find an equation of a line that is perpendicular to #math.equation(block: false, alt: "y equals −4")[$y = −4$] that contains the point #math.equation(block: false, alt: "open parenthesis −4 , 2 close parenthesis")[$( −4 , 2 )$]. Write the equation in slope–intercept form. #solutionbox[ The line #math.equation(block: false, alt: "y equals −4")[$y = −4$] is a horizontal line. Any line perpendicular to it must be vertical, in the form #math.equation(block: false, alt: "x equals a")[$x = a$]. Since the perpendicular line is vertical and passes through #math.equation(block: false, alt: "open parenthesis −4 , 2 close parenthesis")[$( −4 , 2 )$], every point on it has an #emph[x]-coordinate of #math.equation(block: false, alt: "−4")[$−4$]. The equation of the perpendicular line is #math.equation(block: false, alt: "x equals −4")[$x = −4$]. You may want to sketch the lines. Do they appear perpendicular? ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find an equation of a line that is perpendicular to the line #math.equation(block: false, alt: "y equals 1")[$y = 1$] that contains the point #math.equation(block: false, alt: "open parenthesis −5 , 1 close parenthesis")[$( −5 , 1 )$]. Write the equation in slope–intercept form. #solutionbox[ #math.equation(block: true, alt: "x equals −5")[$x = −5$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find an equation of a line that is perpendicular to the line #math.equation(block: false, alt: "y equals −5")[$y = −5$] that contains the point #math.equation(block: false, alt: "open parenthesis −4 , −5 close parenthesis")[$( −4 , −5 )$]. #solutionbox[ #math.equation(block: true, alt: "x equals −4")[$x = −4$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Access this online resource for additional instruction and practice with finding the equation of a line. - #link("https://openstax.org/l/25PointSlopeForm")[Use the Point-Slope Form of an Equation of a Line] ] === Key Concepts - #strong[To Find an Equation of a Line Given the Slope and a Point] + Identify the slope. + Identify the point. + Substitute the values into the point-slope form, #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis")[$y − y_(1) = m ( x − x_(1) )$]. + Write the equation in slope-intercept form. #linebreak() - #strong[To Find an Equation of a Line Given Two Points] + Find the slope using the given points. + Choose one point. + Substitute the values into the point-slope form, #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis")[$y − y_(1) = m ( x − x_(1) )$]. + Write the equation in slope-intercept form. #linebreak() - #strong[To Write and Equation of a Line] - If given slope and #emph[y]-intercept, use slope–intercept form #math.equation(block: false, alt: "y equals m x plus b")[$y = m x + b$]. - If given slope and a point, use point–slope form #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis")[$y − y_(1) = m ( x − x_(1) )$]. - If given two points, use point–slope form #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis")[$y − y_(1) = m ( x − x_(1) )$]. #linebreak() - #strong[To Find an Equation of a Line Parallel to a Given Line] + Find the slope of the given line. + Find the slope of the parallel line. + Identify the point. + Substitute the values into the point-slope form, #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis")[$y − y_(1) = m ( x − x_(1) )$]. + Write the equation in slope-intercept form. #linebreak() - #strong[To Find an Equation of a Line Perpendicular to a Given Line] + Find the slope of the given line. + Find the slope of the perpendicular line. + Identify the point. + Substitute the values into the point-slope form, #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis")[$y − y_(1) = m ( x − x_(1) )$]. + Write the equation in slope-intercept form. ==== Practice Makes Perfect #strong[Find an Equation of the Line Given the Slope and #emph[y]-Intercept] In the following exercises, find the equation of a line with given slope and y-intercept. Write the equation in slope–intercept form. slope 3 and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis")[$( 0 , 5 )$] slope 4 and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$] #solutionbox[ #math.equation(block: true, alt: "y equals 4 x plus 1")[$y = 4 x + 1$] ] slope 6 and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , −4 close parenthesis")[$( 0 , −4 )$] slope 8 and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , −6 close parenthesis")[$( 0 , −6 )$] #solutionbox[ #math.equation(block: true, alt: "y equals 8 x minus 6")[$y = 8 x − 6$] ] slope #math.equation(block: false, alt: "−1")[$−1$] and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$( 0 , 3 )$] slope #math.equation(block: false, alt: "−1")[$−1$] and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 7 close parenthesis")[$( 0 , 7 )$] #solutionbox[ #math.equation(block: true, alt: "y equals − x plus 7")[$y = "−" x + 7$] ] slope #math.equation(block: false, alt: "−2")[$−2$] and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , −3 close parenthesis")[$( 0 , −3 )$] slope #math.equation(block: false, alt: "−3")[$−3$] and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$( 0 , −1 )$] #solutionbox[ #math.equation(block: true, alt: "y equals −3 x minus 1")[$y = −3 x − 1$] ] slope #math.equation(block: false, alt: "the fraction 3 over 5")[$frac(3, 5)$] and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$( 0 , −1 )$] slope #math.equation(block: false, alt: "the fraction 1 over 5")[$frac(1, 5)$] and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , −5 close parenthesis")[$( 0 , −5 )$] #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 1 over 5 x minus 5")[$y = frac(1, 5) x − 5$] ] slope #math.equation(block: false, alt: "minus the fraction 3 over 4")[$− frac(3, 4)$] and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , −2 close parenthesis")[$( 0 , −2 )$] slope #math.equation(block: false, alt: "minus the fraction 2 over 3")[$− frac(2, 3)$] and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , −3 close parenthesis")[$( 0 , −3 )$] #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 2 over 3 x minus 3")[$y = − frac(2, 3) x − 3$] ] slope 0 and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$( 0 , −1 )$] slope 0 and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$] #solutionbox[ #math.equation(block: true, alt: "y equals 2")[$y = 2$] ] slope #math.equation(block: false, alt: "−3")[$−3$] and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] slope #math.equation(block: false, alt: "−4")[$−4$] and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] #solutionbox[ #math.equation(block: true, alt: "y equals −4 x")[$y = −4 x$] ] In the following exercises, find the equation of the line shown in each graph. Write the equation in slope–intercept form. #figure(figph[The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (1, negative 2) is plotted. A line intercepts the y-axis at (0, negative 5), passes through the point (1, negative 2), and intercepts the x-axis at (5 thirds, 0).], alt: "The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (1, negative 2) is plotted. A line intercepts the y-axis at (0, negative 5), passes through the point (1, negative 2), and intercepts the x-axis at (5 thirds, 0).", caption: none) #figure(figph[The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (2, 0) is plotted. A line intercepts the y-axis at (0, 4) and intercepts the x-axis at (2, 0).], alt: "The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (2, 0) is plotted. A line intercepts the y-axis at (0, 4) and intercepts the x-axis at (2, 0).", caption: none) #solutionbox[ #math.equation(block: true, alt: "y equals −2 x plus 4")[$y = −2 x + 4$] ] #figure(figph[The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (6, 0) is plotted. A line intercepts the y-axis at (0, negative 3) and intercepts the x-axis at (6, 0).], alt: "The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (6, 0) is plotted. A line intercepts the y-axis at (0, negative 3) and intercepts the x-axis at (6, 0).", caption: none) #figure(figph[The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (4, 5) is plotted. A line intercepts the x-axis at (negative 8 thirds, 0), intercepts the y-axis at (0, 2), and passes through the point (4, 5).], alt: "The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (4, 5) is plotted. A line intercepts the x-axis at (negative 8 thirds, 0), intercepts the y-axis at (0, 2), and passes through the point (4, 5).", caption: none) #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 3 over 4 x plus 2")[$y = frac(3, 4) x + 2$] ] #figure(figph[The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (3, negative 1) is plotted. A line intercepts the y-axis at (0, 2), intercepts the x-axis at (9 fourths, 0), and passes through the point (3, negative 1).], alt: "The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (3, negative 1) is plotted. A line intercepts the y-axis at (0, 2), intercepts the x-axis at (9 fourths, 0), and passes through the point (3, negative 1).", caption: none) #figure(figph[The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (2, negative 4) is plotted. A line intercepts the x-axis at (negative 2 thirds, 0), intercepts the y-axis at (0, negative 1), and passes through the point (2, negative 4).], alt: "The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (2, negative 4) is plotted. A line intercepts the x-axis at (negative 2 thirds, 0), intercepts the y-axis at (0, negative 1), and passes through the point (2, negative 4).", caption: none) #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 3 over 2 x minus 1")[$y = − frac(3, 2) x − 1$] ] #figure(figph[The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (2, negative 2) is plotted. A line running parallel to the x-axis intercepts the y-axis at (0, negative 2) and passes through the point (2, negative 2).], alt: "The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (2, negative 2) is plotted. A line running parallel to the x-axis intercepts the y-axis at (0, negative 2) and passes through the point (2, negative 2).", caption: none) #figure(figph[The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (negative 3, 6) is plotted. A line running parallel to the x-axis passes through (negative 3, 6) and intercepts the y-axis at (0, 6).], alt: "The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (negative 3, 6) is plotted. A line running parallel to the x-axis passes through (negative 3, 6) and intercepts the y-axis at (0, 6).", caption: none) #solutionbox[ #math.equation(block: true, alt: "y equals 6")[$y = 6$] ] #strong[Find an Equation of the Line Given the Slope and a Point] In the following exercises, find the equation of a line with given slope and containing the given point. Write the equation in slope–intercept form. #math.equation(block: false, alt: "m equals the fraction 5 over 8")[$m = frac(5, 8)$], point #math.equation(block: false, alt: "open parenthesis 8 , 3 close parenthesis")[$( 8 , 3 )$] #math.equation(block: false, alt: "m equals the fraction 3 over 8")[$m = frac(3, 8)$], point #math.equation(block: false, alt: "open parenthesis 8 , 2 close parenthesis")[$( 8 , 2 )$] #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 3 over 8 x minus 1")[$y = frac(3, 8) x − 1$] ] #math.equation(block: false, alt: "m equals the fraction 1 over 6")[$m = frac(1, 6)$], point #math.equation(block: false, alt: "open parenthesis 6 , 1 close parenthesis")[$( 6 , 1 )$] #math.equation(block: false, alt: "m equals the fraction 5 over 6")[$m = frac(5, 6)$], point #math.equation(block: false, alt: "open parenthesis 6 , 7 close parenthesis")[$( 6 , 7 )$] #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 5 over 6 x plus 2")[$y = frac(5, 6) x + 2$] ] #math.equation(block: false, alt: "m equals minus the fraction 3 over 4")[$m = − frac(3, 4)$], point #math.equation(block: false, alt: "open parenthesis 8 , −5 close parenthesis")[$( 8 , −5 )$] #math.equation(block: false, alt: "m equals minus the fraction 3 over 5")[$m = − frac(3, 5)$], point #math.equation(block: false, alt: "open parenthesis 10 , −5 close parenthesis")[$( 10 , −5 )$] #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 3 over 5 x plus 1")[$y = − frac(3, 5) x + 1$] ] #math.equation(block: false, alt: "m equals minus the fraction 1 over 4")[$m = − frac(1, 4)$], point #math.equation(block: false, alt: "open parenthesis −12 , −6 close parenthesis")[$( −12 , −6 )$] #math.equation(block: false, alt: "m equals minus the fraction 1 over 3")[$m = − frac(1, 3)$], point #math.equation(block: false, alt: "open parenthesis −9 , −8 close parenthesis")[$( −9 , −8 )$] #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 1 over 3 x minus 11")[$y = − frac(1, 3) x − 11$] ] Horizontal line containing #math.equation(block: false, alt: "open parenthesis −2 , 5 close parenthesis")[$( −2 , 5 )$] Horizontal line containing #math.equation(block: false, alt: "open parenthesis −1 , 4 close parenthesis")[$( −1 , 4 )$] #solutionbox[ #math.equation(block: true, alt: "y equals 4")[$y = 4$] ] Horizontal line containing #math.equation(block: false, alt: "open parenthesis −2 , −3 close parenthesis")[$( −2 , −3 )$] Horizontal line containing #math.equation(block: false, alt: "open parenthesis −1 , −7 close parenthesis")[$( −1 , −7 )$] #solutionbox[ #math.equation(block: true, alt: "y equals −7")[$y = −7$] ] #math.equation(block: false, alt: "m equals minus the fraction 3 over 2")[$m = − frac(3, 2)$], point #math.equation(block: false, alt: "open parenthesis −4 , −3 close parenthesis")[$( −4 , −3 )$] #math.equation(block: false, alt: "m equals minus the fraction 5 over 2")[$m = − frac(5, 2)$], point #math.equation(block: false, alt: "open parenthesis −8 , −2 close parenthesis")[$( −8 , −2 )$] #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 5 over 2 x minus 22")[$y = − frac(5, 2) x − 22$] ] #math.equation(block: false, alt: "m equals −7")[$m = −7$], point #math.equation(block: false, alt: "open parenthesis −1 , −3 close parenthesis")[$( −1 , −3 )$] #math.equation(block: false, alt: "m equals −4")[$m = −4$], point #math.equation(block: false, alt: "open parenthesis −2 , −3 close parenthesis")[$( −2 , −3 )$] #solutionbox[ #math.equation(block: true, alt: "y equals −4 x minus 11")[$y = −4 x − 11$] ] Horizontal line containing #math.equation(block: false, alt: "open parenthesis 2 , −3 close parenthesis")[$( 2 , −3 )$] Horizontal line containing #math.equation(block: false, alt: "open parenthesis 4 , −8 close parenthesis")[$( 4 , −8 )$] #solutionbox[ #math.equation(block: true, alt: "y equals −8")[$y = −8$] ] #strong[Find an Equation of the Line Given Two Points] In the following exercises, find the equation of a line containing the given points. Write the equation in slope–intercept form. #math.equation(block: false, alt: "open parenthesis 2 , 6 close parenthesis")[$( 2 , 6 )$] and #math.equation(block: false, alt: "open parenthesis 5 , 3 close parenthesis")[$( 5 , 3 )$] #math.equation(block: false, alt: "open parenthesis 3 , 1 close parenthesis")[$( 3 , 1 )$] and #math.equation(block: false, alt: "open parenthesis 2 , 5 close parenthesis")[$( 2 , 5 )$] #solutionbox[ #math.equation(block: true, alt: "y equals −4 x plus 13")[$y = −4 x + 13$] ] #math.equation(block: false, alt: "open parenthesis 4 , 3 close parenthesis")[$( 4 , 3 )$] and #math.equation(block: false, alt: "open parenthesis 8 , 1 close parenthesis")[$( 8 , 1 )$] #math.equation(block: false, alt: "open parenthesis 2 , 7 close parenthesis")[$( 2 , 7 )$] and #math.equation(block: false, alt: "open parenthesis 3 , 8 close parenthesis")[$( 3 , 8 )$] #solutionbox[ #math.equation(block: true, alt: "y equals x plus 5")[$y = x + 5$] ] #math.equation(block: false, alt: "open parenthesis −3 , −4 close parenthesis")[$( −3 , −4 )$] and #math.equation(block: false, alt: "open parenthesis 5 minus 2 close parenthesis")[$( 5 − 2 )$] #math.equation(block: false, alt: "open parenthesis −5 , −3 close parenthesis")[$( −5 , −3 )$] and #math.equation(block: false, alt: "open parenthesis 4 , −6 close parenthesis")[$( 4 , −6 )$] #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 1 over 3 x minus the fraction 14 over 3")[$y = − frac(1, 3) x − frac(14, 3)$] ] #math.equation(block: false, alt: "open parenthesis −1 , 3 close parenthesis")[$( −1 , 3 )$] and #math.equation(block: false, alt: "open parenthesis −6 , −7 close parenthesis")[$( −6 , −7 )$] #math.equation(block: false, alt: "open parenthesis −2 , 8 close parenthesis")[$( −2 , 8 )$] and #math.equation(block: false, alt: "open parenthesis −4 , −6 close parenthesis")[$( −4 , −6 )$] #solutionbox[ #math.equation(block: true, alt: "y equals 7 x plus 22")[$y = 7 x + 22$] ] #math.equation(block: false, alt: "open parenthesis 6 , −4 close parenthesis")[$( 6 , −4 )$] and #math.equation(block: false, alt: "open parenthesis −2 , 5 close parenthesis")[$( −2 , 5 )$] #math.equation(block: false, alt: "open parenthesis 3 , −2 close parenthesis")[$( 3 , −2 )$] and #math.equation(block: false, alt: "open parenthesis −4 , 4 close parenthesis")[$( −4 , 4 )$] #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 6 over 7 x plus the fraction 4 over 7")[$y = − frac(6, 7) x + frac(4, 7)$] ] #math.equation(block: false, alt: "open parenthesis 0 , 4 close parenthesis")[$( 0 , 4 )$] and #math.equation(block: false, alt: "open parenthesis 2 , −3 close parenthesis")[$( 2 , −3 )$] #math.equation(block: false, alt: "open parenthesis 0 , −2 close parenthesis")[$( 0 , −2 )$] and #math.equation(block: false, alt: "open parenthesis −5 , −3 close parenthesis")[$( −5 , −3 )$] #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 1 over 5 x minus 2")[$y = frac(1, 5) x − 2$] ] #math.equation(block: false, alt: "open parenthesis 7 , 2 close parenthesis")[$( 7 , 2 )$] and #math.equation(block: false, alt: "open parenthesis 7 , −2 close parenthesis")[$( 7 , −2 )$] #math.equation(block: false, alt: "open parenthesis 4 , 2 close parenthesis")[$( 4 , 2 )$] and #math.equation(block: false, alt: "open parenthesis 4 , −3 close parenthesis")[$( 4 , −3 )$] #solutionbox[ #math.equation(block: true, alt: "x equals 4")[$x = 4$] ] #math.equation(block: false, alt: "open parenthesis −7 , −1 close parenthesis")[$( −7 , −1 )$] and #math.equation(block: false, alt: "open parenthesis −7 , −4 close parenthesis")[$( −7 , −4 )$] #math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis")[$( −2 , 1 )$] and #math.equation(block: false, alt: "open parenthesis −2 , −4 close parenthesis")[$( −2 , −4 )$] #solutionbox[ #math.equation(block: true, alt: "x equals −2")[$x = −2$] ] #math.equation(block: false, alt: "open parenthesis 6 , 1 close parenthesis")[$( 6 , 1 )$] and #math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$] #math.equation(block: false, alt: "open parenthesis 6 , 2 close parenthesis")[$( 6 , 2 )$] and #math.equation(block: false, alt: "open parenthesis −3 , 2 close parenthesis")[$( −3 , 2 )$] #solutionbox[ #math.equation(block: true, alt: "y equals 2")[$y = 2$] ] #math.equation(block: false, alt: "open parenthesis 3 , −4 close parenthesis")[$( 3 , −4 )$] and #math.equation(block: false, alt: "open parenthesis 5 , −4 close parenthesis")[$( 5 , −4 )$] #math.equation(block: false, alt: "open parenthesis −6 , −3 close parenthesis")[$( −6 , −3 )$] and #math.equation(block: false, alt: "open parenthesis −1 , −3 close parenthesis")[$( −1 , −3 )$] #solutionbox[ #math.equation(block: true, alt: "y equals −3")[$y = −3$] ] #math.equation(block: false, alt: "open parenthesis 4 , 3 close parenthesis")[$( 4 , 3 )$] and #math.equation(block: false, alt: "open parenthesis 8 , 0 close parenthesis")[$( 8 , 0 )$] #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] and #math.equation(block: false, alt: "open parenthesis 1 , 4 close parenthesis")[$( 1 , 4 )$] #solutionbox[ #math.equation(block: true, alt: "y equals 4 x")[$y = 4 x$] ] #math.equation(block: false, alt: "open parenthesis −2 , −3 close parenthesis")[$( −2 , −3 )$] and #math.equation(block: false, alt: "open parenthesis −5 , −6 close parenthesis")[$( −5 , −6 )$] #math.equation(block: false, alt: "open parenthesis −3 , 0 close parenthesis")[$( −3 , 0 )$] and #math.equation(block: false, alt: "open parenthesis −7 , −2 close parenthesis")[$( −7 , −2 )$] #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 1 over 2 x plus the fraction 3 over 2")[$y = frac(1, 2) x + frac(3, 2)$] ] #math.equation(block: false, alt: "open parenthesis 8 , −1 close parenthesis")[$( 8 , −1 )$] and #math.equation(block: false, alt: "open parenthesis 8 , −5 close parenthesis")[$( 8 , −5 )$] #math.equation(block: false, alt: "open parenthesis 3 , 5 close parenthesis")[$( 3 , 5 )$] and #math.equation(block: false, alt: "open parenthesis −7 , 5 close parenthesis")[$( −7 , 5 )$] #solutionbox[ #math.equation(block: true, alt: "y equals 5")[$y = 5$] ] #strong[Find an Equation of a Line Parallel to a Given Line] In the following exercises, find an equation of a line parallel to the given line and contains the given point. Write the equation in slope–intercept form. line #math.equation(block: false, alt: "y equals 4 x plus 2")[$y = 4 x + 2$], point #math.equation(block: false, alt: "open parenthesis 1 , 2 close parenthesis")[$( 1 , 2 )$] line #math.equation(block: false, alt: "y equals 3 x plus 4")[$y = 3 x + 4$], point #math.equation(block: false, alt: "open parenthesis 2 , 5 close parenthesis")[$( 2 , 5 )$] #solutionbox[ #math.equation(block: true, alt: "y equals 3 x minus 1")[$y = 3 x − 1$] ] line #math.equation(block: false, alt: "y equals −2 x minus 3")[$y = −2 x − 3$], point #math.equation(block: false, alt: "open parenthesis −1 , 3 close parenthesis")[$( −1 , 3 )$] line #math.equation(block: false, alt: "y equals −3 x minus 1")[$y = −3 x − 1$], point #math.equation(block: false, alt: "open parenthesis 2 , −3 close parenthesis")[$( 2 , −3 )$] #solutionbox[ #math.equation(block: true, alt: "y equals −3 x plus 3")[$y = −3 x + 3$] ] line #math.equation(block: false, alt: "3 x minus y equals 4")[$3 x − y = 4$], point #math.equation(block: false, alt: "open parenthesis 3 , 1 close parenthesis")[$( 3 , 1 )$] line #math.equation(block: false, alt: "2 x minus y equals 6")[$2 x − y = 6$], point #math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis")[$( 3 , 0 )$] #solutionbox[ #math.equation(block: true, alt: "y equals 2 x minus 6")[$y = 2 x − 6$] ] line #math.equation(block: false, alt: "4 x plus 3 y equals 6")[$4 x + 3 y = 6$], point #math.equation(block: false, alt: "open parenthesis 0 , −3 close parenthesis")[$( 0 , −3 )$] line #math.equation(block: false, alt: "2 x plus 3 y equals 6")[$2 x + 3 y = 6$], point #math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis")[$( 0 , 5 )$] #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 2 over 3 x plus 5")[$y = − frac(2, 3) x + 5$] ] line #math.equation(block: false, alt: "x equals −3")[$x = −3$], point #math.equation(block: false, alt: "open parenthesis −2 , −1 close parenthesis")[$( −2 , −1 )$] line #math.equation(block: false, alt: "x equals −4")[$x = −4$], point #math.equation(block: false, alt: "open parenthesis −3 , −5 close parenthesis")[$( −3 , −5 )$] #solutionbox[ #math.equation(block: true, alt: "x equals −3")[$x = −3$] ] line #math.equation(block: false, alt: "x minus 2 equals 0")[$x − 2 = 0$], point #math.equation(block: false, alt: "open parenthesis 1 , −2 close parenthesis")[$( 1 , −2 )$] line #math.equation(block: false, alt: "x minus 6 equals 0")[$x − 6 = 0$], point #math.equation(block: false, alt: "open parenthesis 4 , −3 close parenthesis")[$( 4 , −3 )$] #solutionbox[ #math.equation(block: true, alt: "x equals 4")[$x = 4$] ] line #math.equation(block: false, alt: "y equals 5")[$y = 5$], point #math.equation(block: false, alt: "open parenthesis 2 , −2 close parenthesis")[$( 2 , −2 )$] line #math.equation(block: false, alt: "y equals 1")[$y = 1$], point #math.equation(block: false, alt: "open parenthesis 3 , −4 close parenthesis")[$( 3 , −4 )$] #solutionbox[ #math.equation(block: true, alt: "y equals −4")[$y = −4$] ] line #math.equation(block: false, alt: "y plus 2 equals 0")[$y + 2 = 0$], point #math.equation(block: false, alt: "open parenthesis 3 , −3 close parenthesis")[$( 3 , −3 )$] line #math.equation(block: false, alt: "y plus 7 equals 0")[$y + 7 = 0$], point #math.equation(block: false, alt: "open parenthesis 1 , −1 close parenthesis")[$( 1 , −1 )$] #solutionbox[ #math.equation(block: true, alt: "y equals −1")[$y = −1$] ] #strong[Find an Equation of a Line Perpendicular to a Given Line] In the following exercises, find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope–intercept form. line #math.equation(block: false, alt: "y equals −2 x plus 3")[$y = −2 x + 3$], point #math.equation(block: false, alt: "open parenthesis 2 , 2 close parenthesis")[$( 2 , 2 )$] line #math.equation(block: false, alt: "y equals − x plus 5")[$y = "−" x + 5$], point #math.equation(block: false, alt: "open parenthesis 3 , 3 close parenthesis")[$( 3 , 3 )$] #solutionbox[ #math.equation(block: true, alt: "y equals x")[$y = x$] ] line #math.equation(block: false, alt: "y equals the fraction 3 over 4 x minus 2")[$y = frac(3, 4) x − 2$], point #math.equation(block: false, alt: "open parenthesis −3 , 4 close parenthesis")[$( −3 , 4 )$] line #math.equation(block: false, alt: "y equals the fraction 2 over 3 x minus 4")[$y = frac(2, 3) x − 4$], point #math.equation(block: false, alt: "open parenthesis 2 , −4 close parenthesis")[$( 2 , −4 )$] #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 3 over 2 x minus 1")[$y = − frac(3, 2) x − 1$] ] line #math.equation(block: false, alt: "2 x minus 3 y equals 8")[$2 x − 3 y = 8$], point #math.equation(block: false, alt: "open parenthesis 4 , −1 close parenthesis")[$( 4 , −1 )$] line #math.equation(block: false, alt: "4 x minus 3 y equals 5")[$4 x − 3 y = 5$], point #math.equation(block: false, alt: "open parenthesis −3 , 2 close parenthesis")[$( −3 , 2 )$] #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 3 over 4 x minus the fraction 1 over 4")[$y = − frac(3, 4) x − frac(1, 4)$] ] line #math.equation(block: false, alt: "2 x plus 5 y equals 6")[$2 x + 5 y = 6$], point #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] line #math.equation(block: false, alt: "4 x plus 5 y equals −3")[$4 x + 5 y = −3$], point #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 5 over 4 x")[$y = frac(5, 4) x$] ] line #math.equation(block: false, alt: "y minus 3 equals 0")[$y − 3 = 0$], point #math.equation(block: false, alt: "open parenthesis −2 , −4 close parenthesis")[$( −2 , −4 )$] line #math.equation(block: false, alt: "y minus 6 equals 0")[$y − 6 = 0$], point #math.equation(block: false, alt: "open parenthesis −5 , −3 close parenthesis")[$( −5 , −3 )$] #solutionbox[ #math.equation(block: true, alt: "x equals −5")[$x = −5$] ] line #emph[y]-axis, point #math.equation(block: false, alt: "open parenthesis 3 , 4 close parenthesis")[$( 3 , 4 )$] line #emph[y]-axis, point #math.equation(block: false, alt: "open parenthesis 2 , 1 close parenthesis")[$( 2 , 1 )$] #solutionbox[ #math.equation(block: true, alt: "y equals 1")[$y = 1$] ] ==== Mixed Practice In the following exercises, find the equation of each line. Write the equation in slope–intercept form. Containing the points #math.equation(block: false, alt: "open parenthesis 4 , 3 close parenthesis")[$( 4 , 3 )$] and #math.equation(block: false, alt: "open parenthesis 8 , 1 close parenthesis")[$( 8 , 1 )$] Containing the points #math.equation(block: false, alt: "open parenthesis 2 , 7 close parenthesis")[$( 2 , 7 )$] and #math.equation(block: false, alt: "open parenthesis 3 , 8 close parenthesis")[$( 3 , 8 )$] #solutionbox[ #math.equation(block: true, alt: "y equals x plus 5")[$y = x + 5$] ] #math.equation(block: false, alt: "m equals the fraction 1 over 6")[$m = frac(1, 6)$], containing point #math.equation(block: false, alt: "open parenthesis 6 , 1 close parenthesis")[$( 6 , 1 )$] #math.equation(block: false, alt: "m equals the fraction 5 over 6")[$m = frac(5, 6)$], containing point #math.equation(block: false, alt: "open parenthesis 6 , 7 close parenthesis")[$( 6 , 7 )$] #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 5 over 6 x plus 2")[$y = frac(5, 6) x + 2$] ] Parallel to the line #math.equation(block: false, alt: "4 x plus 3 y equals 6")[$4 x + 3 y = 6$], containing point #math.equation(block: false, alt: "open parenthesis 0 , −3 close parenthesis")[$( 0 , −3 )$] Parallel to the line #math.equation(block: false, alt: "2 x plus 3 y equals 6")[$2 x + 3 y = 6$], containing point #math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis")[$( 0 , 5 )$] #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 2 over 3 x plus 5")[$y = − frac(2, 3) x + 5$] ] #math.equation(block: false, alt: "m equals minus the fraction 3 over 4")[$m = − frac(3, 4)$], containing point #math.equation(block: false, alt: "open parenthesis 8 , −5 close parenthesis")[$( 8 , −5 )$] #math.equation(block: false, alt: "m equals minus the fraction 3 over 5")[$m = − frac(3, 5)$], containing point #math.equation(block: false, alt: "open parenthesis 10 , −5 close parenthesis")[$( 10 , −5 )$] #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 3 over 5 x plus 1")[$y = − frac(3, 5) x + 1$] ] Perpendicular to the line #math.equation(block: false, alt: "y minus 1 equals 0")[$y − 1 = 0$], point #math.equation(block: false, alt: "open parenthesis −2 , 6 close parenthesis")[$( −2 , 6 )$] Perpendicular to the line #emph[y]-axis, point #math.equation(block: false, alt: "open parenthesis −6 , 2 close parenthesis")[$( −6 , 2 )$] #solutionbox[ #math.equation(block: true, alt: "y equals 2")[$y = 2$] ] Containing the points #math.equation(block: false, alt: "open parenthesis 4 , 3 close parenthesis")[$( 4 , 3 )$] and #math.equation(block: false, alt: "open parenthesis 8 , 1 close parenthesis")[$( 8 , 1 )$] Containing the points #math.equation(block: false, alt: "open parenthesis −2 , 0 close parenthesis")[$( −2 , 0 )$] and #math.equation(block: false, alt: "open parenthesis −3 , −2 close parenthesis")[$( −3 , −2 )$] #solutionbox[ #math.equation(block: true, alt: "y equals 2 x plus 4")[$y = 2 x + 4$] ] Parallel to the line #math.equation(block: false, alt: "x equals −3")[$x = −3$], containing point #math.equation(block: false, alt: "open parenthesis −2 , −1 close parenthesis")[$( −2 , −1 )$] Parallel to the line #math.equation(block: false, alt: "x equals −4")[$x = −4$], containing point #math.equation(block: false, alt: "open parenthesis −3 , −5 close parenthesis")[$( −3 , −5 )$] #solutionbox[ #math.equation(block: true, alt: "x equals −3")[$x = −3$] ] Containing the points #math.equation(block: false, alt: "open parenthesis −3 , −4 close parenthesis")[$( −3 , −4 )$] and #math.equation(block: false, alt: "open parenthesis 2 , −5 close parenthesis")[$( 2 , −5 )$] Containing the points #math.equation(block: false, alt: "open parenthesis −5 , −3 close parenthesis")[$( −5 , −3 )$] and #math.equation(block: false, alt: "open parenthesis 4 , −6 close parenthesis")[$( 4 , −6 )$] #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 1 over 3 x minus the fraction 14 over 3")[$y = − frac(1, 3) x − frac(14, 3)$] ] Perpendicular to the line #math.equation(block: false, alt: "x minus 2 y equals 5")[$x − 2 y = 5$], containing point #math.equation(block: false, alt: "open parenthesis −2 , 2 close parenthesis")[$( −2 , 2 )$] Perpendicular to the line #math.equation(block: false, alt: "4 x plus 3 y equals 1")[$4 x + 3 y = 1$], containing point #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 3 over 4 x")[$y = frac(3, 4) x$] ] ==== Everyday Math #strong[Cholesterol.] The age, #math.equation(block: false, alt: "x")[$x$], and LDL cholesterol level, #math.equation(block: false, alt: "y")[$y$], of two men are given by the points #math.equation(block: false, alt: "open parenthesis 18 , 68 close parenthesis")[$( 18 , 68 )$] and #math.equation(block: false, alt: "open parenthesis 27 , 122 close parenthesis")[$( 27 , 122 )$]. Find a linear equation that models the relationship between age and LDL cholesterol level. #strong[Fuel consumption.] The city mpg, #math.equation(block: false, alt: "x")[$x$], and highway mpg, #math.equation(block: false, alt: "y")[$y$], of two cars are given by the points #math.equation(block: false, alt: "open parenthesis 29 , 40 close parenthesis")[$( 29 , 40 )$] and#math.equation(block: false, alt: "open parenthesis 19 , 28 close parenthesis")[$( 19 , 28 )$]. Find a linear equation that models the relationship between city mpg and highway mpg. #solutionbox[ #math.equation(block: true, alt: "y equals 1.2 x plus 5.2")[$y = 1.2 x + 5.2$] ] ==== Writing Exercises Why are all horizontal lines parallel? Explain in your own words why the slopes of two perpendicular lines must have opposite signs. #solutionbox[ Answers will vary. ] ==== Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. #figure(figph[Table assessing student understanding of finding linear equations. Columns: "I can...", "confidently", "with some help", "no-I don’t get it!". Rows describe skills, including finding equations from slope/intercept, slope/point, two points, parallel lines, and perpendicular lines.], alt: "Table assessing student understanding of finding linear equations. Columns: \"I can...\", \"confidently\", \"with some help\", \"no-I don’t get it!\". Rows describe skills, including finding equations from slope/intercept, slope/point, two points, parallel lines, and perpendicular lines.", caption: none) ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?