#set document(title: "4.3 Graph with Intercepts", author: "OpenStax / XYZ Homework")
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== 4.3#h(0.6em)Graph with Intercepts

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

Solve: #math.equation(block: false, alt: "3 times 0 plus 4 y equals −2 .")[$3 · 0 + 4 y = −2 .$] #linebreak() 
If you missed this problem, review .

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


=== Identify the #emph[x]- and #emph[y]- Intercepts on a Graph

Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that when graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing the line might use different sets of three points.

At first glance, their two lines might not appear to be the same, since they would have different points labeled. But if all the work was done correctly, the lines should be exactly the same. One way to recognize that they are indeed the same line is to look at where the line crosses the #emph[x]- axis and the #emph[y]- axis. These points are called the #emph[intercepts] of the line.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Intercepts of a Line]

The points where a line crosses the #emph[x]- axis and the #emph[y]- axis are called the #strong[intercepts of a line].
]

Let’s look at the graphs of the lines in .

#figure(figph[Four figures, each showing a different straight line on the x y- coordinate plane. The x- axis of the planes runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. Figure a shows a straight line crossing the x- axis at the point (3, 0) and crossing the y- axis at the point (0, 6). The graph is labeled with the equation 2x plus y equals 6. Figure b shows a straight line crossing the x- axis at the point (4, 0) and crossing the y- axis at the point (0, negative 3). The graph is labeled with the equation 3x minus 4y equals 12. Figure c shows a straight line crossing the x- axis at the point (5, 0) and crossing the y- axis at the point (0, negative 5). The graph is labeled with the equation x minus y equals 5. Figure d shows a straight line crossing the x- axis and y- axis at the point (0, 0). The graph is labeled with the equation y equals negative 2x.], alt: "Four figures, each showing a different straight line on the x y- coordinate plane. The x- axis of the planes runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. Figure a shows a straight line crossing the x- axis at the point (3, 0) and crossing the y- axis at the point (0, 6). The graph is labeled with the equation 2x plus y equals 6. Figure b shows a straight line crossing the x- axis at the point (4, 0) and crossing the y- axis at the point (0, negative 3). The graph is labeled with the equation 3x minus 4y equals 12. Figure c shows a straight line crossing the x- axis at the point (5, 0) and crossing the y- axis at the point (0, negative 5). The graph is labeled with the equation x minus y equals 5. Figure d shows a straight line crossing the x- axis and y- axis at the point (0, 0). The graph is labeled with the equation y equals negative 2x.", caption: [Examples of graphs crossing the x-axis.])

First, notice where each of these lines crosses the #math.equation(block: false, alt: "x")[$x$]-axis.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#strong[Figure]], [#strong[The line crosses the #emph[x]- axis at:]], [#strong[Ordered pair of this point]]),
  [Figure (a)], [3], [#math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis")[$( 3 , 0 )$]],
  [Figure (b)], [4], [#math.equation(block: false, alt: "open parenthesis 4 , 0 close parenthesis")[$( 4 , 0 )$]],
  [Figure (c)], [5], [#math.equation(block: false, alt: "open parenthesis 5 , 0 close parenthesis")[$( 5 , 0 )$]],
  [Figure (d)], [0], [#math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]],
))

Do you see a pattern?

For each row, the #emph[y]- coordinate of the point where the line crosses the #emph[x]- axis is zero. The point where the line crosses the #emph[x]- axis has the form #math.equation(block: false, alt: "open parenthesis a , 0 close parenthesis")[$( a , 0 )$] and is called the #strong[#emph[x]- intercept of a line]. The #emph[x]- intercept occurs when #math.equation(block: false, alt: "y")[$y$] is zero.

Now, let’s look at the points where these lines cross the #emph[y]- axis.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#strong[Figure]], [#strong[The line crosses the #emph[y]-axis at:]], [#strong[Ordered pair for this point]]),
  [Figure (a)], [6], [#math.equation(block: false, alt: "open parenthesis 0 , 6 close parenthesis")[$( 0 , 6 )$]],
  [Figure (b)], [#math.equation(block: false, alt: "−3")[$−3$]], [#math.equation(block: false, alt: "open parenthesis 0 , −3 close parenthesis")[$( 0 , −3 )$]],
  [Figure (c)], [#math.equation(block: false, alt: "−5")[$−5$]], [#math.equation(block: false, alt: "open parenthesis 0 , −5 close parenthesis")[$( 0 , −5 )$]],
  [Figure (d)], [0], [#math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]],
))

What is the pattern here?

In each row, the #emph[x]- coordinate of the point where the line crosses the #emph[y]- axis is zero. The point where the line crosses the #emph[y]- axis has the form #math.equation(block: false, alt: "open parenthesis 0 , b close parenthesis")[$( 0 , b )$] and is called the #emph[y- intercept] of the line. The #emph[y]- intercept occurs when #math.equation(block: false, alt: "x")[$x$] is zero.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[#emph[x]- intercept and #strong[#emph[y]- intercept of a line]]

The #emph[x]- intercept is the point #math.equation(block: false, alt: "open parenthesis a , 0 close parenthesis")[$( a , 0 )$] where the line crosses the #emph[x]- axis.

The #emph[y]- intercept is the point #math.equation(block: false, alt: "open parenthesis 0 , b close parenthesis")[$( 0 , b )$] where the line crosses the #emph[y]- axis. #linebreak() 
#figure(figph[The image shows a table that is 2 columns and 3 rows. Column 1 reads: x; a; 0. Column 2 reads: y; 0; b. The image indicates that x-intercept occurs when y is zero and y-intercept occurs when x is zero.], alt: "The image shows a table that is 2 columns and 3 rows. Column 1 reads: x; a; 0. Column 2 reads: y; 0; b. The image indicates that x-intercept occurs when y is zero and y-intercept occurs when x is zero.", caption: none)
]

#examplebox("Example 1")[][
Find the #emph[x]- and #emph[y]- intercepts on each graph.

#figure(figph[Three figures, each showing a different straight line on the x y- coordinate plane. The x- axis of the planes runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. Figure a shows a straight line going through the points (negative 6, 5), (negative 4, 4), (negative 2, 3), (0, 2), (2, 1), (4, 0), and (6, negative 1). Figure b shows a straight line going through the points (0, negative 6), (1, negative 3), (2, 0), (3, 3), and (4, 6). Figure c shows a straight line going through the points (negative 6, 1), (negative 5, 0), (negative 4, negative 1), (negative 3, negative 2), (negative 2, negative 3), (negative 1, negative 4), (0, negative 5), and (1, negative 6).], alt: "Three figures, each showing a different straight line on the x y- coordinate plane. The x- axis of the planes runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. Figure a shows a straight line going through the points (negative 6, 5), (negative 4, 4), (negative 2, 3), (0, 2), (2, 1), (4, 0), and (6, negative 1). Figure b shows a straight line going through the points (0, negative 6), (1, negative 3), (2, 0), (3, 3), and (4, 6). Figure c shows a straight line going through the points (negative 6, 1), (negative 5, 0), (negative 4, negative 1), (negative 3, negative 2), (negative 2, negative 3), (negative 1, negative 4), (0, negative 5), and (1, negative 6).", caption: none)

#solutionbox[
+ ⓐ The graph crosses the #emph[x]- axis at the point #math.equation(block: false, alt: "open parenthesis 4 , 0 close parenthesis")[$( 4 , 0 )$]. The #emph[x]- intercept is #math.equation(block: false, alt: "open parenthesis 4 , 0 close parenthesis")[$( 4 , 0 )$]. #linebreak() 
The graph crosses the #emph[y]- axis at the point #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$]. The #emph[y]- intercept is #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$]. #linebreak()  #linebreak()
+ ⓑ The graph crosses the #emph[x]- axis at the point #math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$]. The #emph[x]- intercept is #math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$] #linebreak() 
The graph crosses the #emph[y]- axis at the point #math.equation(block: false, alt: "open parenthesis 0 , −6 close parenthesis")[$( 0 , −6 )$]. The #emph[y]- intercept is #math.equation(block: false, alt: "open parenthesis 0 , −6 close parenthesis")[$( 0 , −6 )$]. #linebreak()  #linebreak()
+ ⓒ The graph crosses the #emph[x]- axis at the point #math.equation(block: false, alt: "open parenthesis −5 , 0 close parenthesis")[$( −5 , 0 )$]. The #emph[x]- intercept is #math.equation(block: false, alt: "open parenthesis −5 , 0 close parenthesis")[$( −5 , 0 )$]. #linebreak() 
The graph crosses the #emph[y]- axis at the point #math.equation(block: false, alt: "open parenthesis 0 , −5 close parenthesis")[$( 0 , −5 )$]. The #emph[y]- intercept is #math.equation(block: false, alt: "open parenthesis 0 , −5 close parenthesis")[$( 0 , −5 )$].
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the #emph[x]- and #emph[y]- intercepts on the graph.

#figure(figph[A figure showing a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 8, negative 10), (negative 6, negative 8), (negative 4, negative 6), (negative 2, negative 4), (0, negative 2), (2, 0), (4, 2), (6, 4), (8, 6), and (10, 8).], alt: "A figure showing a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 8, negative 10), (negative 6, negative 8), (negative 4, negative 6), (negative 2, negative 4), (0, negative 2), (2, 0), (4, 2), (6, 4), (8, 6), and (10, 8).", caption: none)

#solutionbox[
#emph[x]- intercept: #math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$]; #emph[y]- intercept: #math.equation(block: false, alt: "open parenthesis 0 , −2 close parenthesis")[$( 0 , −2 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the #emph[x]- and #emph[y]- intercepts on the graph.

#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 9, 8), (negative 6, 6), (negative 3, 4), (0, 2), (3, 0), (6, negative 2), and (9, negative 4).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 9, 8), (negative 6, 6), (negative 3, 4), (0, 2), (3, 0), (6, negative 2), and (9, negative 4).", caption: none)

#solutionbox[
#emph[x]- intercept: #math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis")[$( 3 , 0 )$], #emph[y]- intercept: #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$]
]
]


=== Find the #emph[x]- and #emph[y]- Intercepts from an Equation of a Line

Recognizing that the #strong[#emph[x]- intercept] occurs when #emph[y] is zero and that the #emph[y]- intercept occurs when #emph[x] is zero, gives us a method to find the intercepts of a line from its equation. To find the #emph[x]- intercept, let #math.equation(block: false, alt: "y equals 0")[$y = 0$] and solve for #emph[x]. To find the #strong[#emph[y]- intercept], let #math.equation(block: false, alt: "x equals 0")[$x = 0$] and solve for #emph[y].

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Find the #emph[x]- and #emph[y]- Intercepts from the Equation of a Line]

Use the equation of the line. To find:

- the #emph[x]- intercept of the line, let #math.equation(block: false, alt: "y equals 0")[$y = 0$] and solve for #math.equation(block: false, alt: "x")[$x$].
- the #emph[y]- intercept of the line, let #math.equation(block: false, alt: "x equals 0")[$x = 0$] and solve for #math.equation(block: false, alt: "y")[$y$].
]

#examplebox("Example 2")[][
Find the intercepts of #math.equation(block: false, alt: "2 x plus y equals 6")[$2 x + y = 6$].

#solutionbox[
We will let #math.equation(block: false, alt: "y equals 0")[$y = 0$] to find the #emph[x]- intercept, and let #math.equation(block: false, alt: "x equals 0")[$x = 0$] to find the #emph[y]- intercept. We will fill in the table, which reminds us of what we need to find.

#figure(figph[The figure shows a table with four rows and two columns. The first row is a title row and it labels the table with the equation 2 x plus y equals 6. The second row is a header row and it labels each column. The first column header is “x” and the second is "y". The third row is labeled “x- intercept” and has the first column blank and a 0 in the second column. The fourth row is labeled “y- intercept” and has a 0 in the first column with the second column blank.], alt: "The figure shows a table with four rows and two columns. The first row is a title row and it labels the table with the equation 2 x plus y equals 6. The second row is a header row and it labels each column. The first column header is “x” and the second is \"y\". The third row is labeled “x- intercept” and has the first column blank and a 0 in the second column. The fourth row is labeled “y- intercept” and has a 0 in the first column with the second column blank.", caption: none)

To find the #emph[x]- intercept, let #math.equation(block: false, alt: "y equals 0")[$y = 0$].

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The image displays the linear equation 2x + y = 6 in black text on a white background.], alt: "The image displays the linear equation 2x + y = 6 in black text on a white background.", caption: none)]),
  [Let #emph[y] = 0.], [#figure(figph[The mathematical equation '2x + 0 = 6' is displayed in black text on a white background, with the number '0' highlighted in red.], alt: "The mathematical equation '2x + 0 = 6' is displayed in black text on a white background, with the number '0' highlighted in red.", caption: none)],
  [Simplify.], [#figure(figph[The image shows the mathematical equation 2x = 6, which is an algebraic expression involving a variable x and constant numbers, set up as a simple linear equation to be solved.], alt: "The image shows the mathematical equation 2x = 6, which is an algebraic expression involving a variable x and constant numbers, set up as a simple linear equation to be solved.", caption: none)],
  [], [#figure(figph[The image displays the simple algebraic equation 'x = 3' in black text on a white background.], alt: "The image displays the simple algebraic equation 'x = 3' in black text on a white background.", caption: none)],
  [The #emph[x]-intercept is], [(3, 0)],
  [To find the #emph[y]-intercept, let #emph[x] = 0.], [],
  [], [#figure(figph[A mathematical equation is displayed on a white background: 2x + y = 6.], alt: "A mathematical equation is displayed on a white background: 2x + y = 6.", caption: none)],
  [Let #emph[x] = 0.], [#figure(figph[A mathematical equation shows '2 multiplied by 0 plus y equals 6'. The '0' in the equation is highlighted in red, indicating a substitution or a specific point of interest in the problem.], alt: "A mathematical equation shows '2 multiplied by 0 plus y equals 6'. The '0' in the equation is highlighted in red, indicating a substitution or a specific point of interest in the problem.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation is displayed on a white background, reading '0 + y = 6'.], alt: "A mathematical equation is displayed on a white background, reading '0 + y = 6'.", caption: none)],
  [], [#figure(figph[The image displays the equation y = 6 in black text against a white background.], alt: "The image displays the equation y = 6 in black text against a white background.", caption: none)],
  [The #emph[y]-intercept is], [(0, 6)],
))

The intercepts are the points #math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis")[$( 3 , 0 )$] and #math.equation(block: false, alt: "open parenthesis 0 , 6 close parenthesis")[$( 0 , 6 )$] as shown.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  [#strong[#math.equation(block: false, alt: "2 x plus y equals 6")[$2 x + y = 6$]]], [],
  [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]],
  [3], [0],
  [0], [6],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the intercepts of #math.equation(block: false, alt: "3 x plus y equals 12 .")[$3 x + y = 12 .$]

#solutionbox[
#emph[x]- intercept: #math.equation(block: false, alt: "open parenthesis 4 , 0 close parenthesis")[$( 4 , 0 )$], #emph[y]- intercept: #math.equation(block: false, alt: "open parenthesis 0 , 12 close parenthesis")[$( 0 , 12 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the intercepts of #math.equation(block: false, alt: "x plus 4 y equals 8 .")[$x + 4 y = 8 .$]

#solutionbox[
#emph[x]- intercept: #math.equation(block: false, alt: "open parenthesis 8 , 0 close parenthesis")[$( 8 , 0 )$], #emph[y]- intercept: #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$]
]
]

#examplebox("Example 3")[][
Find the intercepts of #math.equation(block: false, alt: "4 x minus 3 y equals 12")[$4 x − 3 y = 12$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([To find the #emph[x]-intercept, let #emph[y] = 0.], []),
  [], [#figure(figph[The equation 4x - 3y = 12 is shown on a white background.], alt: "The equation 4x - 3y = 12 is shown on a white background.", caption: none)],
  [Let #emph[y] = 0.], [#figure(figph[The mathematical equation 4x - 3 \* 0 = 12 is displayed, with the number 0 highlighted in red.], alt: "The mathematical equation 4x - 3 * 0 = 12 is displayed, with the number 0 highlighted in red.", caption: none)],
  [Simplify.], [#figure(figph[The image displays a mathematical equation '4x - 0 = 12' presented in a clear, dark font against a plain white background.], alt: "The image displays a mathematical equation '4x - 0 = 12' presented in a clear, dark font against a plain white background.", caption: none)],
  [], [#figure(figph[A mathematical equation, '4x = 12', displayed in bold, dark grey text on a white background.], alt: "A mathematical equation, '4x = 12', displayed in bold, dark grey text on a white background.", caption: none)],
  [], [#figure(figph[The image shows the mathematical equation 'x = 3' in black text against a white background.], alt: "The image shows the mathematical equation 'x = 3' in black text against a white background.", caption: none)],
  [The #emph[x]-intercept is], [(3, 0)],
  [To find the #emph[y]-intercept, let #emph[x] = 0.], [],
  [], [#figure(figph[A mathematical equation is displayed on a white background, which reads 4x - 3y = 12. The text is black and rendered in a standard font for mathematical notation.], alt: "A mathematical equation is displayed on a white background, which reads 4x - 3y = 12. The text is black and rendered in a standard font for mathematical notation.", caption: none)],
  [Let #emph[x] = 0.], [#figure(figph[An algebraic equation is shown: '4 \* 0 - 3y = 12', with the number 0 highlighted in red, indicating a step in solving for 'y' when 'x' is 0.], alt: "An algebraic equation is shown: '4 * 0 - 3y = 12', with the number 0 highlighted in red, indicating a step in solving for 'y' when 'x' is 0.", caption: none)],
  [Simplify.], [#figure(figph[The image displays the linear equation '0 - 3y = 12' in black text on a white background.], alt: "The image displays the linear equation '0 - 3y = 12' in black text on a white background.", caption: none)],
  [], [#figure(figph[The image shows a mathematical equation, '-3y = 12', displayed in a clear, digital font against a plain white background. The equation involves a variable 'y' and integers.], alt: "The image shows a mathematical equation, '-3y = 12', displayed in a clear, digital font against a plain white background. The equation involves a variable 'y' and integers.", caption: none)],
  [], [#figure(figph[The image displays the mathematical equation 'y = -4' centered against a white background.], alt: "The image displays the mathematical equation 'y = -4' centered against a white background.", caption: none)],
  [The #emph[y]-intercept is], [(0, −4)],
))

The intercepts are the points (3, 0) and (0, −4) as shown in the following table.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  [#strong[#math.equation(block: false, alt: "4 x minus 3 y equals 12")[$4 x − 3 y = 12$]]], [],
  [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]],
  [3], [0],
  [0], [#math.equation(block: false, alt: "−4")[$−4$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the intercepts of #math.equation(block: false, alt: "3 x minus 4 y equals 12 .")[$3 x − 4 y = 12 .$]

#solutionbox[
#emph[x]- intercept: #math.equation(block: false, alt: "open parenthesis 4 , 0 close parenthesis")[$( 4 , 0 )$], #emph[y]- intercept: #math.equation(block: false, alt: "open parenthesis 0 , −3 close parenthesis")[$( 0 , −3 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the intercepts of #math.equation(block: false, alt: "2 x minus 4 y equals 8 .")[$2 x − 4 y = 8 .$]

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


=== Graph a Line Using the Intercepts

To graph a linear equation by plotting points, you need to find three points whose coordinates are solutions to the equation. You can use the #emph[x]- and #emph[y]- intercepts as two of your three points. Find the intercepts, and then find a third point to ensure accuracy. Make sure the points line up—then draw the line. This method is often the quickest way to graph a line.

#examplebox("Example 4")[How to Graph a Line Using Intercepts][
Graph #math.equation(block: false, alt: "minus x plus 2 y equals 6")[$− x + 2 y = 6$] using the intercepts.

#solutionbox[
#figure(figph[The figure shows a table with the general procedure for graphing a line using the intercepts along with a specific example using the equation negative x plus 2y equals 6. Step 1 of the general procedure is “Find the x and y- intercepts of the line. Let y equals 0 and solve for x. Let x equals 0 and solve for y”. Step 1 for the example is a series of statements and equations: “Find the x- intercept. Let y equals 0”, negative x plus 2y equals 6, negative x plus 2(0) equals 6 (where the 0 is red), negative x equals 6, x equals negative 6, “The x- intercept is (negative 6, 0)”, “Find the y- intercept. Let x equals 0”, negative x plus 2y equals 6, negative 0 plus 2y equals 6 (where the 0 is red), 2y equals 6, y equals 3, and “The y- intercept is (0, 3)”.], alt: "The figure shows a table with the general procedure for graphing a line using the intercepts along with a specific example using the equation negative x plus 2y equals 6. Step 1 of the general procedure is “Find the x and y- intercepts of the line. Let y equals 0 and solve for x. Let x equals 0 and solve for y”. Step 1 for the example is a series of statements and equations: “Find the x- intercept. Let y equals 0”, negative x plus 2y equals 6, negative x plus 2(0) equals 6 (where the 0 is red), negative x equals 6, x equals negative 6, “The x- intercept is (negative 6, 0)”, “Find the y- intercept. Let x equals 0”, negative x plus 2y equals 6, negative 0 plus 2y equals 6 (where the 0 is red), 2y equals 6, y equals 3, and “The y- intercept is (0, 3)”.", caption: none)

#figure(figph[Step 2 of the general procedure is “Find another solution to the equation.” Step 2 for the example is a series of statements and equations: “We’ll use x equals 2”, “Let x equals 2”, negative x plus 2y equals 6, negative 2 plus 2y equals 6 (where the first 2 is red), 2y equals 8, y equals 4, and “A third point is (2, 4)”. Step 3 of the general procedure is “Plot the three points. Check that the points line up.”], alt: "Step 2 of the general procedure is “Find another solution to the equation.” Step 2 for the example is a series of statements and equations: “We’ll use x equals 2”, “Let x equals 2”, negative x plus 2y equals 6, negative 2 plus 2y equals 6 (where the first 2 is red), 2y equals 8, y equals 4, and “A third point is (2, 4)”. Step 3 of the general procedure is “Plot the three points. Check that the points line up.”", caption: none)

#figure(figph[Step 3 for the example is a table and a graph. The table has four rows and three columns. The first row is a header row and it labels each column. The first column header is “x”, the second is "y", and the third is “(x,y)”. Under the first column are the numbers negative 6, 0 and 2. Under the second column are the numbers 0, 3, and 4. Under the third column are the ordered pairs (negative 6, 0), (0, 3), and (2, 4). The graph has three points on the x- y coordinate plane. The x- axis of the plane runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. Three points are marked at (negative 6, 0), (0, 3), and (2, 4).], alt: "Step 3 for the example is a table and a graph. The table has four rows and three columns. The first row is a header row and it labels each column. The first column header is “x”, the second is \"y\", and the third is “(x,y)”. Under the first column are the numbers negative 6, 0 and 2. Under the second column are the numbers 0, 3, and 4. Under the third column are the ordered pairs (negative 6, 0), (0, 3), and (2, 4). The graph has three points on the x- y coordinate plane. The x- axis of the plane runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. Three points are marked at (negative 6, 0), (0, 3), and (2, 4).", caption: none)

#figure(figph[Step 4 of the general procedure is “Draw the line.” For the specific example, there is the statement “See the graph” and a graph of a straight line going through three points on the x y- coordinate plane. The x- axis of the plane runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. Three points are marked at (negative 6, 0), (0, 3), and (2, 4). The straight line is drawn through the points (negative 6, 0), (negative 4, 1), (negative 2, 2), (0, 3), (2, 4), (4, 5), and (6, 6).], alt: "Step 4 of the general procedure is “Draw the line.” For the specific example, there is the statement “See the graph” and a graph of a straight line going through three points on the x y- coordinate plane. The x- axis of the plane runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. Three points are marked at (negative 6, 0), (0, 3), and (2, 4). The straight line is drawn through the points (negative 6, 0), (negative 4, 1), (negative 2, 2), (0, 3), (2, 4), (4, 5), and (6, 6).", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Graph #math.equation(block: false, alt: "x minus 2 y equals 4")[$x − 2 y = 4$] using the intercepts.

#solutionbox[
#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 10, negative 7), (negative 8, negative 6), (negative 6, negative 5), (negative 4, negative 4), (negative 2, negative 3), (0, negative 2), (2, negative 1), (4, 0), (6, 1), (8, 2), and (10, 3).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 10, negative 7), (negative 8, negative 6), (negative 6, negative 5), (negative 4, negative 4), (negative 2, negative 3), (0, negative 2), (2, negative 1), (4, 0), (6, 1), (8, 2), and (10, 3).", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Graph #math.equation(block: false, alt: "minus x plus 3 y equals 6")[$− x + 3 y = 6$] using the intercepts.

#solutionbox[
#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 12, negative 2), (negative 9, negative 1), (negative 6, 0), (negative 3, 1), (0, 2), (3, 3), (6, 4), (9, 5), and (12, 6).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 12, negative 2), (negative 9, negative 1), (negative 6, 0), (negative 3, 1), (0, 2), (3, 3), (6, 4), (9, 5), and (12, 6).", caption: none)
]
]

The steps to graph a linear equation using the intercepts are summarized below.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Graph a linear equation using the intercepts.]

+ Find the #emph[x]- and #emph[y]- intercepts of the line.
- Let #math.equation(block: false, alt: "y equals 0")[$y = 0$] and solve for #math.equation(block: false, alt: "x")[$x$]
- Let #math.equation(block: false, alt: "x equals 0")[$x = 0$] and solve for #math.equation(block: false, alt: "y")[$y$].
+ Find a third solution to the equation.
+ Plot the three points and check that they line up.
+ Draw the line.
]

#examplebox("Example 5")[][
Graph #math.equation(block: false, alt: "4 x minus 3 y equals 12")[$4 x − 3 y = 12$] using the intercepts.

#solutionbox[
Find the intercepts and a third point.

#figure(figph[The figure shows a series of statements and equations: “Find the x- intercept. Let y equals 0”, 4x minus 3y equals 12, 4x minus 3(0) equals 12 (where the 0 is red), 4x equals 12, x equals 3, “Find the y- intercept. Let x equals 0”, 4x minus 3y equals 12, 4(0) minus 3y equals 12 (where the 0 is red), negative 3y equals 12, y equals negative 4, “third point, let y equals 4”, 4x minus 3y equals 12, 4x minus 3(4) equals 12 (where the second 4 is red), 4x minus 12 equals 12, 4x equals 24, and x equals 6.], alt: "The figure shows a series of statements and equations: “Find the x- intercept. Let y equals 0”, 4x minus 3y equals 12, 4x minus 3(0) equals 12 (where the 0 is red), 4x equals 12, x equals 3, “Find the y- intercept. Let x equals 0”, 4x minus 3y equals 12, 4(0) minus 3y equals 12 (where the 0 is red), negative 3y equals 12, y equals negative 4, “third point, let y equals 4”, 4x minus 3y equals 12, 4x minus 3(4) equals 12 (where the second 4 is red), 4x minus 12 equals 12, 4x equals 24, and x equals 6.", caption: none)

We list the points in  and show the graph below.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [#strong[#math.equation(block: false, alt: "4 x minus 3 y equals 12")[$4 x − 3 y = 12$]]], [], [],
  [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]],
  [3], [0], [#math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis")[$( 3 , 0 )$]],
  [0], [#math.equation(block: false, alt: "−4")[$−4$]], [#math.equation(block: false, alt: "open parenthesis 0 , −4 close parenthesis")[$( 0 , −4 )$]],
  [6], [4], [#math.equation(block: false, alt: "open parenthesis 6 , 4 close parenthesis")[$( 6 , 4 )$]],
))

#figure(figph[The figure shows the graph of a straight line going through three points on the x y- coordinate plane. The x- axis of the plane runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. Three points are marked at (0, negative 4), (3, 0), and (6, 4). The straight line is drawn through the points (0, negative 4), (3, 0), and (6, 4).], alt: "The figure shows the graph of a straight line going through three points on the x y- coordinate plane. The x- axis of the plane runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. Three points are marked at (0, negative 4), (3, 0), and (6, 4). The straight line is drawn through the points (0, negative 4), (3, 0), and (6, 4).", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Graph #math.equation(block: false, alt: "5 x minus 2 y equals 10")[$5 x − 2 y = 10$] using the intercepts.

#solutionbox[
#figure(figph[The figure shows the graph of a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. The straight line goes through the points (0, negative 5), (2, 0), and (4, 5).], alt: "The figure shows the graph of a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. The straight line goes through the points (0, negative 5), (2, 0), and (4, 5).", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Graph #math.equation(block: false, alt: "3 x minus 4 y equals 12")[$3 x − 4 y = 12$] using the intercepts.

#solutionbox[
#figure(figph[The figure shows the graph of a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. The straight line goes through the points (negative 4, negative 6), (0, negative 3), and (4, 0).], alt: "The figure shows the graph of a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. The straight line goes through the points (negative 4, negative 6), (0, negative 3), and (4, 0).", caption: none)
]
]

#examplebox("Example 6")[][
Graph #math.equation(block: false, alt: "y equals 5 x")[$y = 5 x$] using the intercepts.

#solutionbox[
#figure(figph[The figure shows two sets of statements and equations to find the intercepts from an equation. The first set of statements and equations is “x- intercept”, “let y equals 0”, y equals 5x, 0 equals 5x (where the 0 is red), 0 equals x, (0, 0). The second set of statements and equations is “y- intercept”, “let x equals 0”, y equals 5x, y equals 5(0) (where the 0 is red), y equals 0, (0, 0).], alt: "The figure shows two sets of statements and equations to find the intercepts from an equation. The first set of statements and equations is “x- intercept”, “let y equals 0”, y equals 5x, 0 equals 5x (where the 0 is red), 0 equals x, (0, 0). The second set of statements and equations is “y- intercept”, “let x equals 0”, y equals 5x, y equals 5(0) (where the 0 is red), y equals 0, (0, 0).", caption: none)

This line has only one intercept. It is the point #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$].

To ensure accuracy we need to plot three points. Since the #emph[x]- and #emph[y]- intercepts are the same point, we need #emph[two] more points to graph the line.

#figure(figph[The figure shows two sets of statements and equations to find two points from an equation. The first set of statements and equations is “Let x equals 1”, y equals 5x, y equals 5(1) (where the 1 is red), y equals 5. The second set of statements and equations is “Let x equals negative 1”, y equals 5x, y equals 5(negative 1) (where the negative 1 is red), y equals negative 5.], alt: "The figure shows two sets of statements and equations to find two points from an equation. The first set of statements and equations is “Let x equals 1”, y equals 5x, y equals 5(1) (where the 1 is red), y equals 5. The second set of statements and equations is “Let x equals negative 1”, y equals 5x, y equals 5(negative 1) (where the negative 1 is red), y equals negative 5.", caption: none)

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [#strong[#math.equation(block: false, alt: "y equals 5 x")[$y = 5 x$]]], [], [],
  [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]],
  [0], [0], [#math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]],
  [1], [5], [#math.equation(block: false, alt: "open parenthesis 1 , 5 close parenthesis")[$( 1 , 5 )$]],
  [#math.equation(block: false, alt: "−1")[$−1$]], [#math.equation(block: false, alt: "−5")[$−5$]], [#math.equation(block: false, alt: "open parenthesis −1 , −5 close parenthesis")[$( −1 , −5 )$]],
))

Plot the three points, check that they line up, and draw the line.

#figure(figph[The figure shows the graph of a straight line going through three points on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. Three points are marked and labeled with their coordinates at (negative 1, negative 5), (0, 0), and (1, 5). The straight line is drawn through the points (negative 1, negative 5), (0, 0), and (1, 5).], alt: "The figure shows the graph of a straight line going through three points on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. Three points are marked and labeled with their coordinates at (negative 1, negative 5), (0, 0), and (1, 5). The straight line is drawn through the points (negative 1, negative 5), (0, 0), and (1, 5).", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Graph #math.equation(block: false, alt: "y equals 4 x")[$y = 4 x$] using the intercepts.

#solutionbox[
#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 4, negative 12), (negative 3, negative 9), (negative 2, negative 6), (negative 1, negative 3), (0, 0), (1, 3), (2, 6), (3, 9), and (4, 12).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 4, negative 12), (negative 3, negative 9), (negative 2, negative 6), (negative 1, negative 3), (0, 0), (1, 3), (2, 6), (3, 9), and (4, 12).", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Graph #math.equation(block: false, alt: "y equals − x")[$y = "−" x$] the intercepts.

#solutionbox[
#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 10, 10), (negative 9, 9), (negative 8, 8), (negative 7, 7), (negative 6, 6), (negative 5, 5), (negative 4, 4), (negative 3, 3), (negative 2, 2), (negative 1, 1), (0, 0), (1, negative 1), (2, negative 2), (3, negative 3), (4, negative 4), (5, negative 5), (6, negative 6), (7, negative 7), (8, negative 8), (9, negative 9), and (10, negative 10).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 10, 10), (negative 9, 9), (negative 8, 8), (negative 7, 7), (negative 6, 6), (negative 5, 5), (negative 4, 4), (negative 3, 3), (negative 2, 2), (negative 1, 1), (0, 0), (1, negative 1), (2, negative 2), (3, negative 3), (4, negative 4), (5, negative 5), (6, negative 6), (7, negative 7), (8, negative 8), (9, negative 9), and (10, negative 10).", caption: none)
]
]


=== Key Concepts

- #strong[Find the #emph[x]- and #emph[y]- Intercepts from the Equation of a Line]
- Use the equation of the line to find the #emph[x]- intercept of the line, let #math.equation(block: false, alt: "y equals 0")[$y = 0$] and solve for #emph[x].
- Use the equation of the line to find the #emph[y]- intercept of the line, let #math.equation(block: false, alt: "x equals 0")[$x = 0$] and solve for #emph[y].
- #strong[Graph a Linear Equation using the Intercepts]
+ Find the #emph[x]- and #emph[y]- intercepts of the line. #linebreak() 
Let #math.equation(block: false, alt: "y equals 0")[$y = 0$] and solve for #emph[x]. #linebreak() 
Let #math.equation(block: false, alt: "x equals 0")[$x = 0$] and solve for #emph[y].
+ Find a third solution to the equation.
+ Plot the three points and then check that they line up.
+ Draw the 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 #emph[y] is isolated on one side of the equation, graph by plotting points.
- Choose any three values for #emph[x] and then solve for the corresponding #emph[y]- values.
- 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 and then a third point.


==== Practice Makes Perfect

#strong[Identify the #emph[x]- and #emph[y]- Intercepts on a Graph]

In the following exercises, find the #emph[x]- and #emph[y]- intercepts on each graph.

#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 5, 8), (negative 4, 7), (negative 3, 6), (negative 2, 5), (negative 1, 4), (0, 3), (1, 2), (2, 1), (3, 0), (4, negative 1), (5, negative 2) and (6, negative 3).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 5, 8), (negative 4, 7), (negative 3, 6), (negative 2, 5), (negative 1, 4), (0, 3), (1, 2), (2, 1), (3, 0), (4, negative 1), (5, negative 2) and (6, negative 3).", caption: none)

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 3 , 0 close parenthesis , open parenthesis 0 , 3 close parenthesis")[$( 3 , 0 ) , ( 0 , 3 )$]
]

#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 6, 8), (negative 5, 7), (negative 4, 6), (negative 3, 5), (negative 2, 4), (negative 1, 3), (0, 2), (1, 1), (2, 0), (3, negative 1), (4, negative 2), (5, negative 3) and (6, negative 4).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 6, 8), (negative 5, 7), (negative 4, 6), (negative 3, 5), (negative 2, 4), (negative 1, 3), (0, 2), (1, 1), (2, 0), (3, negative 1), (4, negative 2), (5, negative 3) and (6, negative 4).", caption: none)

#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 5, negative 10), (negative 4, negative 9), (negative 3, negative 8), (negative 2, negative 7), (negative 1, negative 6), (0, negative 5), (1, negative 4), (2, negative 3), (3, negative 2), (4, negative 1), (5, 0), (6, 1), (7, 2), and (8, 3).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 5, negative 10), (negative 4, negative 9), (negative 3, negative 8), (negative 2, negative 7), (negative 1, negative 6), (0, negative 5), (1, negative 4), (2, negative 3), (3, negative 2), (4, negative 1), (5, 0), (6, 1), (7, 2), and (8, 3).", caption: none)

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 5 , 0 close parenthesis , open parenthesis 0 , −5 close parenthesis")[$( 5 , 0 ) , ( 0 , −5 )$]
]

#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 6, negative 7), (negative 5, negative 6), (negative 4, negative 5), (negative 3, negative 4), (negative 2, negative 3), (negative 1, negative 2), (0, negative 1), (1, 0), (2, 1), (3, 2), (4, 3), (5, 4), (6, 5), (7, 6), and (8, 7).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 6, negative 7), (negative 5, negative 6), (negative 4, negative 5), (negative 3, negative 4), (negative 2, negative 3), (negative 1, negative 2), (0, negative 1), (1, 0), (2, 1), (3, 2), (4, 3), (5, 4), (6, 5), (7, 6), and (8, 7).", caption: none)

#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 6, negative 7), (negative 5, negative 6), (negative 4, negative 5), (negative 3, negative 4), (negative 2, negative 3), (negative 1, negative 2), (0, negative 1), (1, 0), (2, 1), (3, 2), (4, 3), (5, 4), (6, 5), (7, 6), and (8, 7).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 6, negative 7), (negative 5, negative 6), (negative 4, negative 5), (negative 3, negative 4), (negative 2, negative 3), (negative 1, negative 2), (0, negative 1), (1, 0), (2, 1), (3, 2), (4, 3), (5, 4), (6, 5), (7, 6), and (8, 7).", caption: none)

#solutionbox[
#math.equation(block: true, alt: "open parenthesis −2 , 0 close parenthesis , open parenthesis 0 , −2 close parenthesis")[$( −2 , 0 ) , ( 0 , −2 )$]
]

#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 6, 3), (negative 5, 2), (negative 4, 1), (negative 3, 0), (negative 2, negative 1), (negative 1, negative 2), (0, negative 3), (1, negative 4), (2, negative 5), (3, negative 6), (4, negative 7), (5, negative 8), and (6, negative 9).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 6, 3), (negative 5, 2), (negative 4, 1), (negative 3, 0), (negative 2, negative 1), (negative 1, negative 2), (0, negative 3), (1, negative 4), (2, negative 5), (3, negative 6), (4, negative 7), (5, negative 8), and (6, negative 9).", caption: none)

#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 6, negative 5), (negative 5, negative 4), (negative 4, negative 3), (negative 3, negative 2), (negative 2, negative 1), (negative 1, 0), (0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 8), and (8, 9).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 6, negative 5), (negative 5, negative 4), (negative 4, negative 3), (negative 3, negative 2), (negative 2, negative 1), (negative 1, 0), (0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 8), and (8, 9).", caption: none)

#solutionbox[
#math.equation(block: true, alt: "open parenthesis −1 , 0 close parenthesis , open parenthesis 0 , 1 close parenthesis")[$( −1 , 0 ) , ( 0 , 1 )$]
]

#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 8, negative 3), (negative 7, negative 2), (negative 6, negative 1), (negative 5, 0), (negative 4, 1), (negative 3, 2), (negative 2, 3), (negative 1, 4), (0, 5), (1, 6), (2, 7), (3, 8), (4, 9), and (5, 10).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 8, negative 3), (negative 7, negative 2), (negative 6, negative 1), (negative 5, 0), (negative 4, 1), (negative 3, 2), (negative 2, 3), (negative 1, 4), (0, 5), (1, 6), (2, 7), (3, 8), (4, 9), and (5, 10).", caption: none)

#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 10, 8), (negative 8, 7), (negative 6, 6), (negative 4, 5), (negative 2, 4), (0, 3), (2, 2), (4, 1), (6, 0), (8, negative 1), and (10, negative 2).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 10, 8), (negative 8, 7), (negative 6, 6), (negative 4, 5), (negative 2, 4), (0, 3), (2, 2), (4, 1), (6, 0), (8, negative 1), and (10, negative 2).", caption: none)

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 6 , 0 close parenthesis , open parenthesis 0 , 3 close parenthesis")[$( 6 , 0 ) , ( 0 , 3 )$]
]

#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. The straight line goes through the points (negative 6, 5), (negative 4, 4), (negative 2, 3), (0, 2), (2, 1), (4, 0), and (6, negative 1).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. The straight line goes through the points (negative 6, 5), (negative 4, 4), (negative 2, 3), (0, 2), (2, 1), (4, 0), and (6, negative 1).", caption: none)

#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the plotted point (0, 0).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the plotted point (0, 0).", caption: none)

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

#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. The straight line goes through the plotted point (0, 0).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. The straight line goes through the plotted point (0, 0).", caption: none)

#strong[Find the #emph[x]- and #emph[y]- Intercepts from an Equation of a Line]

In the following exercises, find the intercepts for each equation.

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

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

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

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

#solutionbox[
#math.equation(block: true, alt: "open parenthesis −2 , 0 close parenthesis , open parenthesis 0 , −2 close parenthesis")[$( −2 , 0 ) , ( 0 , −2 )$]
]

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

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

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 5 , 0 close parenthesis , open parenthesis 0 , −5 close parenthesis")[$( 5 , 0 ) , ( 0 , −5 )$]
]

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

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

#solutionbox[
#math.equation(block: true, alt: "open parenthesis −3 , 0 close parenthesis , open parenthesis 0 , 3 close parenthesis")[$( −3 , 0 ) , ( 0 , 3 )$]
]

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

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

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

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

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

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 2 , 0 close parenthesis , open parenthesis 0 , 6 close parenthesis")[$( 2 , 0 ) , ( 0 , 6 )$]
]

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

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

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 12 , 0 close parenthesis , open parenthesis 0 , −4 close parenthesis")[$( 12 , 0 ) , ( 0 , −4 )$]
]

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

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

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 2 , 0 close parenthesis , open parenthesis 0 , −8 close parenthesis")[$( 2 , 0 ) , ( 0 , −8 )$]
]

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

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

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 5 , 0 close parenthesis , open parenthesis 0 , 2 close parenthesis")[$( 5 , 0 ) , ( 0 , 2 )$]
]

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

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

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 4 , 0 close parenthesis , open parenthesis 0 , −6 close parenthesis")[$( 4 , 0 ) , ( 0 , −6 )$]
]

#math.equation(block: true, alt: "3 x minus 5 y equals 30")[$3 x − 5 y = 30$]

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

#solutionbox[
#math.equation(block: true, alt: "open parenthesis −3 , 0 close parenthesis , open parenthesis 0 , 1 close parenthesis")[$( −3 , 0 ) , ( 0 , 1 )$]
]

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

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

#solutionbox[
#math.equation(block: true, alt: "open parenthesis −10 , 0 close parenthesis , open parenthesis 0 , 2 close parenthesis")[$( −10 , 0 ) , ( 0 , 2 )$]
]

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

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

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

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

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

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

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

#strong[Graph a Line Using the Intercepts]

In the following exercises, graph using the intercepts.

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

#solutionbox[
#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The line graphed is negative x plus 5 y equals 10.], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The line graphed is negative x plus 5 y equals 10.", caption: none)
]

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

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

#solutionbox[
#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 8, 6), (negative 6, 5), (negative 4, 4), (negative 2, 3), (0, 2), (2, 1), (4, 0), (6, negative 1), (8, negative 2), and (10, negative 3).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 8, 6), (negative 6, 5), (negative 4, 4), (negative 2, 3), (0, 2), (2, 1), (4, 0), (6, negative 1), (8, negative 2), and (10, negative 3).", caption: none)
]

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

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

#solutionbox[
#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 8, 10), (negative 7, 9), (negative 6, 8),(negative 5, 7), (negative 4, 6), (negative 3, 5), (negative 2, 4), (negative 1, 3), (0, 2), (1, 1), (2, 0), (3, negative 1), (4, negative 2), (5, negative 3), (6, negative 4), (7, negative 5), (8, negative 6), (9, negative 7), and (10, negative 8).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 8, 10), (negative 7, 9), (negative 6, 8),(negative 5, 7), (negative 4, 6), (negative 3, 5), (negative 2, 4), (negative 1, 3), (0, 2), (1, 1), (2, 0), (3, negative 1), (4, negative 2), (5, negative 3), (6, negative 4), (7, negative 5), (8, negative 6), (9, negative 7), and (10, negative 8).", caption: none)
]

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

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

#solutionbox[
#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. The straight line goes through the points (negative 7, 4), (negative 6, 3), (negative 5, 2),(negative 4, 1), (negative 3, 0), (negative 2, negative 1), (negative 1, negative 2), (0, negative 3), (1, negative 4), (2, negative 5), (3, negative 6), and (4, negative 7).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. The straight line goes through the points (negative 7, 4), (negative 6, 3), (negative 5, 2),(negative 4, 1), (negative 3, 0), (negative 2, negative 1), (negative 1, negative 2), (0, negative 3), (1, negative 4), (2, negative 5), (3, negative 6), and (4, negative 7).", caption: none)
]

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

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

#solutionbox[
#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 8, negative 9), (negative 7, negative 8), (negative 6, negative 7),(negative 5, negative 6), (negative 4, negative 5), (negative 3, negative 4), (negative 2, negative 3), (negative 1, negative 2), (0, negative 1), (1, 0), (2, 1), (3, 2), (4, 3), (5, 4), (6, 5), (7, 6), (8, 7), (9, 8), and (10, 9).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 8, negative 9), (negative 7, negative 8), (negative 6, negative 7),(negative 5, negative 6), (negative 4, negative 5), (negative 3, negative 4), (negative 2, negative 3), (negative 1, negative 2), (0, negative 1), (1, 0), (2, 1), (3, 2), (4, 3), (5, 4), (6, 5), (7, 6), (8, 7), (9, 8), and (10, 9).", caption: none)
]

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

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

#solutionbox[
#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 8, negative 4), (negative 7, negative 3), (negative 6, negative 2),(negative 5, negative 1), (negative 4, 0), (negative 3, 1), (negative 2, 2), (negative 1, 3), (0, 4), (1, 5), (2, 6), (3, 7), (4, 8), (5, 9), (6, 10), (7, 11), and (8, 12).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 8, negative 4), (negative 7, negative 3), (negative 6, negative 2),(negative 5, negative 1), (negative 4, 0), (negative 3, 1), (negative 2, 2), (negative 1, 3), (0, 4), (1, 5), (2, 6), (3, 7), (4, 8), (5, 9), (6, 10), (7, 11), and (8, 12).", caption: none)
]

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

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

#solutionbox[
#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 2, 12), (negative 1, 8), (0, 4), (1, 0), (2, negative 4), (3, negative 8), and (4, negative 12).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 2, 12), (negative 1, 8), (0, 4), (1, 0), (2, negative 4), (3, negative 8), and (4, negative 12).", caption: none)
]

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

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

#solutionbox[
#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. The straight line goes through the points (negative 6, 6), (negative 4, 5), (negative 2, 4), (0, 3), (2, 2), (4, 1), and (6, 0).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. The straight line goes through the points (negative 6, 6), (negative 4, 5), (negative 2, 4), (0, 3), (2, 2), (4, 1), and (6, 0).", caption: none)
]

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

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

#solutionbox[
#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 6, negative 12), (negative 4, negative 9), (negative 2, negative 6), (0, negative 3), (2, 0), (4, 3), (6, 6), (8, 9), and (10, 12).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 6, negative 12), (negative 4, negative 9), (negative 2, negative 6), (0, negative 3), (2, 0), (4, 3), (6, 6), (8, 9), and (10, 12).", caption: none)
]

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

#math.equation(block: true, alt: "2 x minus 5 y equals −20")[$2 x − 5 y = −20$]

#solutionbox[
#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 10, 0), (negative 5, 2), (0, 4), (5, 6), and (10, 8).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 10, 0), (negative 5, 2), (0, 4), (5, 6), and (10, 8).", caption: none)
]

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

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

#solutionbox[
#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 6, negative 12), (negative 5, negative 9), (negative 4, negative 6), (negative 3, negative 3), (negative 2, 0), (1, 3), (2, 6), (3, 9), and (4, 12).], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 6, negative 12), (negative 5, negative 9), (negative 4, negative 6), (negative 3, negative 3), (negative 2, 0), (1, 3), (2, 6), (3, 9), and (4, 12).", caption: none)
]

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

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

#solutionbox[
#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 5, 10), (negative 4, 8), (negative 3, 6), (negative 2, 4), (negative 1, 2), (0, 0), (1, negative 2), (2, negative 4), (3, negative 6), (4, negative 8), (5, negative 10), and (6, negative 12)], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 5, 10), (negative 4, 8), (negative 3, 6), (negative 2, 4), (negative 1, 2), (0, 0), (1, negative 2), (2, negative 4), (3, negative 6), (4, negative 8), (5, negative 10), and (6, negative 12)", caption: none)
]

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

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

#solutionbox[
#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 10, 10), (negative 9, 9), (negative 8, 8), (negative 7, 7), (negative 6, 6), (negative 5, 5), (negative 4, 4), (negative 3, 3), (negative 2, 2), (negative 1, 1), (0, 0), (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6), (7, 7), (8, 8), (9, 9), and (10, 10)], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 12 to 12. The y- axis of the planes runs from negative 12 to 12. The straight line goes through the points (negative 10, 10), (negative 9, 9), (negative 8, 8), (negative 7, 7), (negative 6, 6), (negative 5, 5), (negative 4, 4), (negative 3, 3), (negative 2, 2), (negative 1, 1), (0, 0), (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6), (7, 7), (8, 8), (9, 9), and (10, 10)", caption: none)
]

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


==== Everyday Math

#strong[Road trip.] Damien is driving from Chicago to Denver, a distance of 1000 miles. The #emph[x]- axis on the graph below shows the time in hours since Damien left Chicago. The #emph[y]- axis represents the distance he has left to drive.

#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from 0 to 16. The y- axis of the planes runs from 0 to 1200 in increments of 200. The straight line goes through the points (0, 1000), (3, 800), (6, 600), (9, 400), (12, 200), and (15, 0). The points (0, 1000) and (15, 0) are marked and labeled with their coordinates.], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from 0 to 16. The y- axis of the planes runs from 0 to 1200 in increments of 200. The straight line goes through the points (0, 1000), (3, 800), (6, 600), (9, 400), (12, 200), and (15, 0). The points (0, 1000) and (15, 0) are marked and labeled with their coordinates.", caption: none)

+ ⓐ Find the #emph[x]- and #emph[y]- intercepts.
+ ⓑ Explain what the #emph[x]- and #emph[y]- intercepts mean for Damien.

#solutionbox[
ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 1000 close parenthesis , open parenthesis 15 , 0 close parenthesis")[$( 0 , 1000 ) , ( 15 , 0 )$] #linebreak() 
ⓑ At #math.equation(block: false, alt: "open parenthesis 0 , 1000 close parenthesis")[$( 0 , 1000 )$], he has been gone 0 hours and has 1000 miles left. At #math.equation(block: false, alt: "open parenthesis 15 , 0 close parenthesis")[$( 15 , 0 )$], he has been gone 15 hours and has 0 miles left to go.
]

#strong[Road trip.] Ozzie filled up the gas tank of his truck and headed out on a road trip. The #emph[x]- axis on the graph below shows the number of miles Ozzie drove since filling up. The #emph[y]- axis represents the number of gallons of gas in the truck’s gas tank.

#figure(figph[The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from 0 to 350 in increments of 50. The y- axis of the planes runs from 0 to 18 in increments of 2. The straight line goes through the points (0, 16), (150, 8), and (300, 0). The points (0, 16) and (300, 0) are marked and labeled with their coordinates], alt: "The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from 0 to 350 in increments of 50. The y- axis of the planes runs from 0 to 18 in increments of 2. The straight line goes through the points (0, 16), (150, 8), and (300, 0). The points (0, 16) and (300, 0) are marked and labeled with their coordinates", caption: none)

+ ⓐ Find the #emph[x]- and #emph[y]- intercepts.
+ ⓑ Explain what the #emph[x]- and #emph[y]- intercepts mean for Ozzie.


==== Writing Exercises

How do you find the #emph[x]- intercept of the graph of #math.equation(block: false, alt: "3 x minus 2 y equals 6")[$3 x − 2 y = 6$]?

#solutionbox[
Answers will vary.
]

Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation #math.equation(block: false, alt: "4 x plus y equals −4")[$4 x + y = −4$]? Why?

Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation #math.equation(block: false, alt: "y equals the fraction 2 over 3 x minus 2")[$y = frac(2, 3) x − 2$]? Why?

#solutionbox[
Answers will vary.
]

Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation #math.equation(block: false, alt: "y equals 6")[$y = 6$]? Why?


==== Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

#figure(figph[The figure shows a table with four rows and four columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is "confidently", the third is “with some help”, “no minus I don’t get it!”. Under the first column are the phrases “identify the x and y intercepts of a graph”, “find the x and y intercepts from an equation of a line”, and “graph a line using intercepts”. Under the second, third, fourth columns are blank spaces where the learner can check what level of mastery they have achieved.], alt: "The figure shows a table with four rows and four columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is \"confidently\", the third is “with some help”, “no minus I don’t get it!”. Under the first column are the phrases “identify the x and y intercepts of a graph”, “find the x and y intercepts from an equation of a line”, and “graph a line using intercepts”. Under the second, third, fourth columns are blank spaces where the learner can check what level of mastery they have achieved.", caption: none)

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?