#set document(title: "4.5 Use the Slope-Intercept Form of an 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"))


== 4.5#h(0.6em)Use the Slope-Intercept Form of an Equation of a Line

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Before you get started, take this readiness quiz.

Add: #math.equation(block: false, alt: "the fraction x over 4 plus the fraction 1 over 4 .")[$frac(x, 4) + frac(1, 4) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "the fraction x plus 1 over 4")[$frac(x + 1, 4)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the reciprocal of #math.equation(block: false, alt: "the fraction 3 over 7 .")[$frac(3, 7) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "the fraction 7 over 3")[$frac(7, 3)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve #math.equation(block: false, alt: "2 x minus 3 y equals 12 for y")[$2 x − 3 y = 12 #h(0.2em) "for" #h(0.2em) y$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "y equals the fraction 12 minus 2 x over −3")[$y = frac(12 − 2 x, −3)$]
]
]


=== Recognize the Relation Between the Graph and the Slope–Intercept Form of an Equation of a Line

We have graphed linear equations by plotting points, using intercepts, recognizing horizontal and vertical lines, and using the point–slope method. Once we see how an equation in slope–intercept form and its graph are related, we’ll have one more method we can use to graph lines.

In Graph Linear Equations in Two Variables, we graphed the line of the equation #math.equation(block: false, alt: "y equals the fraction 1 over 2 x plus 3")[$y = frac(1, 2) x + 3$] by plotting points. Let’s find the slope of this line.

#figure(figph[This figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. The line is labeled with the equation y equals one half x, plus 3. The points (0, 3), (2, 4) and (4, 5) are labeled also. A red vertical line begins at the point (2, 4) and ends one unit above the point. It is labeled “Rise equals 1”. A red horizontal line begins at the end of the vertical line and ends at the point (4, 5). It is labeled “Run equals 2. The red lines create a right triangle with the line y equals one half x, plus 3 as the hypotenuse.], alt: "This figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. The line is labeled with the equation y equals one half x, plus 3. The points (0, 3), (2, 4) and (4, 5) are labeled also. A red vertical line begins at the point (2, 4) and ends one unit above the point. It is labeled “Rise equals 1”. A red horizontal line begins at the end of the vertical line and ends at the point (4, 5). It is labeled “Run equals 2. The red lines create a right triangle with the line y equals one half x, plus 3 as the hypotenuse.", caption: none)

The red lines show us the rise is 1 and the run is 2. Substituting into the slope formula:

#math.equation(block: true, alt: "m, equals, the fraction rise over run; m, equals, the fraction 1 over 2")[$m & = & frac("rise", "run") \ m & = & frac(1, 2)$]What is the #emph[y]-intercept of the line? The #emph[y]-intercept is where the line crosses the #emph[y]-axis, so #emph[y]-intercept is #math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$( 0 , 3 )$]. The equation of this line is:

#figure(figph[The figure shows the equation y equals one half x, plus 3. The fraction one half is colored red and the number 3 is colored blue.], alt: "The figure shows the equation y equals one half x, plus 3. The fraction one half is colored red and the number 3 is colored blue.", caption: none)

Notice, the line has:

#figure(figph[The figure shows the statement “slope m equals one half and y-intercept (0, 3). The slope, one half, is colored red and the number 3 in the y-intercept is colored blue.], alt: "The figure shows the statement “slope m equals one half and y-intercept (0, 3). The slope, one half, is colored red and the number 3 in the y-intercept is colored blue.", caption: none)

When a linear equation is solved for #math.equation(block: false, alt: "y")[$y$], the coefficient of the #math.equation(block: false, alt: "x")[$x$] term is the slope and the constant term is the #emph[y]-coordinate of the #emph[y]-intercept. We say that the equation #math.equation(block: false, alt: "y equals the fraction 1 over 2 x plus 3")[$y = frac(1, 2) x + 3$] is in slope–intercept form.

#figure(figph[The figure shows the statement “m equals one half; y-intercept is (0, 3). The slope, one half, is colored red and the number 3 in the y-intercept is colored blue. Below that statement is the equation y equals one half x, plus 3. The fraction one half is colored red and the number 3 is colored blue. Below the equation is another equation y equals m x, plus b. The variable m is colored red and the variable b is colored blue.], alt: "The figure shows the statement “m equals one half; y-intercept is (0, 3). The slope, one half, is colored red and the number 3 in the y-intercept is colored blue. Below that statement is the equation y equals one half x, plus 3. The fraction one half is colored red and the number 3 is colored blue. Below the equation is another equation y equals m x, plus b. The variable m is colored red and the variable b is colored blue.", caption: none)

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Slope-Intercept Form of an Equation of a Line]

The #strong[slope–intercept form] of an equation of a line with slope #math.equation(block: false, alt: "m")[$m$] and #emph[y]-intercept, #math.equation(block: false, alt: "open parenthesis 0 , b close parenthesis")[$( 0 , b )$] is,

#math.equation(block: true, alt: "y equals m x plus b")[$y = m x + b$]
]

Sometimes the slope–intercept form is called the “#emph[y]-form.”

#examplebox("Example 1")[][
Use the graph to find the slope and #emph[y]-intercept of the line, #math.equation(block: false, alt: "y equals 2 x plus 1")[$y = 2 x + 1$].

Compare these values to the equation#math.equation(block: false, alt: "y equals m x plus b")[$y = m x + b$].

#solutionbox[
To find the slope of the line, we need to choose two points on the line. We’ll use the points #math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$] and #math.equation(block: false, alt: "open parenthesis 1 , 3 close parenthesis")[$( 1 , 3 )$].

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A graph on a coordinate plane shows the linear equation y = 2x + 1. The line, with arrows, passes through points (0, 1) and (1, 3). The x and y axes are marked from -4 to 4.], alt: "A graph on a coordinate plane shows the linear equation y = 2x + 1. The line, with arrows, passes through points (0, 1) and (1, 3). The x and y axes are marked from -4 to 4.", caption: none)]),
  [Find the rise and run.], [#figure(figph[A mathematical equation for slope reads 'm = rise / run' on a plain white background.], alt: "A mathematical equation for slope reads 'm = rise / run' on a plain white background.", caption: none)],
  [], [#figure(figph[A mathematical equation displays 'm = 2/1' on a white background, representing a slope or a ratio where m is equal to two divided by one.], alt: "A mathematical equation displays 'm = 2/1' on a white background, representing a slope or a ratio where m is equal to two divided by one.", caption: none)],
  [], [#figure(figph[A mathematical expression 'm = 2' is displayed centrally on a white background.], alt: "A mathematical expression 'm = 2' is displayed centrally on a white background.", caption: none)],
  [Find the #emph[y]-intercept of the line.], [The #emph[y]-intercept is the point (0, 1).],
  [#figure(figph[We found slope m = 2 and y-intercept (0, 1).], alt: "We found slope m = 2 and y-intercept (0, 1).", caption: none)], [#figure(figph[Comparing y=2x+1 with the slope-intercept form y=mx+b, where m=2 (red) is the slope and b=1 (blue) is the y-intercept.], alt: "Comparing y=2x+1 with the slope-intercept form y=mx+b, where m=2 (red) is the slope and b=1 (blue) is the y-intercept.", caption: none)],
))

The slope is the same as the coefficient of #math.equation(block: false, alt: "x")[$x$] and the #emph[y]-coordinate of the #emph[y]-intercept is the same as the constant term.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the graph to find the slope and #emph[y]-intercept of the line #math.equation(block: false, alt: "y equals the fraction 2 over 3 x minus 1")[$y = frac(2, 3) x − 1$]. Compare these values to the equation #math.equation(block: false, alt: "y equals m x plus b")[$y = m x + b$].

#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. The line goes through the points (0, negative 1) and (6, 3).], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. The line goes through the points (0, negative 1) and (6, 3).", caption: none)

#solutionbox[
slope #math.equation(block: false, alt: "m equals the fraction 2 over 3")[$m = frac(2, 3)$] and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$( 0 , −1 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the graph to find the slope and #emph[y]-intercept of the line #math.equation(block: false, alt: "y equals the fraction 1 over 2 x plus 3")[$y = frac(1, 2) x + 3$]. Compare these values to the equation #math.equation(block: false, alt: "y equals m x plus b")[$y = m x + b$].

#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. The line goes through the points (0, 3) and (negative 6, 0).], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. The line goes through the points (0, 3) and (negative 6, 0).", caption: none)

#solutionbox[
slope #math.equation(block: false, alt: "m equals the fraction 1 over 2")[$m = frac(1, 2)$] and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$( 0 , 3 )$]
]
]


=== Identify the Slope and #emph[y]-Intercept From an Equation of a Line

In Understand Slope of a Line, we graphed a line using the slope and a point. When we are given an equation in slope–intercept form, we can use the #emph[y]-intercept as the point, and then count out the slope from there. Let’s practice finding the values of the slope and #emph[y]-intercept from the equation of a line.

#examplebox("Example 2")[][
Identify the slope and #emph[y]-intercept of the line with equation #math.equation(block: false, alt: "y equals −3 x plus 5")[$y = −3 x + 5$].

#solutionbox[
We compare our equation to the slope–intercept form of the equation.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The image displays the linear equation y = mx + b, which is a fundamental formula in algebra representing the slope-intercept form of a straight line. The 'm' is colored red, and 'b' is colored light blue.], alt: "The image displays the linear equation y = mx + b, which is a fundamental formula in algebra representing the slope-intercept form of a straight line. The 'm' is colored red, and 'b' is colored light blue.", caption: none)]),
  [Write the equation of the line.], [#figure(figph[The image displays a mathematical equation, 'y = -3x + 5,' written in a clear, digital font against a white background.], alt: "The image displays a mathematical equation, 'y = -3x + 5,' written in a clear, digital font against a white background.", caption: none)],
  [Identify the slope.], [#figure(figph[The mathematical equation 'm = -3' is displayed against a white background, with the number '-3' highlighted in red.], alt: "The mathematical equation 'm = -3' is displayed against a white background, with the number '-3' highlighted in red.", caption: none)],
  [Identify the #emph[y]-intercept.], [#figure(figph[The image displays the text 'y-intercept is (0, 5)' in a horizontal line. The number 5 is highlighted in a light blue color, while the rest of the text is in gray.], alt: "The image displays the text 'y-intercept is (0, 5)' in a horizontal line. The number 5 is highlighted in a light blue color, while the rest of the text is in gray.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Identify the slope and #emph[y]-intercept of the line #math.equation(block: false, alt: "y equals the fraction 2 over 5 x minus 1")[$y = frac(2, 5) x − 1$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 over 5 ; open parenthesis 0 , −1 close parenthesis")[$frac(2, 5) ; ( 0 , −1 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Identify the slope and #emph[y]-intercept of the line #math.equation(block: false, alt: "y equals minus the fraction 4 over 3 x plus 1")[$y = − frac(4, 3) x + 1$].

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 4 over 3 ; open parenthesis 0 , 1 close parenthesis")[$− frac(4, 3) ; ( 0 , 1 )$]
]
]

When an equation of a line is not given in slope–intercept form, our first step will be to solve the equation for #math.equation(block: false, alt: "y")[$y$].

#examplebox("Example 3")[][
Identify the slope and #emph[y]-intercept of the line with equation #math.equation(block: false, alt: "x plus 2 y equals 6")[$x + 2 y = 6$].

#solutionbox[
This equation is not in slope–intercept form. In order to compare it to the slope–intercept form we must first solve the equation for#math.equation(block: false, alt: "y")[$y$].

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Solve for #emph[y].], [#math.equation(block: false, alt: "x plus 2 y equals 6")[$x + 2 y = 6$]]),
  [Subtract #emph[x] from each side.], [#figure(figph[A mathematical equation is displayed on a white background, reading '2y = -x + 6' in black text.], alt: "A mathematical equation is displayed on a white background, reading '2y = -x + 6' in black text.", caption: none)],
  [Divide both sides by 2.], [#figure(figph[A mathematical equation shows '2y divided by 2 equals the quantity of negative x plus 6, all divided by 2'.], alt: "A mathematical equation shows '2y divided by 2 equals the quantity of negative x plus 6, all divided by 2'.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation is displayed: 2y/2 = -x/2 + 6/2, representing an algebraic step towards simplifying or rearranging a linear equation.], alt: "A mathematical equation is displayed: 2y/2 = -x/2 + 6/2, representing an algebraic step towards simplifying or rearranging a linear equation.", caption: none)],
  [#math.equation(block: false, alt: "open parenthesis Remember: the fraction a plus b over c equals the fraction a over c plus the fraction b over c close parenthesis")[$( "Remember:" #h(0.2em) frac(a + b, c) = frac(a, c) + frac(b, c) )$]], [],
  [Simplify.], [#figure(figph[The image shows the linear equation y = -1/2x + 3, representing a line with a negative slope of -1/2 and a y-intercept of 3.], alt: "The image shows the linear equation y = -1/2x + 3, representing a line with a negative slope of -1/2 and a y-intercept of 3.", caption: none)],
  [Write the slope–intercept form of the equation of the line.], [#figure(figph[The slope-intercept form equation of a straight line, y = mx + b, is displayed on a white background, with 'm' highlighted in red and 'b' in light blue.], alt: "The slope-intercept form equation of a straight line, y = mx + b, is displayed on a white background, with 'm' highlighted in red and 'b' in light blue.", caption: none)],
  [Write the equation of the line.], [#figure(figph[A mathematical equation, y = -1/2x + 3, is displayed on a white background. The fraction '-1/2' is written in red, and the number '3' is in blue, while the rest of the equation is in black.], alt: "A mathematical equation, y = -1/2x + 3, is displayed on a white background. The fraction '-1/2' is written in red, and the number '3' is in blue, while the rest of the equation is in black.", caption: none)],
  [Identify the slope.], [#figure(figph[The image displays the mathematical equation m = -1/2, where 'm' represents a variable and the value assigned to it is negative one half.], alt: "The image displays the mathematical equation m = -1/2, where 'm' represents a variable and the value assigned to it is negative one half.", caption: none)],
  [Identify the #emph[y]-intercept.], [#figure(figph[The image shows the text 'y-intercept is (0, 3)'.], alt: "The image shows the text 'y-intercept is (0, 3)'.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Identify the slope and #emph[y]-intercept of the line #math.equation(block: false, alt: "x plus 4 y equals 8")[$x + 4 y = 8$].

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 1 over 4 ; open parenthesis 0 , 2 close parenthesis")[$− frac(1, 4) ; ( 0 , 2 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Identify the slope and #emph[y]-intercept of the line #math.equation(block: false, alt: "3 x plus 2 y equals 12")[$3 x + 2 y = 12$].

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 3 over 2 ; open parenthesis 0 , 6 close parenthesis")[$− frac(3, 2) ; ( 0 , 6 )$]
]
]


=== Graph a Line Using its Slope and Intercept

Now that we know how to find the slope and #emph[y]-intercept of a line from its equation, we can graph the line by plotting the #emph[y]-intercept and then using the slope to find another point.

#examplebox("Example 4")[How to Graph a Line Using its Slope and Intercept][
Graph the line of the equation #math.equation(block: false, alt: "y equals 4 x minus 2")[$y = 4 x − 2$] using its slope and #emph[y]-intercept.

#solutionbox[
#figure(figph[The figure shows the steps to graph the equation y equals 4x minus 2. Step 1 is to find the slope intercept form of the equation. The equation is already in slope intercept form.], alt: "The figure shows the steps to graph the equation y equals 4x minus 2. Step 1 is to find the slope intercept form of the equation. The equation is already in slope intercept form.", caption: none)

#figure(figph[Step 2 is to identify the slope and y-intercept. Use the equation y equals m x, plus b. The equation y equals m x, plus b is shown with the variable m colored red and the variable b colored blue. Below that is the equation y equals 4 x, plus -2. The number 4 is colored red and -2 is colored blue. From this equation we can see that m equals 4 and b equals -2 so the slope is 4 and the y-intercept is the point (0, negative 2).], alt: "Step 2 is to identify the slope and y-intercept. Use the equation y equals m x, plus b. The equation y equals m x, plus b is shown with the variable m colored red and the variable b colored blue. Below that is the equation y equals 4 x, plus -2. The number 4 is colored red and -2 is colored blue. From this equation we can see that m equals 4 and b equals -2 so the slope is 4 and the y-intercept is the point (0, negative 2).", caption: none)

#figure(figph[Step 3 is to plot the y-intercept. An x y-coordinate plane is shown with the x-axis of the plane running from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. The point (0, negative 2) is plotted.], alt: "Step 3 is to plot the y-intercept. An x y-coordinate plane is shown with the x-axis of the plane running from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. The point (0, negative 2) is plotted.", caption: none)

#figure(figph[Step 4 is to use the slope formula m equals rise over run to identify the rise and the run. Since m equals 4, rise over run equals 4 over 1. From this we can determine that the rise is 4 and the run is 1.], alt: "Step 4 is to use the slope formula m equals rise over run to identify the rise and the run. Since m equals 4, rise over run equals 4 over 1. From this we can determine that the rise is 4 and the run is 1.", caption: none)

#figure(figph[Step 5 is to start at they-intercept, count out the rise and run to mark the second point. So start at the point (0, negative 2) and count the rise and the run. The rise is up 4 and the run is right 1. On the x y-coordinate plane is a red vertical line starts at the point (0, negative 2) and rises 4 units at its end a red horizontal line runs 1 unit to end at the point (1, 2). The point (1, 2)  is plotted.], alt: "Step 5 is to start at they-intercept, count out the rise and run to mark the second point. So start at the point (0, negative 2) and count the rise and the run. The rise is up 4 and the run is right 1. On the x y-coordinate plane is a red vertical line starts at the point (0, negative 2) and rises 4 units at its end a red horizontal line runs 1 unit to end at the point (1, 2). The point (1, 2) is plotted.", caption: none)

#figure(figph[Step 6 is to connect the points with a line. On the x y-coordinate plane the points (0, negative 2) and (1, 2) are plotted and a line runs through the two points. The line is the graph of y equals 4 x, minus 2.], alt: "Step 6 is to connect the points with a line. On the x y-coordinate plane the points (0, negative 2) and (1, 2) are plotted and a line runs through the two points. The line is the graph of y equals 4 x, minus 2.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Graph the line of the equation #math.equation(block: false, alt: "y equals 4 x plus 1")[$y = 4 x + 1$] using its slope and #emph[y]-intercept.

#solutionbox[
#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The points (0, 1) and (1, 5) are plotted on the line.], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The points (0, 1) and (1, 5) are plotted on the line.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Graph the line of the equation #math.equation(block: false, alt: "y equals 2 x minus 3")[$y = 2 x − 3$] using its slope and #emph[y]-intercept.

#solutionbox[
#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The points (0, negative 3) and (1, negative 1) are plotted on the line.], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The points (0, negative 3) and (1, negative 1) are plotted on the line.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Graph a line using its slope and #emph[y]-intercept.]

+ Find the slope-intercept form of the equation of the line.
+ Identify the slope and #emph[y]-intercept.
+ Plot the #emph[y]-intercept.
+ Use the slope formula #math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$] to identify the rise and the run.
+ Starting at the #emph[y]-intercept, count out the rise and run to mark the second point.
+ Connect the points with a line.
]

#examplebox("Example 5")[][
Graph the line of the equation #math.equation(block: false, alt: "y equals − x plus 4")[$y = "−" x + 4$] using its slope and #emph[y]-intercept.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  [], [#math.equation(block: false, alt: "y equals m x plus b")[$y = m x + b$]],
  [The equation is in slope–intercept form.], [#math.equation(block: false, alt: "y equals − x plus 4")[$y = "−" x + 4$]],
  [Identify the slope and #emph[y]-intercept.], [#math.equation(block: false, alt: "m equals −1")[$m = −1$]],
  [], [#emph[y]-intercept is (0, 4)],
  [Plot the #emph[y]-intercept.], [See graph below.],
  [Identify the rise and the run.], [#math.equation(block: false, alt: "m equals the fraction −1 over 1")[$m = frac(−1, 1)$]],
  [Count out the rise and run to mark the second point.], [rise −1, run 1],
  [Draw the line.], [#figure(figph[A linear graph on a coordinate plane, showing a downward-sloping line that intersects the y-axis at 4 and the x-axis at 4. Points (0, 4) and (1, 3) are marked.], alt: "A linear graph on a coordinate plane, showing a downward-sloping line that intersects the y-axis at 4 and the x-axis at 4. Points (0, 4) and (1, 3) are marked.", caption: none)],
  [To check your work, you can find another point on the line and make sure it is a solution of the equation. In the graph we see the line goes through (4, 0).], [],
  [Check. #linebreak() #math.equation(block: false, alt: "y, equals, − x plus 4; 0, equals ?, minus 4 plus 4; 0, equals, 0 ✓")[$y & = & "−" x + 4 \ 0 & limits(=)^(?) & − 4 + 4 \ 0 & = & 0 ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Graph the line of the equation #math.equation(block: false, alt: "y equals − x minus 3")[$y = "−" x − 3$] using its slope and #emph[y]-intercept.

#solutionbox[
#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The points (0, negative 3) and (1, negative 4) are plotted on the line.], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The points (0, negative 3) and (1, negative 4) are plotted on the line.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Graph the line of the equation #math.equation(block: false, alt: "y equals − x minus 1")[$y = "−" x − 1$] using its slope and #emph[y]-intercept.

#solutionbox[
#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The points (0, negative 1) and (1, negative 2) are plotted on the line.], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The points (0, negative 1) and (1, negative 2) are plotted on the line.", caption: none)
]
]

#examplebox("Example 6")[][
Graph the line of the equation #math.equation(block: false, alt: "y equals minus the fraction 2 over 3 x minus 3")[$y = − frac(2, 3) x − 3$] using its slope and #emph[y]-intercept.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "y equals m x plus b")[$y = m x + b$]]),
  [The equation is in slope–intercept form.], [#math.equation(block: false, alt: "y equals minus the fraction 2 over 3 x minus 3")[$y = − frac(2, 3) x − 3$]],
  [Identify the slope and #emph[y]-intercept.], [#math.equation(block: false, alt: "m equals minus the fraction 2 over 3")[$m = − frac(2, 3)$]; #emph[y]-intercept is (0, −3)],
  [Plot the #emph[y]-intercept.], [See graph below.],
  [Identify the rise and the run.], [],
  [Count out the rise and run to mark the second point.], [],
  [Draw the line.], [#figure(figph[A Cartesian coordinate system displays a straight line with a negative slope, passing through the points (0, -3) and (3, -5).], alt: "A Cartesian coordinate system displays a straight line with a negative slope, passing through the points (0, -3) and (3, -5).", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Graph the line of the equation #math.equation(block: false, alt: "y equals minus the fraction 5 over 2 x plus 1")[$y = − frac(5, 2) x + 1$] using its slope and #emph[y]-intercept.

#solutionbox[
#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The points (0,1) and (2, negative 4) are plotted on the line.], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The points (0,1) and (2, negative 4) are plotted on the line.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Graph the line of the equation #math.equation(block: false, alt: "y equals minus the fraction 3 over 4 x minus 2")[$y = − frac(3, 4) x − 2$] using its slope and #emph[y]-intercept.

#solutionbox[
#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The points (0, negative 2) and (4, negative 5) are plotted on the line.], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The points (0, negative 2) and (4, negative 5) are plotted on the line.", caption: none)
]
]

#examplebox("Example 7")[][
Graph the line of the equation #math.equation(block: false, alt: "4 x minus 3 y equals 12")[$4 x − 3 y = 12$] using its slope and #emph[y]-intercept.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "4 x minus 3 y equals 12")[$4 x − 3 y = 12$]]),
  [Find the slope–intercept form of the equation.], [#math.equation(block: false, alt: "−3 y equals −4 x plus 12")[$#h(1.43em) −3 y = −4 x + 12$]],
  [], [#math.equation(block: false, alt: "minus the fraction 3 y over 3 equals the fraction −4 x plus 12 over −3")[$#h(1.26em) − frac(3 y, 3) = frac(−4 x + 12, −3)$]],
  [The equation is now in slope–intercept form.], [#math.equation(block: false, alt: "y equals the fraction 4 over 3 x minus 4")[$#h(2.63em) y = frac(4, 3) x − 4$]],
  [Identify the slope and #emph[y]-intercept.], [#math.equation(block: false, alt: "m equals the fraction 4 over 3")[$#h(2.37em) m = frac(4, 3)$]],
  [], [#emph[y]-intercept is (0, −4)],
  [Plot the #emph[y]-intercept.], [See graph below.],
  [Identify the rise and the run; count out the rise and run to mark the second point.], [],
  [Draw the line.], [#figure(figph[A Cartesian graph displays a linear equation as a blue line with a positive slope. The line intersects the x-axis at (3, 0) and the y-axis at (0, -4), passing through quadrants I, III, and IV.], alt: "A Cartesian graph displays a linear equation as a blue line with a positive slope. The line intersects the x-axis at (3, 0) and the y-axis at (0, -4), passing through quadrants I, III, and IV.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Graph the line of the equation #math.equation(block: false, alt: "2 x minus y equals 6")[$2 x − y = 6$] using its slope and #emph[y]-intercept.

#solutionbox[
#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The points (0, negative 6) and (1, negative 4) are plotted on the line.], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The points (0, negative 6) and (1, negative 4) are plotted on the line.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Graph the line of the equation #math.equation(block: false, alt: "3 x minus 2 y equals 8")[$3 x − 2 y = 8$] using its slope and #emph[y]-intercept.

#solutionbox[
#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The points (0, negative 4) and (2, negative 1) are plotted on the line.], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The points (0, negative 4) and (2, negative 1) are plotted on the line.", caption: none)
]
]

We have used a grid with #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$] both going from about #math.equation(block: false, alt: "−10")[$−10$] to 10 for all the equations we’ve graphed so far. Not all linear equations can be graphed on this small grid. Often, especially in applications with real-world data, we’ll need to extend the axes to bigger positive or smaller negative numbers.

#examplebox("Example 8")[][
Graph the line of the equation #math.equation(block: false, alt: "y equals 0.2 x plus 45")[$y = 0.2 x + 45$] using its slope and #emph[y]-intercept.

#solutionbox[
We’ll use a grid with the axes going from about #math.equation(block: false, alt: "−80")[$−80$] to 80.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "y equals m x plus b")[$y = m x + b$]]),
  [The equation is in slope–intercept form.], [#math.equation(block: false, alt: "y equals 0.2 x plus 45")[$y = 0.2 x + 45$]],
  [Identify the slope and #emph[y]-intercept.], [#math.equation(block: false, alt: "m equals 0.2")[$m = 0.2$]],
  [], [The #emph[y]-intercept is (0, 45)],
  [Plot the #emph[y]-intercept.], [See graph below.],
  [Count out the rise and run to mark the second point. The slope is #math.equation(block: false, alt: "m equals 0.2")[$m = 0.2$]; in fraction form this means #math.equation(block: false, alt: "m equals the fraction 2 over 10")[$m = frac(2, 10)$]. Given the scale of our graph, it would be easier to use the equivalent fraction #math.equation(block: false, alt: "m equals the fraction 10 over 50")[$m = frac(10, 50)$].], [],
  [Draw the line.], [#figure(figph[A graph displays a straight line on a Cartesian coordinate system. The line passes through two distinct points: (0, 45) and (50, 55), exhibiting a positive slope as it extends across the plane.], alt: "A graph displays a straight line on a Cartesian coordinate system. The line passes through two distinct points: (0, 45) and (50, 55), exhibiting a positive slope as it extends across the plane.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Graph the line of the equation #math.equation(block: false, alt: "y equals 0.5 x plus 25")[$y = 0.5 x + 25$] using its slope and #emph[y]-intercept.

#solutionbox[
#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 70 to 30. The y-axis of the plane runs from negative 20 to 40. The points (0, 25) and (10, 30) are plotted on the line.], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 70 to 30. The y-axis of the plane runs from negative 20 to 40. The points (0, 25) and (10, 30) are plotted on the line.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Graph the line of the equation #math.equation(block: false, alt: "y equals 0.1 x minus 30")[$y = 0.1 x − 30$] using its slope and #emph[y]-intercept.

#solutionbox[
#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 50 to 350. The y-axis of the plane runs from negative 40 to 40. The points (0, negative 30) and (100, negative 20) are plotted on the line.], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 50 to 350. The y-axis of the plane runs from negative 40 to 40. The points (0, negative 30) and (100, negative 20) are plotted on the line.", caption: none)
]
]

Now that we have graphed lines by using the slope and #emph[y]-intercept, let’s summarize all the methods we have used to graph lines.

#figure(figph[The table has two rows and four columns. The first row spans all four columns and is a header row. The header is “Methods to Graph Lines”. The second row is made up of four columns. The first column is labeled “Plotting Points” and shows a smaller table with four rows and two columns. The first row is a header row with the first column labeled “x” and the second labeled “y”. The rest of the table is blank. Below the table it reads “Find three points. Plot the points, make sure they line up, then draw the line.” The Second column is labeled “Slope–Intercept” and shows the equation y equals m x, plus b. Below the equation it reads “Find the slope and y-intercept. Start at the y-intercept, then count the slope to get a second point.” The third column is labeled “Intercepts” and shows a smaller table with four rows and two columns. The first row is a header row with the first column labeled “x” and the second labeled “y”. The second row has a 0 in the “x” column and the “y” column is blank. The second row is blank in the “x” column and has a 0 in the “y” column. The third row is blank. Below the table it reads “Find the intercepts and a third point. Plot the points, make sure they line up, then draw the line.” The fourth column is labeled “Recognize Vertical and Horizontal Lines”. Below that it reads “The equation has only one variable.” The equation x equals a is a vertical line and the equation y equals b is a horizontal line.], alt: "The table has two rows and four columns. The first row spans all four columns and is a header row. The header is “Methods to Graph Lines”. The second row is made up of four columns. The first column is labeled “Plotting Points” and shows a smaller table with four rows and two columns. The first row is a header row with the first column labeled “x” and the second labeled “y”. The rest of the table is blank. Below the table it reads “Find three points. Plot the points, make sure they line up, then draw the line.” The Second column is labeled “Slope–Intercept” and shows the equation y equals m x, plus b. Below the equation it reads “Find the slope and y-intercept. Start at the y-intercept, then count the slope to get a second point.” The third column is labeled “Intercepts” and shows a smaller table with four rows and two columns. The first row is a header row with the first column labeled “x” and the second labeled “y”. The second row has a 0 in the “x” column and the “y” column is blank. The second row is blank in the “x” column and has a 0 in the “y” column. The third row is blank. Below the table it reads “Find the intercepts and a third point. Plot the points, make sure they line up, then draw the line.” The fourth column is labeled “Recognize Vertical and Horizontal Lines”. Below that it reads “The equation has only one variable.” The equation x equals a is a vertical line and the equation y equals b is a horizontal line.", caption: none)


=== Choose the Most Convenient Method to Graph a Line

Now that we have seen several methods we can use to graph lines, how do we know which method to use for a given equation?

While we could plot points, use the slope–intercept form, or find the intercepts for #emph[any] equation, if we recognize the most convenient way to graph a certain type of equation, our work will be easier. Generally, plotting points is not the most efficient way to graph a line. We saw better methods in sections 4.3, 4.4, and earlier in this section. Let’s look for some patterns to help determine the most convenient method to graph a line.

Here are six equations we graphed in this chapter, and the method we used to graph each of them.

#math.equation(block: true, alt: "Equation, , Method; #1, x equals 2, , Vertical line; #2, y equals 4, , Horizontal line; #3, − x plus 2 y equals 6, , Intercepts; #4, 4 x minus 3 y equals 12, , Intercepts; #5, y equals 4 x minus 2, , Slope–intercept; #6, y equals − x plus 4, , Slope–intercept")[$&  &  & "Equation" &  & #h(5em) &  & "Method" \ "#1" &  &  & x = 2 &  & #h(5em) &  & "Vertical line" \ "#2" &  &  & y = 4 &  & #h(5em) &  & "Horizontal line" \ "#3" &  &  & "−" x + 2 y = 6 &  & #h(5em) &  & "Intercepts" \ "#4" &  &  & 4 x − 3 y = 12 &  & #h(5em) &  & "Intercepts" \ "#5" &  &  & y = 4 x − 2 &  & #h(5em) &  & "Slope–intercept" \ "#6" &  &  & y = "−" x + 4 &  & #h(5em) &  & "Slope–intercept"$]Equations \#1 and \#2 each have just one variable. Remember, in equations of this form the value of that one variable is constant; it does not depend on the value of the other variable. Equations of this form have graphs that are vertical or horizontal lines.

In equations \#3 and \#4, both #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$] are on the same side of the equation. These two equations are of the form #math.equation(block: false, alt: "A x plus B y equals C")[$A x + B y = C$]. We substituted #math.equation(block: false, alt: "y equals 0")[$y = 0$] to find the #emph[x]-intercept and #math.equation(block: false, alt: "x equals 0")[$x = 0$] to find the #emph[y]-intercept, and then found a third point by choosing another value for #math.equation(block: false, alt: "x")[$x$] or #math.equation(block: false, alt: "y")[$y$].

Equations \#5 and \#6 are written in slope–intercept form. After identifying the slope and #emph[y]-intercept from the equation we used them to graph the line.

This leads to the following strategy.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Strategy for Choosing the Most Convenient Method to Graph a Line]

Consider the form of the equation.

- If it only has one variable, it is a vertical or horizontal line.
- #math.equation(block: false, alt: "x equals a")[$x = a$] is a vertical line passing through the #emph[x]-axis at #math.equation(block: false, alt: "a")[$a$].
- #math.equation(block: false, alt: "y equals b")[$y = b$] is a horizontal line passing through the #emph[y]-axis at #math.equation(block: false, alt: "b")[$b$].
- If #math.equation(block: false, alt: "y")[$y$] is isolated on one side of the equation, in the form #math.equation(block: false, alt: "y equals m x plus b")[$y = m x + b$], graph by using the slope and y-intercept.
- Identify the slope and #emph[y]-intercept and then graph.
- If the equation is of the form #math.equation(block: false, alt: "A x plus B y equals C")[$A x + B y = C$], find the intercepts.
- Find the #emph[x]- and #emph[y]-intercepts, a third point, and then graph.
]

#examplebox("Example 9")[][
Determine the most convenient method to graph each line.

ⓐ #math.equation(block: false, alt: "y equals −6")[$y = −6$] ⓑ #math.equation(block: false, alt: "5 x minus 3 y equals 15")[$5 x − 3 y = 15$] ⓒ #math.equation(block: false, alt: "x equals 7")[$x = 7$] ⓓ #math.equation(block: false, alt: "y equals the fraction 2 over 5 x minus 1")[$y = frac(2, 5) x − 1$].

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "y equals −6")[$y = −6$] #linebreak() 
This equation has only one variable,#math.equation(block: false, alt: "y")[$y$]. Its graph is a horizontal line crossing the #emph[y]-axis at #math.equation(block: false, alt: "−6")[$−6$].
+ ⓑ #math.equation(block: false, alt: "5 x minus 3 y equals 15")[$5 x − 3 y = 15$] #linebreak() 
This equation is of the form #math.equation(block: false, alt: "A x plus B y equals C")[$A x + B y = C$]. The easiest way to graph it will be to find the intercepts and one more point.
+ ⓒ #math.equation(block: false, alt: "x equals 7")[$x = 7$] #linebreak() 
There is only one variable, #math.equation(block: false, alt: "x")[$x$]. The graph is a vertical line crossing the #emph[x]-axis at 7.
+ ⓓ #math.equation(block: false, alt: "y equals the fraction 2 over 5 x minus 1")[$y = frac(2, 5) x − 1$] #linebreak() 
Since this equation is in #math.equation(block: false, alt: "y equals m x plus b")[$y = m x + b$] form, it will be easiest to graph this line by using the slope and #emph[y]-intercept.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Determine the most convenient method to graph each line: ⓐ #math.equation(block: false, alt: "3 x plus 2 y equals 12")[$3 x + 2 y = 12$] ⓑ #math.equation(block: false, alt: "y equals 4")[$y = 4$] ⓒ #math.equation(block: false, alt: "y equals the fraction 1 over 5 x minus 4")[$y = frac(1, 5) x − 4$] ⓓ #math.equation(block: false, alt: "x equals −7")[$x = −7$].

#solutionbox[
ⓐ intercepts ⓑ horizontal line ⓒ slope–intercept ⓓ vertical line
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Determine the most convenient method to graph each line: ⓐ #math.equation(block: false, alt: "x equals 6")[$x = 6$] ⓑ #math.equation(block: false, alt: "y equals minus the fraction 3 over 4 x plus 1")[$y = − frac(3, 4) x + 1$] ⓒ #math.equation(block: false, alt: "y equals −8")[$y = −8$] ⓓ #math.equation(block: false, alt: "4 x minus 3 y equals −1")[$4 x − 3 y = −1$].

#solutionbox[
ⓐ vertical line ⓑ slope–intercept ⓒ horizontal line ⓓ intercepts
]
]


=== Graph and Interpret Applications of Slope–Intercept

Many real-world applications are modeled by linear equations. We will take a look at a few applications here so you can see how equations written in slope–intercept form relate to real-world situations.

Usually when a linear equation models a real-world situation, different letters are used for the variables, instead of #emph[x] and #emph[y]. The variable names remind us of what quantities are being measured.

#examplebox("Example 10")[][
The equation #math.equation(block: false, alt: "F equals the fraction 9 over 5 C plus 32")[$F = frac(9, 5) C + 32$] is used to convert temperatures, #math.equation(block: false, alt: "C")[$C$], on the Celsius scale to temperatures, #math.equation(block: false, alt: "F")[$F$], on the Fahrenheit scale.

ⓐ Find the Fahrenheit temperature for a Celsius temperature of 0. #linebreak() 
ⓑ Find the Fahrenheit temperature for a Celsius temperature of 20. #linebreak() 
ⓒ Interpret the slope and #emph[F]-intercept of the equation. #linebreak() 
ⓓ Graph the equation.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ 
     #linebreak() Find the Fahrenheit temperature for a Celsius temperature of 0.
     #linebreak() Find #math.equation(block: false, alt: "F")[$F$] when #math.equation(block: false, alt: "C equals 0")[$C = 0$].
     #linebreak() Simplify.], [#math.equation(block: false, alt: "F equals the fraction 9 over 5 C plus 32; F equals the fraction 9 over 5 open parenthesis 0 close parenthesis plus 32; F equals 32")[$F = frac(9, 5) C + 32 \ F = frac(9, 5) ( 0 ) + 32 \ F = 32$]]),
  [ⓑ 
     #linebreak() Find the Fahrenheit temperature for a Celsius temperature of 20.
     #linebreak() Find #math.equation(block: false, alt: "F")[$F$] when #math.equation(block: false, alt: "C equals 20")[$C = 20$].
     #linebreak() Simplify.
	 #linebreak() Simplify.], [#math.equation(block: false, alt: "F equals the fraction 9 over 5 C plus 32; F equals the fraction 9 over 5 open parenthesis 20 close parenthesis plus 32; F equals 36 plus 32; F equals 68")[$F = frac(9, 5) C + 32 \ F = frac(9, 5) ( 20 ) + 32 \ F = 36 + 32 \ F = 68$]],
))

ⓒ Interpret the slope and #emph[F]-intercept of the equation.

Even though this equation uses #math.equation(block: false, alt: "F")[$F$]and #math.equation(block: false, alt: "C")[$C$], it is still in slope–intercept form.

#figure(figph[This image shows three lines of equations. The first line reads y equals m x plus b. The second line reads F equals m C plus b and the third line reads F equals nine fifths times C plus 32.], alt: "This image shows three lines of equations. The first line reads y equals m x plus b. The second line reads F equals m C plus b and the third line reads F equals nine fifths times C plus 32.", caption: none)

The slope, #math.equation(block: false, alt: "the fraction 9 over 5")[$frac(9, 5)$], means that the temperature Fahrenheit (#emph[F]) increases 9 degrees when the temperature Celsius (#emph[C]) increases 5 degrees.

The #emph[F]-intercept means that when the temperature is #math.equation(block: false, alt: "0 °")[$0 "°"$] on the Celsius scale, it is #math.equation(block: false, alt: "32 °")[$32 "°"$] on the Fahrenheit scale.

ⓓ Graph the equation.

We’ll need to use a larger scale than our usual. Start at the #emph[F]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 32 close parenthesis")[$( 0 , 32 )$] then count out the rise of 9 and the run of 5 to get a second point.

#figure(figph[A graph with x-axis ranging from –20 to 50, and y-axis ranging from –20 to 50. A line runs upwards, intercepting at points (0, 32) and a second point which the student must find.], alt: "A graph with x-axis ranging from –20 to 50, and y-axis ranging from –20 to 50. A line runs upwards, intercepting at points (0, 32) and a second point which the student must find.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The equation #math.equation(block: false, alt: "h equals 2 s plus 50")[$h = 2 s + 50$] is used to estimate a woman’s height in inches, #emph[h], based on her shoe size, #emph[s].

+ ⓐ Estimate the height of a child who wears women’s shoe size 0.
+ ⓑ Estimate the height of a woman with shoe size 8.
+ ⓒ Interpret the slope and #emph[h]-intercept of the equation.
+ ⓓ Graph the equation.

#solutionbox[
+ ⓐ 50 inches
+ ⓑ 66 inches
+ ⓒ The slope, 2, means that the height, #emph[h], increases by 2 inches when the shoe size, #emph[s], increases by 1. The #emph[h]-intercept means that when the shoe size is 0, the height is 50 inches.
+ ⓓ  #linebreak() 
#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane represents the variable s and runs from negative 2 to 15. The y-axis of the plane represents the variable h and runs from negative 1 to 80. The line begins at the point (0, 50) and goes through the points (8, 66).], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane represents the variable s and runs from negative 2 to 15. The y-axis of the plane represents the variable h and runs from negative 1 to 80. The line begins at the point (0, 50) and goes through the points (8, 66).", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The equation #math.equation(block: false, alt: "T equals the fraction 1 over 4 n plus 40")[$T = frac(1, 4) n + 40$] is used to estimate the temperature in degrees Fahrenheit, #emph[T], based on the number of cricket chirps, #emph[n], in one minute.

+ ⓐ Estimate the temperature when there are no chirps.
+ ⓑ Estimate the temperature when the number of chirps in one minute is 100.
+ ⓒ Interpret the slope and #emph[T]-intercept of the equation.
+ ⓓ Graph the equation.

#solutionbox[
+ ⓐ 40 degrees
+ ⓑ 65 degrees
+ ⓒ The slope, #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$], means that the temperature Fahrenheit (#emph[F]) increases 1 degree when the number of chirps, #emph[n], increases by 4. The #emph[T]-intercept means that when the number of chirps is 0, the temperature is #math.equation(block: false, alt: "40 °")[$40 "°"$].
+ ⓓ  #linebreak() 
#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane represents the variable n and runs from 10 to 140 The y-axis of the plane represents the variable T and runs from negative 5 to 75. The line begins at the point (0, 40) and goes through the point (100, 65).], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane represents the variable n and runs from 10 to 140 The y-axis of the plane represents the variable T and runs from negative 5 to 75. The line begins at the point (0, 40) and goes through the point (100, 65).", caption: none)
]
]

The cost of running some types business has two components—a #emph[fixed cost] and a #emph[variable cost]. The fixed cost is always the same regardless of how many units are produced. This is the cost of rent, insurance, equipment, advertising, and other items that must be paid regularly. The variable cost depends on the number of units produced. It is for the material and labor needed to produce each item.

#examplebox("Example 11")[][
Stella has a home business selling gourmet pizzas. The equation #math.equation(block: false, alt: "C equals 4 p plus 25")[$C = 4 p + 25$] models the relation between her weekly cost, #emph[C], in dollars and the number of pizzas, #emph[p], that she sells.

ⓐ Find Stella’s cost for a week when she sells no pizzas. #linebreak() 
ⓑ Find the cost for a week when she sells 15 pizzas. #linebreak() 
ⓒ Interpret the slope and #emph[C]-intercept of the equation. #linebreak() 
ⓓ Graph the equation.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ Find Stella's cost for a week when she sells no pizzas.], [#figure(figph[A mathematical equation C = 4p + 25 is displayed in a digital format on a white background.], alt: "A mathematical equation C = 4p + 25 is displayed in a digital format on a white background.", caption: none)]),
  [Find #emph[C] when #math.equation(block: false, alt: "p equals 0")[$p = 0$].], [#figure(figph[A mathematical equation is displayed on a white background: C = 4(0) + 25.], alt: "A mathematical equation is displayed on a white background: C = 4(0) + 25.", caption: none)],
  [Simplify.], [#figure(figph[A close-up shot of a mathematical equation displayed in black text against a white background, reading 'C = 25'.], alt: "A close-up shot of a mathematical equation displayed in black text against a white background, reading 'C = 25'.", caption: none)],
  [], [Stella's fixed cost is \$25 when she sells no pizzas.],
  [ⓑ Find the cost for a week when she sells 15 pizzas.], [#figure(figph[A mathematical equation C = 4p + 25 is displayed in black text on a white background.], alt: "A mathematical equation C = 4p + 25 is displayed in black text on a white background.", caption: none)],
  [Find #emph[C] when #math.equation(block: false, alt: "p equals 15")[$p = 15$].], [#figure(figph[A mathematical equation shows C equals 4 multiplied by 15, plus 25.], alt: "A mathematical equation shows C equals 4 multiplied by 15, plus 25.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation shows 'C = 60 + 25' displayed on a white background.], alt: "A mathematical equation shows 'C = 60 + 25' displayed on a white background.", caption: none)],
  [], [#figure(figph[The text 'C = 85' is displayed in a sans-serif font against a white background.], alt: "The text 'C = 85' is displayed in a sans-serif font against a white background.", caption: none)],
  [], [Stella's costs are \$85 when she sells 15 pizzas.],
  [ⓒ Interpret the slope and #emph[C]-intercept of the equation.], [#figure(figph[Two linear equations are displayed: y = mx + b and C = 4p + 25. The variables 'y' and 'C' are highlighted in red, while 'x' and 'p' are highlighted in teal.], alt: "Two linear equations are displayed: y = mx + b and C = 4p + 25. The variables 'y' and 'C' are highlighted in red, while 'x' and 'p' are highlighted in teal.", caption: none)],
  [], [The slope, 4, means that the cost increases by \$4 for each pizza Stella sells. The #emph[C]-intercept means that even when Stella sells no pizzas, her costs for the week are \$25.],
  [ⓓ Graph the equation. We'll need to use a larger scale than our usual. Start at the #emph[C]-intercept (0, 25) then count out the rise of 4 and the run of 1 to get a second point.], [#figure(figph[A graph on a coordinate plane shows a linear function with axes labeled C and P. The blue line passes through the points (0, 25) and (1, 29), indicating a positive slope.], alt: "A graph on a coordinate plane shows a linear function with axes labeled C and P. The blue line passes through the points (0, 25) and (1, 29), indicating a positive slope.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Sam drives a delivery van. The equation #math.equation(block: false, alt: "C equals 0.5 m plus 60")[$C = 0.5 m + 60$] models the relation between his weekly cost, #emph[C], in dollars and the number of miles, #emph[m], that he drives.

ⓐ Find Sam’s cost for a week when he drives 0 miles. #linebreak() 
ⓑ Find the cost for a week when he drives 250 miles. #linebreak() 
ⓒ Interpret the slope and #emph[C]-intercept of the equation. #linebreak() 
ⓓ Graph the equation.

#solutionbox[
+ ⓐ \$60
+ ⓑ \$185
+ ⓒ The slope, 0.5, means that the weekly cost, #emph[C], increases by \$0.50 when the number of miles driven, #emph[n,] increases by 1. The #emph[C]-intercept means that when the number of miles driven is 0, the weekly cost is \$60
+ ⓓ  #linebreak() 
#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane represents the variable m and runs from negative 10 to 400. The y-axis of the plane represents the variable C and runs from negative 10 to 300. The line begins at the point (0, 65) and goes through the point (250, 185).], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane represents the variable m and runs from negative 10 to 400. The y-axis of the plane represents the variable C and runs from negative 10 to 300. The line begins at the point (0, 65) and goes through the point (250, 185).", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Loreen has a calligraphy business. The equation #math.equation(block: false, alt: "C equals 1.8 n plus 35")[$C = 1.8 n + 35$] models the relation between her weekly cost, #emph[C], in dollars and the number of wedding invitations, #emph[n], that she writes.

+ ⓐ Find Loreen’s cost for a week when she writes no invitations.
+ ⓑ Find the cost for a week when she writes 75 invitations.
+ ⓒ Interpret the slope and #emph[C]-intercept of the equation.
+ ⓓ Graph the equation.

#solutionbox[
+ ⓐ \$35
+ ⓑ \$170
+ ⓒ The slope, 1.8, means that the weekly cost, C, increases by \$1.80 when the number of invitations, #emph[n], increases by 1. #linebreak() 
The #emph[C]-intercept means that when the number of invitations is 0, the weekly cost is \$35.;
+ ⓓ  #linebreak() 
#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane represents the variable n and runs from negative 10 to 400. The y-axis of the plane represents the variable C and runs from negative 10 to 300. The line begins at the point (0, 35) and goes through the point (75, 170).], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane represents the variable n and runs from negative 10 to 400. The y-axis of the plane represents the variable C and runs from negative 10 to 300. The line begins at the point (0, 35) and goes through the point (75, 170).", caption: none)
]
]


=== Use Slopes to Identify Parallel Lines

The slope of a line indicates how steep the line is and whether it rises or falls as we read it from left to right. Two lines that have the same slope are called parallel lines. Parallel lines never intersect.

#figure(figph[The figure shows three pairs of lines side-by-side. The pair of lines on the left run diagonally rising from left to right. The pair run side-by-side, not crossing. The pair of lines in the middle run diagonally dropping from left to right. The pair run side-by-side, not crossing. The pair of lines on the right run diagonally also dropping from left to right, but with a lesser slope. The pair run side-by-side, not crossing.], alt: "The figure shows three pairs of lines side-by-side. The pair of lines on the left run diagonally rising from left to right. The pair run side-by-side, not crossing. The pair of lines in the middle run diagonally dropping from left to right. The pair run side-by-side, not crossing. The pair of lines on the right run diagonally also dropping from left to right, but with a lesser slope. The pair run side-by-side, not crossing.", caption: none)

We say this more formally in terms of the rectangular coordinate system. Two lines that have the same slope and different #emph[y]-intercepts are called #strong[parallel lines].

#figure(figph[The figure shows two lines graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. One line goes through the points (negative 5,1) and (5,5). The other line goes through the points (negative 5, negative 4) and (5,0).], alt: "The figure shows two lines graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. One line goes through the points (negative 5,1) and (5,5). The other line goes through the points (negative 5, negative 4) and (5,0).", caption: [Verify that both lines have the same slope, #math.equation(block: false, alt: "m equals the fraction 2 over 5")[$m = frac(2, 5)$], and different #emph[y]-intercepts.])

What about vertical lines? The slope of a vertical line is undefined, so vertical lines don’t fit in the definition above. We say that vertical lines that have different #emph[x]-intercepts are parallel.

#figure(figph[The figure shows two vertical lines graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. One line goes through the points (2,1) and (2,5). The other line goes through the points (5, negative 4) and (5,0).], alt: "The figure shows two vertical lines graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. One line goes through the points (2,1) and (2,5). The other line goes through the points (5, negative 4) and (5,0).", caption: [Vertical lines with diferent #emph[x]-intercepts are parallel.])

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Parallel Lines]

Parallel lines are lines in the same plane that do not intersect.

- Parallel lines have the same slope and different #emph[y]-intercepts.
- If #math.equation(block: false, alt: "m sub 1")[$m_(1)$] and #math.equation(block: false, alt: "m sub 2")[$m_(2)$] are the slopes of two parallel lines then#math.equation(block: false, alt: "m sub 1 equals m sub 2")[$m_(1) = m_(2)$].
- Parallel vertical lines have different #emph[x]-intercepts.
]

Let’s graph the equations #math.equation(block: false, alt: "y equals −2 x plus 3")[$y = −2 x + 3$] and #math.equation(block: false, alt: "2 x plus y equals −1")[$2 x + y = −1$] on the same grid. The first equation is already in slope–intercept form: #math.equation(block: false, alt: "y equals −2 x plus 3")[$y = −2 x + 3$]. We solve the second equation for #math.equation(block: false, alt: "y")[$y$]:

#math.equation(block: true, alt: "2 x plus y, equals, −1; y, equals, −2 x minus 1")[$2 x + y & = & −1 \ y & = & −2 x − 1$]Graph the lines.

#figure(figph[The figure shows two lines graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. One line goes through the points (negative 4, 7) and (3, negative 7). The other line goes through the points (negative 2, 7) and (5, negative 7).], alt: "The figure shows two lines graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. One line goes through the points (negative 4, 7) and (3, negative 7). The other line goes through the points (negative 2, 7) and (5, negative 7).", caption: none)

Notice the lines look parallel. What is the slope of each line? What is the #emph[y]-intercept of each line?

#math.equation(block: true, alt: "y, equals, m x plus b, , y, equals, m x plus b; y, equals, −2 x plus 3, , y, equals, −2 x minus 1; m, equals, −2, , m, equals, −2; b, equals, 3 , (0, 3), , b, equals, −1 , (0, −1)")[$y & = & m x + b &  & #h(5em) &  &  & y & = & m x + b \ y & = & −2 x + 3 &  & #h(5em) &  &  & y & = & −2 x − 1 \ m & = & −2 &  & #h(5em) &  &  & m & = & −2 \ b & = & 3 , "(0, 3)" &  & #h(5em) &  &  & b & = & −1 , "(0, −1)"$]The slopes of the lines are the same and the #emph[y]-intercept of each line is different. So we know these lines are parallel.

Since parallel lines have the same slope and different #emph[y]-intercepts, we can now just look at the slope–intercept form of the equations of lines and decide if the lines are parallel.

#examplebox("Example 12")[][
Use slopes and #emph[y]-intercepts to determine if the lines #math.equation(block: false, alt: "3 x minus 2 y equals 6")[$3 x − 2 y = 6$] and #math.equation(block: false, alt: "y equals the fraction 3 over 2 x plus 1")[$y = frac(3, 2) x + 1$] are parallel.

#solutionbox[
#figure(table(
  columns: 4, align: left, inset: 6pt,
  table.header([Solve the first equation for #math.equation(block: false, alt: "y")[$y$].], [#math.equation(block: false, alt: "3 x minus 2 y, equals, 6; minus 2 y, equals, −3 x plus 6; the fraction −2 y over −2, equals, the fraction −3 x plus 6 over −2")[$3 x − 2 y & = & 6 \ − 2 y & = & −3 x + 6 \ frac(−2 y, −2) & = & frac(−3 x + 6, −2)$]], [and], [#math.equation(block: false, alt: "y equals the fraction 3 over 2 x plus 1")[$y = frac(3, 2) x + 1$]]),
  [The equation is now in slope-intercept form.], [#math.equation(block: false, alt: "y equals the fraction 3 over 2 x minus 3")[$y = frac(3, 2) x − 3$]], [], [],
  [The equation of the second line is already in slope-intercept form.], [], [], [#math.equation(block: false, alt: "y equals the fraction 3 over 2 x plus 1")[$y = frac(3, 2) x + 1$]],
  [Identify the slope and #math.equation(block: false, alt: "y")[$y$]-intercept of both lines.], [#math.equation(block: false, alt: "y, equals, the fraction 3 over 2 x minus 3; y, equals, m x plus b; m, equals, the fraction 3 over 2")[$y & = & frac(3, 2) x − 3 \ y & = & m x + b \ m & = & frac(3, 2)$]], [], [#math.equation(block: false, alt: "y equals the fraction 3 over 2 x plus 1; y equals m x plus b; m equals the fraction 3 over 2")[$#h(0.2em) y = frac(3, 2) x + 1 \ #h(0.2em) y = m x + b \ m = frac(3, 2)$]],
  [], [#math.equation(block: false, alt: "y -intercept is (0, −3)")[$y "-intercept is (0, −3)"$]], [], [#math.equation(block: false, alt: "y -intercept is (0, 1)")[$y "-intercept is (0, 1)"$]],
))

The lines have the same slope and different #emph[y]-intercepts and so they are parallel. You may want to graph the lines to confirm whether they are parallel.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use slopes and #emph[y]-intercepts to determine if the lines #math.equation(block: false, alt: "2 x plus 5 y equals 5 and y equals minus the fraction 2 over 5 x minus 4")[$2 x + 5 y = 5 #h(0.2em) "and" #h(0.2em) y = − frac(2, 5) x − 4$] are parallel.

#solutionbox[
parallel
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use slopes and #emph[y]-intercepts to determine if the lines #math.equation(block: false, alt: "4 x minus 3 y equals 6 and y equals the fraction 4 over 3 x minus 1")[$4 x − 3 y = 6 #h(0.2em) "and" #h(0.2em) y = frac(4, 3) x − 1$] are parallel.

#solutionbox[
parallel
]
]

#examplebox("Example 13")[][
Use slopes and #emph[y]-intercepts to determine if the lines #math.equation(block: false, alt: "y equals −4")[$y = −4$] and #math.equation(block: false, alt: "y equals 3")[$y = 3$] are parallel.

#solutionbox[
#figure(table(
  columns: 4, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "y equals −4; y equals 0 x minus 4")[$y = −4 \ y = 0 x − 4$]], [and], [#math.equation(block: false, alt: "y equals 3; y equals 0 x plus 3")[$y = 3 \ y = 0 x + 3$]]),
  [Write each equation in slope-intercept form.], [#math.equation(block: false, alt: "y equals 0 x minus 4")[$y = 0 x − 4$]], [], [#math.equation(block: false, alt: "y equals 0 x plus 3")[$y = 0 x + 3$]],
  [Since there is no #math.equation(block: false, alt: "x")[$x$] term we write #math.equation(block: false, alt: "0 x")[$0 x$].], [#math.equation(block: false, alt: "y equals m x plus b")[$y = m x + b$]], [], [#math.equation(block: false, alt: "y equals m x plus b")[$y = m x + b$]],
  [Identify the slope and #math.equation(block: false, alt: "y")[$y$]-intercept of both lines.], [#math.equation(block: false, alt: "m equals 0")[$m = 0$]], [], [#math.equation(block: false, alt: "m equals 0")[$m = 0$]],
  [], [#math.equation(block: false, alt: "y -intercept is (0, 4)")[$y "-intercept is (0, 4)"$]], [], [#math.equation(block: false, alt: "y -intercept is (0, 3)")[$y "-intercept is (0, 3)"$]],
))

The lines have the same slope and different #emph[y]-intercepts and so they are parallel.

There is another way you can look at this example. If you recognize right away from the equations that these are horizontal lines, you know their slopes are both 0. Since the horizontal lines cross the #emph[y]-axis at #math.equation(block: false, alt: "y equals −4")[$y = −4$] and at #math.equation(block: false, alt: "y equals 3")[$y = 3$], we know the #emph[y]-intercepts are #math.equation(block: false, alt: "open parenthesis 0 , −4 close parenthesis")[$( 0 , −4 )$] and #math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$( 0 , 3 )$]. The lines have the same slope and different #emph[y]-intercepts and so they are parallel.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use slopes and #emph[y]-intercepts to determine if the lines #math.equation(block: false, alt: "y equals 8 and y equals −6")[$y = 8 #h(0.2em) "and" #h(0.2em) y = −6$] are parallel.

#solutionbox[
parallel
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use slopes and #emph[y]-intercepts to determine if the lines #math.equation(block: false, alt: "y equals 1 and y equals −5")[$y = 1 #h(0.2em) "and" #h(0.2em) y = −5$] are parallel.

#solutionbox[
parallel
]
]

#examplebox("Example 14")[][
Use slopes and #emph[y]-intercepts to determine if the lines #math.equation(block: false, alt: "x equals −2")[$x = −2$] and #math.equation(block: false, alt: "x equals −5")[$x = −5$] are parallel.

#solutionbox[
#math.equation(block: true, alt: "x equals −2 and x equals −5")[$x = −2 #h(0.4em) "and" #h(0.4em) x = −5$]Since there is no#math.equation(block: false, alt: "y")[$y$], the equations cannot be put in slope–intercept form. But we recognize them as equations of vertical lines. Their #emph[x]-intercepts are #math.equation(block: false, alt: "−2")[$−2$] and #math.equation(block: false, alt: "−5")[$−5$]. Since their #emph[x]-intercepts are different, the vertical lines are parallel.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use slopes and #emph[y]-intercepts to determine if the lines #math.equation(block: false, alt: "x equals 1")[$x = 1$] and #math.equation(block: false, alt: "x equals −5")[$x = −5$] are parallel.

#solutionbox[
parallel
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use slopes and #emph[y]-intercepts to determine if the lines #math.equation(block: false, alt: "x equals 8")[$x = 8$] and #math.equation(block: false, alt: "x equals −6")[$x = −6$] are parallel.

#solutionbox[
parallel
]
]

#examplebox("Example 15")[][
Use slopes and #emph[y]-intercepts to determine if the lines #math.equation(block: false, alt: "y equals 2 x minus 3")[$y = 2 x − 3$] and #math.equation(block: false, alt: "−6 x plus 3 y equals −9")[$−6 x + 3 y = −9$] are parallel. You may want to graph these lines, too, to see what they look like.

#solutionbox[
#figure(table(
  columns: 4, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "y equals 2 x minus 3")[$y = 2 x − 3$]], [and], [#math.equation(block: false, alt: "−6 x plus 3 y equals −9")[$−6 x + 3 y = −9$]]),
  [The first equation is already in slope-intercept form.], [#math.equation(block: false, alt: "y equals 2 x minus 3")[$y = 2 x − 3$]], [], [],
  [Solve the second equation for #math.equation(block: false, alt: "y")[$y$].], [], [], [#math.equation(block: false, alt: "minus 6 x plus 3 y, equals, −9; 3 y, equals, 6 x minus 9; the fraction 3 y over 3, equals, the fraction 6 x minus 9 over 3; y, equals, 2 x minus 3")[$− 6 x + 3 y & = & −9 \ 3 y & = & 6 x − 9 \ frac(3 y, 3) & = & frac(6 x − 9, 3) \ y & = & 2 x − 3$]],
  [The second equation is now in slope-intercept form.], [#math.equation(block: false, alt: "y equals 2 x minus 3")[$y = 2 x − 3$]], [], [],
  [Identify the slope and #math.equation(block: false, alt: "y")[$y$]-intercept of both lines.], [#math.equation(block: false, alt: "y, equals, 2 x minus 3; y, equals, m x plus b; m, equals, 2")[$y & = & 2 x − 3 \ y & = & m x + b \ m & = & 2$]], [], [#math.equation(block: false, alt: "y, equals, 2 x minus 3; y, equals, m x plus b; m, equals, 2")[$y & = & 2 x − 3 \ y & = & m x + b \ m & = & 2$]],
  [], [#math.equation(block: false, alt: "y -intercept is (0, −3)")[$y "-intercept is (0, −3)"$]], [], [#math.equation(block: false, alt: "y -intercept is (0, −3)")[$y "-intercept is (0, −3)"$]],
))

The lines have the same slope, but they also have the same #emph[y]-intercepts. Their equations represent the same line. They are not parallel; they are the same line.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use slopes and #emph[y]-intercepts to determine if the lines #math.equation(block: false, alt: "y equals minus the fraction 1 over 2 x minus 1")[$y = − frac(1, 2) x − 1$] and #math.equation(block: false, alt: "x plus 2 y equals 2")[$x + 2 y = 2$] are parallel.

#solutionbox[
parallel
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use slopes and #emph[y]-intercepts to determine if the lines #math.equation(block: false, alt: "y equals the fraction 3 over 4 x minus 3")[$y = frac(3, 4) x − 3$] and #math.equation(block: false, alt: "3 x minus 4 y equals 12")[$3 x − 4 y = 12$] are parallel.

#solutionbox[
not parallel; same line
]
]


=== Use Slopes to Identify Perpendicular Lines

Let’s look at the lines whose equations are #math.equation(block: false, alt: "y equals the fraction 1 over 4 x minus 1")[$y = frac(1, 4) x − 1$] and #math.equation(block: false, alt: "y equals −4 x plus 2")[$y = −4 x + 2$], shown.

#figure(figph[The figure shows two lines graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. One line is labeled with the equation y equals negative 4x plus 2 and goes through the points (0,2) and (1, negative 2). The other line is labeled with the equation y equals one fourth x minus 1 and goes through the points (0, negative 1) and (4,0).], alt: "The figure shows two lines graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. One line is labeled with the equation y equals negative 4x plus 2 and goes through the points (0,2) and (1, negative 2). The other line is labeled with the equation y equals one fourth x minus 1 and goes through the points (0, negative 1) and (4,0).", caption: none)

These lines lie in the same plane and intersect in right angles. We call these lines #strong[perpendicular].

What do you notice about the slopes of these two lines? As we read from left to right, the line #math.equation(block: false, alt: "y equals the fraction 1 over 4 x minus 1")[$y = frac(1, 4) x − 1$] rises, so its slope is positive. The line#math.equation(block: false, alt: "y equals −4 x plus 2")[$y = −4 x + 2$] drops from left to right, so it has a negative slope. Does it make sense to you that the slopes of two perpendicular lines will have opposite signs?

If we look at the slope of the first line, #math.equation(block: false, alt: "m sub 1 equals the fraction 1 over 4")[$m_(1) = frac(1, 4)$], and the slope of the second line, #math.equation(block: false, alt: "m sub 2 equals −4")[$m_(2) = −4$], we can see that they are #emph[negative reciprocals] of each other. If we multiply them, their product is #math.equation(block: false, alt: "−1 .")[$−1 .$]

#math.equation(block: true, alt: "m sub 1 times m sub 2; the fraction 1 over 4 open parenthesis −4 close parenthesis; minus 1")[$m_(1) · m_(2) \ frac(1, 4) ( −4 ) \ − 1$]This is always true for #strong[perpendicular lines] and leads us to this definition.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Perpendicular Lines]

#strong[Perpendicular lines] are lines in the same plane that form a right angle.

If #math.equation(block: false, alt: "m sub 1 and m sub 2")[$m_(1) #h(0.2em) "and" #h(0.2em) m_(2)$] are the slopes of two perpendicular lines, then:

#math.equation(block: true, alt: "m sub 1 times m sub 2 equals −1 and m sub 1 equals the fraction −1 over m sub 2")[$m_(1) · m_(2) = −1 #h(0.5em) "and" #h(0.5em) m_(1) = frac(−1, m_(2))$]Vertical lines and horizontal lines are always perpendicular to each other.
]

We were able to look at the slope–intercept form of linear equations and determine whether or not the lines were parallel. We can do the same thing for perpendicular lines.

We find the slope–intercept form of the equation, and then see if the slopes are negative reciprocals. If the product of the slopes is #math.equation(block: false, alt: "−1")[$−1$], the lines are perpendicular. Perpendicular lines may have the same #emph[y]-intercepts.

#examplebox("Example 16")[][
Use slopes to determine if the lines, #math.equation(block: false, alt: "y equals −5 x minus 4")[$y = −5 x − 4$] and #math.equation(block: false, alt: "x minus 5 y equals 5")[$x − 5 y = 5$] are perpendicular.

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([The first equation is already in slope-intercept form.], [#math.equation(block: false, alt: "y, equals, −5 x minus 4")[$y & = & −5 x − 4$]], []),
  [Solve the second equation for #math.equation(block: false, alt: "y")[$y$].], [#math.equation(block: false, alt: "x minus 5 y, equals, 5; minus 5 y, equals, − x plus 5; the fraction −5 y over −5, equals, the fraction − x plus 5 over −5; y, equals, the fraction 1 over 5 x minus 1")[$x − 5 y & = & 5 \ − 5 y & = & "−" x + 5 \ frac(−5 y, −5) & = & frac("−" x + 5, −5) \ y & = & frac(1, 5) x − 1$]], [],
  [Identify the slope of each line.], [#math.equation(block: false, alt: "y, equals, −5 x minus 4; y, equals, m x plus b; m sub 1, equals, −5")[$y & = & −5 x − 4 \ y & = & m x + b \ m_(1) & = & −5$]], [#math.equation(block: false, alt: "y, equals, the fraction 1 over 5 x minus 1; y, equals, m x plus b; m sub 2, equals, the fraction 1 over 5")[$y & = & frac(1, 5) x − 1 \ y & = & m x + b \ m_(2) & = & frac(1, 5)$]],
))

The slopes are negative reciprocals of each other, so the lines are perpendicular. We check by multiplying the slopes,

#math.equation(block: true, alt: "m sub 1 times m sub 2; minus 5 open parenthesis the fraction 1 over 5 close parenthesis; minus 1 ✓")[$m_(1) · m_(2) \  \ − 5 ( frac(1, 5) ) \ − 1 ✓$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use slopes to determine if the lines #math.equation(block: false, alt: "y equals −3 x plus 2")[$y = −3 x + 2$] and #math.equation(block: false, alt: "x minus 3 y equals 4")[$x − 3 y = 4$] are perpendicular.

#solutionbox[
perpendicular
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use slopes to determine if the lines #math.equation(block: false, alt: "y equals 2 x minus 5")[$y = 2 x − 5$] and #math.equation(block: false, alt: "x plus 2 y equals −6")[$x + 2 y = −6$] are perpendicular.

#solutionbox[
perpendicular
]
]

#examplebox("Example 17")[][
Use slopes to determine if the lines, #math.equation(block: false, alt: "7 x plus 2 y equals 3")[$7 x + 2 y = 3$] and #math.equation(block: false, alt: "2 x plus 7 y equals 5")[$2 x + 7 y = 5$] are perpendicular.

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Solve the equations for #math.equation(block: false, alt: "y")[$y$].], [#math.equation(block: false, alt: "7 x plus 2 y, equals, 3; 2 y, equals, −7 x plus 3; the fraction 2 y over 2, equals, the fraction −7 x plus 3 over 2; y, equals, minus the fraction 7 over 2 x plus the fraction 3 over 2")[$#h(2em) 7 x + 2 y & = & 3 &  \ #h(2em) 2 y & = & −7 x + 3 \ #h(2em) frac(2 y, 2) & = & frac(−7 x + 3, 2) \ #h(2em) y & = & − frac(7, 2) x + frac(3, 2)$]], [#math.equation(block: false, alt: "2 x plus 7 y, equals, 5; 7 y, equals, −2 x plus 5; the fraction 7 y over 7, equals, the fraction −2 x plus 5 over 7; y, equals, minus the fraction 2 over 7 x plus the fraction 5 over 7")[$2 x + 7 y & = & 5 \ 7 y & = & −2 x + 5 \ frac(7 y, 7) & = & frac(−2 x + 5, 7) \ y & = & − frac(2, 7) x + frac(5, 7)$]]),
  [Identify the slope of each line.], [#math.equation(block: false, alt: "y, equals, m x plus b; m sub 1, equals, minus the fraction 7 over 2")[$y & = & m x + b \ #h(2em) m_(1) & = & − frac(7, 2)$]], [#math.equation(block: false, alt: "y, equals, m x plus b; m sub 2, equals, minus the fraction 2 over 7")[$y & = & m x + b \ #h(2em) m_(2) & = & − frac(2, 7)$]],
))

The slopes are reciprocals of each other, but they have the same sign. Since they are not negative reciprocals, the lines are not perpendicular.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use slopes to determine if the lines #math.equation(block: false, alt: "5 x plus 4 y equals 1")[$5 x + 4 y = 1$] and #math.equation(block: false, alt: "4 x plus 5 y equals 3")[$4 x + 5 y = 3$] are perpendicular.

#solutionbox[
not perpendicular
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use slopes to determine if the lines #math.equation(block: false, alt: "2 x minus 9 y equals 3")[$2 x − 9 y = 3$] and #math.equation(block: false, alt: "9 x minus 2 y equals 1")[$9 x − 2 y = 1$] are perpendicular.

#solutionbox[
not perpendicular
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Access this online resource for additional instruction and practice with graphs.

- #link("https://openstax.org/l/25GraphPractice")[Explore the Relation Between a Graph and the Slope–Intercept Form of an Equation of a Line]
]


=== Key Concepts

- The slope–intercept form of an equation of a line with slope #math.equation(block: false, alt: "m")[$m$] and #emph[y]-intercept, #math.equation(block: false, alt: "open parenthesis 0 , b close parenthesis")[$( 0 , b )$] is, #math.equation(block: false, alt: "y equals m x plus b")[$y = m x + b$].
- #strong[Graph a Line Using its Slope and #emph[y]-Intercept]
+ Find the slope-intercept form of the equation of the line.
+ Identify the slope and #emph[y]-intercept.
+ Plot the #emph[y]-intercept.
+ Use the slope formula #math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$] to identify the rise and the run.
+ Starting at the #emph[y]-intercept, count out the rise and run to mark the second point.
+ Connect the points with a line.

#linebreak()
- #strong[Strategy for Choosing the Most Convenient Method to Graph a Line:] Consider the form of the equation.
- If it only has one variable, it is a vertical or horizontal line. #linebreak() #math.equation(block: false, alt: "x equals a")[$x = a$] is a vertical line passing through the #emph[x]-axis at #math.equation(block: false, alt: "a")[$a$]. #linebreak() #math.equation(block: false, alt: "y equals b")[$y = b$] is a horizontal line passing through the #emph[y]-axis at #math.equation(block: false, alt: "b")[$b$].
- If #math.equation(block: false, alt: "y")[$y$] is isolated on one side of the equation, in the form #math.equation(block: false, alt: "y equals m x plus b")[$y = m x + b$], graph by using the slope and #emph[y]-intercept. #linebreak() 
Identify the slope and #emph[y]-intercept and then graph.
- If the equation is of the form #math.equation(block: false, alt: "A x plus B y equals C")[$A x + B y = C$], find the intercepts. #linebreak() 
Find the #emph[x]- and #emph[y]-intercepts, a third point, and then graph.
- Parallel lines are lines in the same plane that do not intersect.
- Parallel lines have the same slope and different #emph[y]-intercepts.
- If #emph[m]#sub[1] and #emph[m]#sub[2] are the slopes of two parallel lines then #math.equation(block: false, alt: "m sub 1 equals m sub 2 .")[$m_(1) = m_(2) .$]
- Parallel vertical lines have different #emph[x]-intercepts.
- Perpendicular lines are lines in the same plane that form a right angle.
- If #math.equation(block: false, alt: "m sub 1 and m sub 2")[$m_(1) #h(0.2em) "and" #h(0.2em) m_(2)$] are the slopes of two perpendicular lines, then #math.equation(block: false, alt: "m sub 1 times m sub 2 equals −1")[$m_(1) · m_(2) = −1$] and #math.equation(block: false, alt: "m sub 1 equals the fraction −1 over m sub 2")[$m_(1) = frac(−1, m_(2))$].
- Vertical lines and horizontal lines are always perpendicular to each other.


==== Practice Makes Perfect

#strong[Recognize the Relation Between the Graph and the Slope–Intercept Form of an Equation of a Line]

In the following exercises, use the graph to find the slope and y-intercept of each line. Compare the values to the equation #math.equation(block: false, alt: "y equals m x plus b")[$y = m x + b$].

#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0, negative 5) and (1, negative 2).], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0, negative 5) and (1, negative 2).", caption: none)

#math.equation(block: true, alt: "y equals 3 x minus 5")[$y = 3 x − 5$]

#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0, negative 2) and (1,2).], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0, negative 2) and (1,2).", caption: none)

#math.equation(block: true, alt: "y equals 4 x minus 2")[$y = 4 x − 2$]

#solutionbox[
slope #math.equation(block: false, alt: "m equals 4")[$m = 4$] and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , −2 close parenthesis")[$( 0 , −2 )$]
]

#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0,4) and (1,3).], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0,4) and (1,3).", caption: none)

#math.equation(block: true, alt: "y equals − x plus 4")[$y = "−" x + 4$]

#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0,1) and (1, negative 2).], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0,1) and (1, negative 2).", caption: none)

#math.equation(block: true, alt: "y equals −3 x plus 1")[$y = −3 x + 1$]

#solutionbox[
slope #math.equation(block: false, alt: "m equals −3")[$m = −3$] and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$]
]

#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0,1) and (3, negative 3).], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0,1) and (3, negative 3).", caption: none)

#math.equation(block: true, alt: "y equals minus the fraction 4 over 3 x plus 1")[$y = − frac(4, 3) x + 1$]

#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0,3) and (1,5).], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0,3) and (1,5).", caption: none)

#math.equation(block: true, alt: "y equals minus the fraction 2 over 5 x plus 3")[$y = − frac(2, 5) x + 3$]

#solutionbox[
slope #math.equation(block: false, alt: "m equals minus the fraction 2 over 5")[$m = − frac(2, 5)$] and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$( 0 , 3 )$]
]

#strong[Identify the Slope and y-Intercept From an Equation of a Line]

In the following exercises, identify the slope and y-intercept of each line.

#math.equation(block: true, alt: "y equals −7 x plus 3")[$y = −7 x + 3$]

#math.equation(block: true, alt: "y equals −9 x plus 7")[$y = −9 x + 7$]

#solutionbox[
#math.equation(block: true, alt: "−9 ; open parenthesis 0 , 7 close parenthesis")[$−9 ; ( 0 , 7 )$]
]

#math.equation(block: true, alt: "y equals 6 x minus 8")[$y = 6 x − 8$]

#math.equation(block: true, alt: "y equals 4 x minus 10")[$y = 4 x − 10$]

#solutionbox[
#math.equation(block: true, alt: "4 ; open parenthesis 0 , −10 close parenthesis")[$4 ; ( 0 , −10 )$]
]

#math.equation(block: true, alt: "3 x plus y equals 5")[$3 x + y = 5$]

#math.equation(block: true, alt: "4 x plus y equals 8")[$4 x + y = 8$]

#solutionbox[
#math.equation(block: true, alt: "−4 ; open parenthesis 0 , 8 close parenthesis")[$−4 ; ( 0 , 8 )$]
]

#math.equation(block: true, alt: "6 x plus 4 y equals 12")[$6 x + 4 y = 12$]

#math.equation(block: true, alt: "8 x plus 3 y equals 12")[$8 x + 3 y = 12$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 8 over 3 ; open parenthesis 0 , 4 close parenthesis")[$− frac(8, 3) ; ( 0 , 4 )$]
]

#math.equation(block: true, alt: "5 x minus 2 y equals 6")[$5 x − 2 y = 6$]

#math.equation(block: true, alt: "7 x minus 3 y equals 9")[$7 x − 3 y = 9$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 7 over 3 ; open parenthesis 0 , −3 close parenthesis")[$frac(7, 3) ; ( 0 , −3 )$]
]

#strong[Graph a Line Using Its Slope and Intercept]

In the following exercises, graph the line of each equation using its slope and y-intercept.

#math.equation(block: true, alt: "y equals x plus 3")[$y = x + 3$]

#math.equation(block: true, alt: "y equals x plus 4")[$y = x + 4$]

#solutionbox[
#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0, 4) and (1, 5).], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0, 4) and (1, 5).", caption: none)
]

#math.equation(block: true, alt: "y equals 3 x minus 1")[$y = 3 x − 1$]

#math.equation(block: true, alt: "y equals 2 x minus 3")[$y = 2 x − 3$]

#solutionbox[
#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0, negative 3) and (1, negative 1).], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0, negative 3) and (1, negative 1).", caption: none)
]

#math.equation(block: true, alt: "y equals − x plus 2")[$y = "−" x + 2$]

#math.equation(block: true, alt: "y equals − x plus 3")[$y = "−" x + 3$]

#solutionbox[
#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0, 3) and (1, 2).], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0, 3) and (1, 2).", caption: none)
]

#math.equation(block: true, alt: "y equals − x minus 4")[$y = "−" x − 4$]

#math.equation(block: true, alt: "y equals − x minus 2")[$y = "−" x − 2$]

#solutionbox[
#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0, negative 2) and (1, negative 3).], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0, negative 2) and (1, negative 3).", caption: none)
]

#math.equation(block: true, alt: "y equals minus the fraction 3 over 4 x minus 1")[$y = − frac(3, 4) x − 1$]

#math.equation(block: true, alt: "y equals minus the fraction 2 over 5 x minus 3")[$y = − frac(2, 5) x − 3$]

#solutionbox[
#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0, negative 3) and (5, negative 5).], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0, negative 3) and (5, negative 5).", caption: none)
]

#math.equation(block: true, alt: "y equals minus the fraction 3 over 5 x plus 2")[$y = − frac(3, 5) x + 2$]

#math.equation(block: true, alt: "y equals minus the fraction 2 over 3 x plus 1")[$y = − frac(2, 3) x + 1$]

#solutionbox[
#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0,1) and (3, negative 1).], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0,1) and (3, negative 1).", caption: none)
]

#math.equation(block: true, alt: "3 x minus 4 y equals 8")[$3 x − 4 y = 8$]

#math.equation(block: true, alt: "4 x minus 3 y equals 6")[$4 x − 3 y = 6$]

#solutionbox[
#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0, negative 2) and (3,2).], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0, negative 2) and (3,2).", caption: none)
]

#math.equation(block: true, alt: "y equals 0.1 x plus 15")[$y = 0.1 x + 15$]

#math.equation(block: true, alt: "y equals 0.3 x plus 25")[$y = 0.3 x + 25$]

#solutionbox[
#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0, 25) and (negative 50, 10).], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0, 25) and (negative 50, 10).", caption: none)
]

#strong[Choose the Most Convenient Method to Graph a Line]

In the following exercises, determine the most convenient method to graph each line.

#math.equation(block: true, alt: "x equals 2")[$x = 2$]

#math.equation(block: true, alt: "y equals 4")[$y = 4$]

#solutionbox[
horizontal line
]

#math.equation(block: true, alt: "y equals 5")[$y = 5$]

#math.equation(block: true, alt: "x equals −3")[$x = −3$]

#solutionbox[
vertical line
]

#math.equation(block: true, alt: "y equals −3 x plus 4")[$y = −3 x + 4$]

#math.equation(block: true, alt: "y equals −5 x plus 2")[$y = −5 x + 2$]

#solutionbox[
slope–intercept
]

#math.equation(block: true, alt: "x minus y equals 5")[$x − y = 5$]

#math.equation(block: true, alt: "x minus y equals 1")[$x − y = 1$]

#solutionbox[
intercepts
]

#math.equation(block: true, alt: "y equals the fraction 2 over 3 x minus 1")[$y = frac(2, 3) x − 1$]

#math.equation(block: true, alt: "y equals the fraction 4 over 5 x minus 3")[$y = frac(4, 5) x − 3$]

#solutionbox[
slope–intercept
]

#math.equation(block: true, alt: "y equals −3")[$y = −3$]

#math.equation(block: true, alt: "y equals −1")[$y = −1$]

#solutionbox[
horizontal line
]

#math.equation(block: true, alt: "3 x minus 2 y equals −12")[$3 x − 2 y = −12$]

#math.equation(block: true, alt: "2 x minus 5 y equals −10")[$2 x − 5 y = −10$]

#solutionbox[
intercepts
]

#math.equation(block: true, alt: "y equals minus the fraction 1 over 4 x plus 3")[$y = − frac(1, 4) x + 3$]

#math.equation(block: true, alt: "y equals minus the fraction 1 over 3 x plus 5")[$y = − frac(1, 3) x + 5$]

#solutionbox[
slope–intercept
]

#strong[Graph and Interpret Applications of Slope–Intercept]

The equation #math.equation(block: false, alt: "P equals 31 plus 1.75 w")[$P = 31 + 1.75 w$] models the relation between the amount of Tuyet’s monthly water bill payment, #emph[P], in dollars, and the number of units of water, #emph[w], used.

+ ⓐ Find Tuyet’s payment for a month when 0 units of water are used.
+ ⓑ Find Tuyet’s payment for a month when 12 units of water are used.
+ ⓒ Interpret the slope and #emph[P]-intercept of the equation.
+ ⓓ Graph the equation.

The equation #math.equation(block: false, alt: "P equals 28 plus 2.54 w")[$P = 28 + 2.54 w$] models the relation between the amount of Randy’s monthly water bill payment, #emph[P], in dollars, and the number of units of water, #emph[w], used.

+ ⓐ Find the payment for a month when Randy used 0 units of water.
+ ⓑ Find the payment for a month when Randy used 15 units of water.
+ ⓒ Interpret the slope and #emph[P]-intercept of the equation.
+ ⓓ Graph the equation.

#solutionbox[
+ ⓐ \$28
+ ⓑ \$66.10
+ ⓒ The slope, 2.54, means that Randy’s payment, #emph[P], increases by \$2.54 when the number of units of water he used, #emph[w,] increases by 1. The #emph[P]–intercept means that if the number units of water Randy used was 0, the payment would be \$28.
+ ⓓ  #linebreak() 
#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane represents the variable w and runs from negative 2 to 20. The y-axis of the plane represents the variable P and runs from negative 1 to 100. The line begins at the point (0, 28) and goes through the point (15, 66.1).], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane represents the variable w and runs from negative 2 to 20. The y-axis of the plane represents the variable P and runs from negative 1 to 100. The line begins at the point (0, 28) and goes through the point (15, 66.1).", caption: none)
]

Bruce drives his car for his job. The equation #math.equation(block: false, alt: "R equals 0.575 m plus 42")[$R = 0.575 m + 42$] models the relation between the amount in dollars, #emph[R], that he is reimbursed and the number of miles, #emph[m], he drives in one day.

+ ⓐ Find the amount Bruce is reimbursed on a day when he drives 0 miles.
+ ⓑ Find the amount Bruce is reimbursed on a day when he drives 220 miles.
+ ⓒ Interpret the slope and #emph[R]-intercept of the equation.
+ ⓓ Graph the equation.

Janelle is planning to rent a car while on vacation. The equation #math.equation(block: false, alt: "C equals 0.32 m plus 15")[$C = 0.32 m + 15$] models the relation between the cost in dollars, #emph[C], per day and the number of miles, #emph[m], she drives in one day.

+ ⓐ Find the cost if Janelle drives the car 0 miles one day.
+ ⓑ Find the cost on a day when Janelle drives the car 400 miles.
+ ⓒ Interpret the slope and #emph[C]–intercept of the equation.
+ ⓓ Graph the equation.

#solutionbox[
+ ⓐ \$15
+ ⓑ \$143
+ ⓒ The slope, 0.32, means that the cost, #emph[C], increases by \$0.32 when the number of miles driven, #emph[m,] increases by 1. The #emph[C]-intercept means that if Janelle drives 0 miles one day, the cost would be \$15.
+ ⓓ  #linebreak() 
#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane represents the variable m and runs from negative 1 to 500. The y-axis of the plane represents the variable C and runs from negative 1 to 200. The line begins at the point (0,15) and goes through the point (400,143).], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane represents the variable m and runs from negative 1 to 500. The y-axis of the plane represents the variable C and runs from negative 1 to 200. The line begins at the point (0,15) and goes through the point (400,143).", caption: none)
]

Cherie works in retail and her weekly salary includes commission for the amount she sells. The equation #math.equation(block: false, alt: "S equals 400 plus 0.15 c")[$S = 400 + 0.15 c$] models the relation between her weekly salary, #emph[S], in dollars and the amount of her sales, #emph[c], in dollars.

+ ⓐ Find Cherie’s salary for a week when her sales were 0.
+ ⓑ Find Cherie’s salary for a week when her sales were 3600.
+ ⓒ Interpret the slope and #emph[S]–intercept of the equation.
+ ⓓ Graph the equation.

Patel’s weekly salary includes a base pay plus commission on his sales. The equation #math.equation(block: false, alt: "S equals 750 plus 0.09 c")[$S = 750 + 0.09 c$] models the relation between his weekly salary, #emph[S], in dollars and the amount of his sales, #emph[c], in dollars.

+ ⓐ Find Patel’s salary for a week when his sales were 0.
+ ⓑ Find Patel’s salary for a week when his sales were 18,540.
+ ⓒ Interpret the slope and #emph[S]-intercept of the equation.
+ ⓓ Graph the equation.

#solutionbox[
+ ⓐ \$750
+ ⓑ \$2418.60
+ ⓒ The slope, 0.09, means that Patel’s salary, #emph[S], increases by \$0.09 for every \$1 increase in his sales. The #emph[S]-intercept means that when his sales are \$0, his salary is \$750.
+ ⓓ  #linebreak() 
#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane represents the variable w and runs from negative 1 to 20000. The y-axis of the plane represents the variable P and runs from negative 1 to 3000. The line begins at the point (0, 750) and goes through the point (18540, 2415).], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane represents the variable w and runs from negative 1 to 20000. The y-axis of the plane represents the variable P and runs from negative 1 to 3000. The line begins at the point (0, 750) and goes through the point (18540, 2415).", caption: none)
]

Costa is planning a lunch banquet. The equation #math.equation(block: false, alt: "C equals 450 plus 28 g")[$C = 450 + 28 g$] models the relation between the cost in dollars, #emph[C], of the banquet and the number of guests, #emph[g].

+ ⓐ Find the cost if the number of guests is 40.
+ ⓑ Find the cost if the number of guests is 80.
+ ⓒ Interpret the slope and #emph[C]-intercept of the equation.
+ ⓓ Graph the equation.

Margie is planning a dinner banquet. The equation #math.equation(block: false, alt: "C equals 750 plus 42 g")[$C = 750 + 42 g$] models the relation between the cost in dollars, #emph[C] of the banquet and the number of guests, #emph[g].

+ ⓐ Find the cost if the number of guests is 50.
+ ⓑ Find the cost if the number of guests is 100.
+ ⓒ Interpret the slope and #emph[C]–intercept of the equation.
+ ⓓ Graph the equation.

#solutionbox[
+ ⓐ \$2850
+ ⓑ \$4950
+ ⓒ The slope, 42, means that the cost, #emph[C], increases by \$42 for when the number of guests increases by 1. The #emph[C]-intercept means that when the number of guests is 0, the cost would be \$750.
+ ⓓ  #linebreak() 
#figure(figph[The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane represents the variable g and runs from negative 1 to 150. The y-axis of the plane represents the variable C and runs from negative 1 to 7000. The line begins at the point (0, 750) and goes through the point (100, 4950).], alt: "The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane represents the variable g and runs from negative 1 to 150. The y-axis of the plane represents the variable C and runs from negative 1 to 7000. The line begins at the point (0, 750) and goes through the point (100, 4950).", caption: none)
]

#strong[Use Slopes to Identify Parallel Lines]

In the following exercises, use slopes and y-intercepts to determine if the lines are parallel.

#math.equation(block: true, alt: "y equals the fraction 3 over 4 x minus 3 ; 3 x minus 4 y equals minus 2")[$y = frac(3, 4) x − 3 ; #h(0.5em) 3 x − 4 y = − 2$]

#math.equation(block: true, alt: "y equals the fraction 2 over 3 x minus 1 ; 2 x minus 3 y equals minus 2")[$y = frac(2, 3) x − 1 ; #h(0.5em) 2 x − 3 y = − 2$]

#solutionbox[
parallel
]

#math.equation(block: true, alt: "2 x minus 5 y equals minus 3 ; y equals the fraction 2 over 5 x plus 1")[$2 x − 5 y = − 3 ; #h(0.5em) y = frac(2, 5) x + 1 #h(0.5em)$]

#math.equation(block: true, alt: "3 x minus 4 y equals minus 2 ; y equals the fraction 3 over 4 x minus 3")[$3 x − 4 y = − 2 ; #h(0.5em) y = frac(3, 4) x − 3 #h(0.5em)$]

#solutionbox[
parallel
]

#math.equation(block: true, alt: "2 x minus 4 y equals 6 ; x minus 2 y equals 3")[$2 x − 4 y = 6 ; #h(0.5em) x − 2 y = 3$]

#math.equation(block: true, alt: "6 x minus 3 y equals 9 ; 2 x minus y equals 3")[$6 x − 3 y = 9 ; #h(0.5em) 2 x − y = 3$]

#solutionbox[
not parallel
]

#math.equation(block: true, alt: "4 x plus 2 y equals 6 ; 6 x plus 3 y equals 3")[$4 x + 2 y = 6 ; #h(0.5em) 6 x + 3 y = 3$]

#math.equation(block: true, alt: "8 x plus 6 y equals 6 ; 12 x plus 9 y equals 12")[$8 x + 6 y = 6 ; #h(0.5em) 12 x + 9 y = 12$]

#solutionbox[
parallel
]

#math.equation(block: true, alt: "x equals 5 ; x equals minus 6")[$x = 5 ; #h(0.5em) x = − 6$]

#math.equation(block: true, alt: "x equals 7 ; x equals minus 8")[$x = 7 ; #h(0.5em) x = − 8$]

#solutionbox[
parallel
]

#math.equation(block: true, alt: "x equals minus 4 ; x equals minus 1")[$x = − 4 ; #h(0.5em) x = − 1$]

#math.equation(block: true, alt: "x equals minus 3 ; x equals minus 2")[$x = − 3 ; #h(0.5em) x = − 2$]

#solutionbox[
parallel
]

#math.equation(block: true, alt: "y equals 2 ; y equals 6")[$y = 2 ; #h(0.5em) y = 6$]

#math.equation(block: true, alt: "y equals 5 ; y equals 1")[$y = 5 ; #h(0.5em) y = 1$]

#solutionbox[
parallel
]

#math.equation(block: true, alt: "y equals minus 4 ; y equals 3")[$y = − 4 ; #h(0.5em) y = 3$]

#math.equation(block: true, alt: "y equals minus 1 ; y equals 2")[$y = − 1 ; #h(0.5em) y = 2$]

#solutionbox[
parallel
]

#math.equation(block: true, alt: "x minus y equals 2 ; 2 x minus 2 y equals 4")[$x − y = 2 ; #h(0.5em) 2 x − 2 y = 4$]

#math.equation(block: true, alt: "4 x plus 4 y equals 8 ; x plus y equals 2")[$4 x + 4 y = 8 ; #h(0.5em) x + y = 2$]

#solutionbox[
not parallel
]

#math.equation(block: true, alt: "x minus 3 y equals 6 ; 2 x minus 6 y equals 12")[$x − 3 y = 6 ; #h(0.5em) 2 x − 6 y = 12$]

#math.equation(block: true, alt: "5 x minus 2 y equals 11 ; 5 x minus y equals 7")[$5 x − 2 y = 11 ; #h(0.5em) 5 x − y = 7$]

#solutionbox[
not parallel
]

#math.equation(block: true, alt: "3 x minus 6 y equals 12 ; 6 x minus 3 y equals 3")[$3 x − 6 y = 12 ; #h(0.5em) 6 x − 3 y = 3$]

#math.equation(block: true, alt: "4 x minus 8 y equals 16 ; x minus 2 y equals 4")[$4 x − 8 y = 16 ; #h(0.5em) x − 2 y = 4$]

#solutionbox[
not parallel
]

#math.equation(block: true, alt: "9 x minus 3 y equals 6 ; 3 x minus y equals 2")[$9 x − 3 y = 6 ; #h(0.5em) 3 x − y = 2$]

#math.equation(block: true, alt: "x minus 5 y equals 10 ; 5 x minus y equals minus 10")[$x − 5 y = 10 ; #h(0.5em) 5 x − y = − 10$]

#solutionbox[
not parallel
]

#math.equation(block: true, alt: "7 x minus 4 y equals 8 ; 4 x plus 7 y equals 14")[$7 x − 4 y = 8 ; #h(0.5em) 4 x + 7 y = 14$]

#math.equation(block: true, alt: "9 x minus 5 y equals 4 ; 5 x plus 9 y equals minus 1")[$9 x − 5 y = 4 ; #h(0.5em) 5 x + 9 y = − 1$]

#solutionbox[
not parallel
]

#strong[Use Slopes to Identify Perpendicular Lines]

In the following exercises, use slopes and y-intercepts to determine if the lines are perpendicular.

#math.equation(block: true, alt: "3 x minus 2 y equals 8 ; 2 x plus 3 y equals 6")[$3 x − 2 y = 8 ; 2 x + 3 y = 6$]

#math.equation(block: true, alt: "x minus 4 y equals 8 ; 4 x plus y equals 2")[$x − 4 y = 8 ; 4 x + y = 2$]

#solutionbox[
perpendicular
]

#math.equation(block: true, alt: "2 x plus 5 y equals 3 ; 5 x minus 2 y equals 6")[$2 x + 5 y = 3 ; 5 x − 2 y = 6$]

#math.equation(block: true, alt: "2 x plus 3 y equals 5 ; 3 x minus 2 y equals 7")[$2 x + 3 y = 5 ; 3 x − 2 y = 7$]

#solutionbox[
perpendicular
]

#math.equation(block: true, alt: "3 x minus 2 y equals 1 ; 2 x minus 3 y equals 2")[$3 x − 2 y = 1 ; 2 x − 3 y = 2$]

#math.equation(block: true, alt: "3 x minus 4 y equals 8 ; 4 x minus 3 y equals 6")[$3 x − 4 y = 8 ; 4 x − 3 y = 6$]

#solutionbox[
not perpendicular
]

#math.equation(block: true, alt: "5 x plus 2 y equals 6 ; 2 x plus 5 y equals 8")[$5 x + 2 y = 6 ; 2 x + 5 y = 8$]

#math.equation(block: true, alt: "2 x plus 4 y equals 3 ; 6 x plus 3 y equals 2")[$2 x + 4 y = 3 ; 6 x + 3 y = 2$]

#solutionbox[
not perpendicular
]

#math.equation(block: true, alt: "4 x minus 2 y equals 5 ; 3 x plus 6 y equals 8")[$4 x − 2 y = 5 ; 3 x + 6 y = 8$]

#math.equation(block: true, alt: "2 x minus 6 y equals 4 ; 12 x plus 4 y equals 9")[$2 x − 6 y = 4 ; 12 x + 4 y = 9$]

#solutionbox[
perpendicular
]

#math.equation(block: true, alt: "6 x minus 4 y equals 5 ; 8 x plus 12 y equals 3")[$6 x − 4 y = 5 ; 8 x + 12 y = 3$]

#math.equation(block: true, alt: "8 x minus 2 y equals 7 ; 3 x plus 12 y equals 9")[$8 x − 2 y = 7 ; 3 x + 12 y = 9$]

#solutionbox[
perpendicular
]


==== Everyday Math

The equation #math.equation(block: false, alt: "C equals the fraction 5 over 9 F minus 17.8")[$C = frac(5, 9) F − 17.8$] can be used to convert temperatures #emph[F], on the Fahrenheit scale to temperatures, #emph[C], on the Celsius scale.

+ ⓐ Explain what the slope of the equation means.
+ ⓑ Explain what the #emph[C]–intercept of the equation means.

The equation #math.equation(block: false, alt: "n equals 4 T minus 160")[$n = 4 T − 160$] is used to estimate the number of cricket chirps, #emph[n], in one minute based on the temperature in degrees Fahrenheit, #emph[T].

+ ⓐ Explain what the slope of the equation means.
+ ⓑ Explain what the #emph[n]–intercept of the equation means. Is this a realistic situation?

#solutionbox[
+ ⓐ For every increase of one degree Fahrenheit, the number of chirps increases by four.
+ ⓑ There would be #math.equation(block: false, alt: "−160")[$−160$] chirps when the Fahrenheit temperature is #math.equation(block: false, alt: "0 °")[$0 "°"$]. (Notice that this does not make sense; this model cannot be used for all possible temperatures.)
]


==== Writing Exercises

Explain in your own words how to decide which method to use to graph a line.

Why are all horizontal lines parallel?

#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[This table has eight rows and four columns. The first row is a header row and it labels each column. The first column is labeled "I can …", the second "Confidently", the third “With some help” and the last "No–I don’t get it". In the “I can…” column the next row reads “recognize the relation between the graph and the slope-intercept form of an equation of a line.” The third row reads “identify the Slope and y-intercept from an equation of a line”. The fourth row reads “graph a line using its slope and intercept”. The fifth row reads “choose the most convenient method to graph a line.” The sixth row reads “graph and interpret applications of slope-intercept”. The seventh row reads “use slopes to identify parallel lines” and the last row reads “use slopes to identify perpendicular lines.” The remaining columns are blank.], alt: "This table has eight rows and four columns. The first row is a header row and it labels each column. The first column is labeled \"I can …\", the second \"Confidently\", the third “With some help” and the last \"No–I don’t get it\". In the “I can…” column the next row reads “recognize the relation between the graph and the slope-intercept form of an equation of a line.” The third row reads “identify the Slope and y-intercept from an equation of a line”. The fourth row reads “graph a line using its slope and intercept”. The fifth row reads “choose the most convenient method to graph a line.” The sixth row reads “graph and interpret applications of slope-intercept”. The seventh row reads “use slopes to identify parallel lines” and the last row reads “use slopes to identify perpendicular lines.” The remaining columns are blank.", caption: none)

ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?