#set document(title: "11.4 Graphing with Intercepts", author: "OpenStax / XYZ Homework")
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== 11.4#h(0.6em)Graphing 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 x plus 4 y equals −12")[$3 x + 4 y = −12$] for #math.equation(block: false, alt: "x")[$x$] when #math.equation(block: false, alt: "y equals 0 .")[$y = 0 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "minus 4")[$− 4$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Is the point #math.equation(block: false, alt: "open parenthesis 0 , −5 close parenthesis")[$( 0 , −5 )$] on the #math.equation(block: false, alt: "x -axis")[$x "-axis"$] or #math.equation(block: false, alt: "y -axis?")[$y "-axis?"$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "y -axis")[$y "-axis"$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Which ordered pairs are solutions to the equation #math.equation(block: false, alt: "2 x minus y equals 6 ?")[$2 x − y = 6 ?$] #linebreak() ⓐ #math.equation(block: false, alt: "open parenthesis 6 , 0 close parenthesis")[$#h(0.2em) ( 6 , 0 ) #h(0.2em)$]ⓑ #math.equation(block: false, alt: "open parenthesis 0 , −6 close parenthesis")[$#h(0.2em) #h(0.2em) ( 0 , −6 ) #h(0.2em)$]ⓒ #math.equation(block: false, alt: "open parenthesis 4 , −2 close parenthesis .")[$#h(0.2em) ( 4 , −2 ) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
b
]
]


=== Identify the Intercepts on a Graph

Every #strong[linear equation] has a unique line that represents all the solutions of the equation. When graphing a line by plotting points, each person who graphs the line can choose any three points, so two people graphing the line might use different sets of points.

At first glance, their two lines might appear different since they would have different points labeled. But if all the work was done correctly, the lines will be exactly the same line. One way to recognize that they are indeed the same line is to focus on where the line crosses the axes. Each of these points is called an #strong[intercept of the line].

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

Each of the points at which a line crosses the #math.equation(block: false, alt: "x -axis")[$x "-axis"$] and the #math.equation(block: false, alt: "y -axis")[$y "-axis"$] is called an intercept of the line.
]

Let’s look at the graph of the lines shown.

#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through two labeled points, “ordered pair 0, 6” and ordered pair 3, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through two labeled points, “ordered pair 0, 6” and ordered pair 3, 0”.", caption: none)

First, notice where each of these lines crosses the #emph[x]- axis:

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#emph[Figure:]], [#emph[The line crosses the x-axis at:]], [#emph[Ordered pair of this point]]),
  [42], [3], [(3,0)],
  [43], [4], [(4,0)],
  [44], [5], [(5,0)],
  [45], [0], [(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 #emph[x-intercept] of the line.  The #emph[x-] intercept occurs when y is zero.

Now, let's look at the points where these lines cross the y-axis.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#emph[Figure:]], [#emph[The line crosses the y-axis at:]], [#emph[Ordered pair for this point]]),
  [42], [6], [(0,6)],
  [43], [-3], [(0,-3)],
  [44], [-5], [(0,-5)],
  [45], [0], [(0,0)],
))

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

The #math.equation(block: false, alt: "x -intercept")[$x "-intercept"$] is the point, #math.equation(block: false, alt: "open parenthesis a , 0 ),")[$( a , 0 ),$] where the graph crosses the #math.equation(block: false, alt: "x -axis .")[$x "-axis" .$] The #math.equation(block: false, alt: "x -intercept")[$x "-intercept"$] occurs when #math.equation(block: false, alt: "y")[$"y"$] is zero.

The #math.equation(block: false, alt: "y -intercept")[$y "-intercept"$] is the point, #math.equation(block: false, alt: "open parenthesis 0 , b ),")[$( 0 , b ),$] where the graph crosses the #math.equation(block: false, alt: "y -axis .")[$y "-axis" .$]

The #math.equation(block: false, alt: "y -intercept")[$y "-intercept"$] occurs when #math.equation(block: false, alt: "x")[$"x"$] is zero.
]

#examplebox("Example 1")[][
Find the #math.equation(block: false, alt: "x - and y -intercepts")[$x "- and" #h(0.2em) y "-intercepts"$] of each line:

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ #math.equation(block: false, alt: "x plus 2 y equals 4")[$#h(0.2em) x + 2 y = 4$]], [#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points “ordered pair 0, 2” and “ordered pair 4, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points “ordered pair 0, 2” and “ordered pair 4, 0”.", caption: none)]),
  [ⓑ #math.equation(block: false, alt: "3 x minus y equals 6")[$#h(0.2em) 3 x − y = 6$]], [#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points “ordered pair 0, -6” and “ordered pair 2, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points “ordered pair 0, -6” and “ordered pair 2, 0”.", caption: none)],
  [ⓒ #math.equation(block: false, alt: "x plus y equals −5")[$#h(0.2em) x + y = −5$]], [#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points “ordered pair 0, -5” and “ordered pair -5, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points “ordered pair 0, -5” and “ordered pair -5, 0”.", caption: none)],
))

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [The graph crosses the #emph[x]-axis at the point (4, 0).], [The #emph[x]-intercept is (4, 0).],
  [The graph crosses the #emph[y]-axis at the point (0, 2).], [The #emph[y]-intercept is (0, 2).],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [The graph crosses the #emph[x]-axis at the point (2, 0).], [The #emph[x]-intercept is (2, 0)],
  [The graph crosses the #emph[y]-axis at the point (0, −6).], [The #emph[y]-intercept is (0, −6).],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓒ], []),
  [The graph crosses the #emph[x]-axis at the point (−5, 0).], [The #emph[x]-intercept is (−5, 0).],
  [The graph crosses the #emph[y]-axis at the point (0, −5).], [The #emph[y]-intercept is (0, −5).],
))
]
]

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

#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points “ordered pair 0, 2” and “ordered pair 2, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points “ordered pair 0, 2” and “ordered pair 2, 0”.", caption: none)

#solutionbox[
#emph[x]-intercept (2,0): #emph[y]-intercept (0,−2)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the #math.equation(block: false, alt: "x -")[$x "-"$] and #math.equation(block: false, alt: "y -intercepts")[$y "-intercepts"$] of the graph: #math.equation(block: false, alt: "2 x plus 3 y equals 6 .")[$2 x + 3 y = 6 .$]

#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points “ordered pair 0, 2” and “ordered pair 3, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points “ordered pair 0, 2” and “ordered pair 3, 0”.", caption: none)

#solutionbox[
#emph[x]-intercept (3,0); #emph[y]-intercept (0,2)
]
]


=== Find the Intercepts from an Equation of a Line

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

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

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

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[x]], [#emph[y]]),
  [], [0],
  [0], [],
))
]

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

#solutionbox[
We'll fill in .

#figure(figph[A table illustrates how to find the x-intercept and y-intercept for the equation 2x + y = 6, by substituting 0 for y to find the x-intercept, and 0 for x to find the y-intercept.], alt: "A table illustrates how to find the x-intercept and y-intercept for the equation 2x + y = 6, by substituting 0 for y to find the x-intercept, and 0 for x to find the y-intercept.", caption: none)

To find the 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[A mathematical equation is shown in black text on a white background, reading '2x + y = 6'.], alt: "A mathematical equation is shown in black text on a white background, reading '2x + y = 6'.", caption: none)]),
  [Substitute 0 for #emph[y].], [#figure(figph[A mathematical equation '2x + 0 = 6' is displayed, with the number '0' highlighted in red to emphasize its role in the expression.], alt: "A mathematical equation '2x + 0 = 6' is displayed, with the number '0' highlighted in red to emphasize its role in the expression.", caption: none)],
  [Add.], [#figure(figph[A simple algebraic equation showing '2x = 6' on a white background. This equation can be solved to find the value of x.], alt: "A simple algebraic equation showing '2x = 6' on a white background. This equation can be solved to find the value of x.", caption: none)],
  [Divide by 2.], [#figure(figph[The image displays the mathematical equation 'x = 3' in black characters against a white background.], alt: "The image displays the mathematical equation 'x = 3' in black characters against a white background.", caption: none)],
  [The #emph[x]-intercept is (3, 0).], [],
))

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

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A linear equation is displayed, which reads '2x + y = 6' in black font against a white background.], alt: "A linear equation is displayed, which reads '2x + y = 6' in black font against a white background.", caption: none)]),
  [Substitute 0 for #emph[x].], [#figure(figph[A mathematical equation shows '2 multiplied by 0 plus y equals 6'. The number 0 is highlighted in red, suggesting it might be a specific value being substituted or emphasized in the equation.], alt: "A mathematical equation shows '2 multiplied by 0 plus y equals 6'. The number 0 is highlighted in red, suggesting it might be a specific value being substituted or emphasized in the equation.", caption: none)],
  [Multiply.], [#figure(figph[A mathematical equation is displayed, showing '0 + y = 6' in black text against a white background.], alt: "A mathematical equation is displayed, showing '0 + y = 6' in black text against a white background.", caption: none)],
  [Add.], [#figure(figph[The mathematical equation y = 6 is displayed on a white background, representing a horizontal line in a coordinate system.], alt: "The mathematical equation y = 6 is displayed on a white background, representing a horizontal line in a coordinate system.", caption: none)],
  [The #emph[y]-intercept is (0, 6).], [],
))

#figure(figph[A two column, four row table. The first row has an equation while the following rows show ordered pairs (3, 0) and (0, 6).], alt: "A two column, four row table. The first row has an equation while the following rows show ordered pairs (3, 0) and (0, 6).", caption: none)

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")[$( 0 , 6 )$].
]
]

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

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

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

#solutionbox[
#math.equation(block: false, alt: "open parenthesis 8 , 0 close parenthesis")[$( 8 , 0 )$] and #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 −3 y equals 12 .")[$4 x −3 y = 12 .$]

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

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "4 x minus 3 y equals 12")[$4 x − 3 y = 12$]]),
  [Substitute 0 for #math.equation(block: false, alt: "y .")[$y .$]], [#math.equation(block: false, alt: "4 x minus 3 times 0 equals 12")[$4 x − 3 · 0 = 12$]],
  [Multiply.], [#math.equation(block: false, alt: "4 x minus 0 equals 12")[$4 x − 0 = 12$]],
  [Subtract.], [#math.equation(block: false, alt: "4 x equals 12")[$4 x = 12$]],
  [Divide by 4.], [#math.equation(block: false, alt: "x equals 3")[$x = 3$]],
))

The #math.equation(block: false, alt: "x -intercept")[$x "-intercept"$] is #math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis .")[$( 3 , 0 ) .$]

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

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "4 x minus 3 y equals 12")[$4 x − 3 y = 12$]]),
  [Substitute 0 for #math.equation(block: false, alt: "x .")[$x .$]], [#math.equation(block: false, alt: "4 times 0 minus 3 y equals 12")[$4 · 0 − 3 y = 12$]],
  [Multiply.], [#math.equation(block: false, alt: "0 minus 3 y equals 12")[$0 − 3 y = 12$]],
  [Simplify.], [#math.equation(block: false, alt: "−3 y equals 12")[$−3 y = 12$]],
  [Divide by −3.], [#math.equation(block: false, alt: "y equals −4")[$y = −4$]],
))

The #math.equation(block: false, alt: "y -intercept")[$y "-intercept"$] is #math.equation(block: false, alt: "open parenthesis 0 , −4 ).")[$( 0 , −4 ).$]

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 , −4 close parenthesis .")[$( 0 , −4 ) .$]

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

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

#solutionbox[
#emph[x]-intercept (4,0); #emph[y]-intercept: (0,−3)
]
]

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

#solutionbox[
#emph[x]-intercept (4,0); #emph[y]-intercept: (0,−2)
]
]


=== Graph a Line Using the Intercepts

To graph a linear equation by plotting points, you can use the intercepts as two of your three points. Find the two intercepts, and then a third point to ensure accuracy, and draw the line. This method is often the quickest way to graph a line.

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

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

#math.equation(block: true, alt: "minus x plus 2 y equals 6; minus x plus 2 open parenthesis 0 close parenthesis equals 6; minus x equals 6; x equals −6")[$#h(0.7em) − x + 2 y = 6 \ − x + 2 ( 0 ) = 6 \ #h(2.8em) − x = 6 \ #h(4.3em) x = −6$]

The #math.equation(block: false, alt: "x -intercept")[$x "-intercept"$] is #math.equation(block: false, alt: "(− 6 , 0 ).")[$(− 6 , 0 ).$]

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

#math.equation(block: true, alt: "minus x plus 2 y equals 6; −0 plus 2 y equals 6; 2 y equals 6; y equals 3")[$− x + 2 y = 6 \ −0 + 2 y = 6 \  \  \ #h(2.4em) 2 y = 6 \ #h(3em) y = 3$]

The #math.equation(block: false, alt: "y -intercept")[$y "-intercept"$] is #math.equation(block: false, alt: "open parenthesis 0 , 3 ).")[$( 0 , 3 ).$]

Find a third point. We’ll use #math.equation(block: false, alt: "x equals 2 ,")[$x = 2 ,$]

#math.equation(block: true, alt: "minus x plus 2 y equals 6; −2 plus 2 y equals 6; 2 y equals 8; y equals 4")[$− x + 2 y = 6 \ −2 + 2 y = 6 \  \  \ #h(2.4em) 2 y = 8 \ #h(3em) y = 4$]

A third solution to the equation is #math.equation(block: false, alt: "open parenthesis 2 , 4 ).")[$( 2 , 4 ).$]

Summarize the three points in a table and then plot them on a graph.

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

#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. Three labeled points are shown at “ordered pair -6, 0”, “ordered pair 0, 3” and “ordered pair 2, 4”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. Three labeled points are shown at “ordered pair -6, 0”, “ordered pair 0, 3” and “ordered pair 2, 4”.", caption: none)

Do the points line up? Yes, so draw line through the points. #linebreak() 
#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. Three labeled points are shown at “ordered pair -6, 0”, “ordered pair 0, 3” and “ordered pair 2, 4”.  A line passes through the three labeled points.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. Three labeled points are shown at “ordered pair -6, 0”, “ordered pair 0, 3” and “ordered pair 2, 4”. A line passes through the three labeled points.", caption: none)
]
]

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

#solutionbox[
#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12.  A line passes through the points “ordered pair 0, -2” and “ordered pair 4, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, -2” and “ordered pair 4, 0”.", caption: none)
]
]

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

#solutionbox[
#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12.  A line passes through the points “ordered pair 0, 2” and “ordered pair -6, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 2” and “ordered pair -6, 0”.", caption: none)
]
]

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

+ Find the #math.equation(block: false, alt: "x minus")[$x −$] and #math.equation(block: false, alt: "y-intercepts")[$"y-intercepts"$] 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 then check that they line up.
+ Draw the line.
]

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

#solutionbox[
Find the intercepts and a third point.

#figure(figph[The figure shows 3 solutions to the equation 4 x - 3 y = 12. The first is titled “x-intercept, let y = 0”. The first line is 4 x - 3 y = 12. The second line shows 0 in red substituted for y, reading 4 x - 3 open parentheses 0 closed parentheses = 12. The third line is 4 x = 12. The last line is x = 3. The second solution is titled “y-intercept, let x = 0”. The first line is 4 x - 3 y = 12. The second line shows 0 in red substituted for x, reading 4 open parentheses 0 closed parentheses - 3 y = 12. The third line is -3 y = 12. The last line is y = -4. The third solution is titled “third point, let y = 4”. The first line is 4 x - 3 y = 12. The second line shows 4 in red substituted for y, reading 4 x - 3 open parentheses 4 closed parentheses = 12. The third line is 4 x - 12 = 12. The last line is x = 6.], alt: "The figure shows 3 solutions to the equation 4 x - 3 y = 12. The first is titled “x-intercept, let y = 0”. The first line is 4 x - 3 y = 12. The second line shows 0 in red substituted for y, reading 4 x - 3 open parentheses 0 closed parentheses = 12. The third line is 4 x = 12. The last line is x = 3. The second solution is titled “y-intercept, let x = 0”. The first line is 4 x - 3 y = 12. The second line shows 0 in red substituted for x, reading 4 open parentheses 0 closed parentheses - 3 y = 12. The third line is -3 y = 12. The last line is y = -4. The third solution is titled “third point, let y = 4”. The first line is 4 x - 3 y = 12. The second line shows 4 in red substituted for y, reading 4 x - 3 open parentheses 4 closed parentheses = 12. The third line is 4 x - 12 = 12. The last line is x = 6.", caption: none)

We list the points and show the graph.

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

#figure(figph[The graph shows the x y-coordinate plane. Both axes run from -7 to 7. Three unlabeled points are drawn at  “ordered pair 0, -4”, “ordered pair 3, 0” and “ordered pair  6, 4”.  A line passes through the points.], alt: "The graph shows the x y-coordinate plane. Both axes run from -7 to 7. Three unlabeled points are drawn at “ordered pair 0, -4”, “ordered pair 3, 0” and “ordered pair 6, 4”. A line passes through the points.", caption: none)
]
]

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

#solutionbox[
#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7.  A line passes through the points “ordered pair 0, -5” and “ordered pair 2, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points “ordered pair 0, -5” and “ordered pair 2, 0”.", caption: none)
]
]

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

#solutionbox[
#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7.  A line passes through the points “ordered pair 0, -3” and “ordered pair 4, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points “ordered pair 0, -3” and “ordered pair 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 2 solutions to y = 5 x. The first solution is titled “x-intercept; Let y = 0.” The first line is y = 5 x. The second line is 0, shown in red, = 5 x. The third line is 0 = x. The fourth line is x = 0. The last line is “The x-intercept is “ordered pair 0, 0”. The second  solution is titled “y-intercept; Let x = 0.” The first line is y = 5 x. The second line is  y = 5 open parentheses 0, shown in red, closed parentheses. The third line is y = 0. The last line is “The y-intercept is “ordered pair 0, 0”.], alt: "The figure shows 2 solutions to y = 5 x. The first solution is titled “x-intercept; Let y = 0.” The first line is y = 5 x. The second line is 0, shown in red, = 5 x. The third line is 0 = x. The fourth line is x = 0. The last line is “The x-intercept is “ordered pair 0, 0”. The second solution is titled “y-intercept; Let x = 0.” The first line is y = 5 x. The second line is y = 5 open parentheses 0, shown in red, closed parentheses. The third line is y = 0. The last line is “The y-intercept is “ordered pair 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 intercepts are the same point, we need two more points to graph the line. As always, we can choose any values for #math.equation(block: false, alt: "x ,")[$x ,$] so we’ll let #math.equation(block: false, alt: "x")[$x$] be #math.equation(block: false, alt: "1")[$1$] and #math.equation(block: false, alt: "−1 .")[$−1 .$] #linebreak() 
#figure(figph[The figure shows two substitutions in the equation y = 5 x. In the first substitution,  the first line is y = 5 x. The second line is y = 5 open parentheses 1, shown in red, closed parentheses. The third line is y =5. The last line is “ordered pair 1, 5”.  In the second substitution,  the first line is y = 5 x. The second line is y = 5 open parentheses -1, shown in red, closed parentheses. The third line is y = -5. The last line is “ordered pair -1, -5”.], alt: "The figure shows two substitutions in the equation y = 5 x. In the first substitution, the first line is y = 5 x. The second line is y = 5 open parentheses 1, shown in red, closed parentheses. The third line is y =5. The last line is “ordered pair 1, 5”. In the second substitution, the first line is y = 5 x. The second line is y = 5 open parentheses -1, shown in red, closed parentheses. The third line is y = -5. The last line is “ordered pair -1, -5”.", caption: none)

Organize the points in a table.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [#math.equation(block: false, alt: "y equals 5 x")[$y = 5 x$]], [], [],
  [#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]],
  [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]],
  [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "5")[$5$]], [#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 graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10.  A line passes through three labeled points, “ordered pair -1, -5”, “ordered pair 0, 0”, and ordered pair 1, 5”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. A line passes through three labeled points, “ordered pair -1, -5”, “ordered pair 0, 0”, and ordered pair 1, 5”.", caption: none)
]
]

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

#solutionbox[
#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12.  A line passes through the points “ordered pair 0, 0” and “ordered pair 1, 3”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 0” and “ordered pair 1, 3”.", caption: none)
]
]

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

#solutionbox[
#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12.  A line passes through the points “ordered pair 0, 0” and “ordered pair 1, -1”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 0” and “ordered pair 1, -1”.", caption: none)
]
]


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

While we could graph any #strong[linear equation] by plotting points, it may not always be the most convenient method. This table shows six of equations we’ve graphed in this chapter, and the methods we used to graph them.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([], [#emph[Equation]], [#emph[Method]]),
  [\#1], [#math.equation(block: false, alt: "y equals 2 x plus 1")[$y = 2 x + 1$]], [Plotting points],
  [\#2], [#math.equation(block: false, alt: "y equals the fraction 1 over 2 x plus 3")[$y = frac(1, 2) x + 3$]], [Plotting points],
  [\#3], [#math.equation(block: false, alt: "x equals −7")[$x = −7$]], [Vertical line],
  [\#4], [#math.equation(block: false, alt: "y equals 4")[$y = 4$]], [Horizontal line],
  [\#5], [#math.equation(block: false, alt: "2 x plus y equals 6")[$2 x + y = 6$]], [Intercepts],
  [\#6], [#math.equation(block: false, alt: "4 x minus 3 y equals 12")[$4 x − 3 y = 12$]], [Intercepts],
))

What is it about the form of equation that can help us choose the most convenient method to graph its line?

Notice that in equations \#1 and \#2, #emph[y] is isolated on one side of the equation, and its coefficient is 1. We found points by substituting values for #emph[x] on the right side of the equation and then simplifying to get the corresponding #emph[y-] values.

Equations \#3 and \#4 each have just one variable. Remember, in this kind of equation 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 \#5 and \#6, both #emph[x] and #emph[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")[$y = 0$] and #math.equation(block: false, alt: "x equals 0")[$x = 0$] to find the #emph[x-] and #emph[y-] intercepts, and then found a third point by choosing a value for #emph[x] or #emph[y].

This leads to the following strategy for choosing the most convenient method to graph a line.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Choose the most convenient method to graph a line.]

+ If the equation has only 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 #math.equation(block: false, alt: "x -axis")[$x "-axis"$] 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 #math.equation(block: false, alt: "y -axis")[$y "-axis"$] at #math.equation(block: false, alt: "b .")[$b .$]
+ If #math.equation(block: false, alt: "y")[$y$] is isolated on one side of the equation. Graph by plotting points.- Choose any three values for #math.equation(block: false, alt: "x")[$x$] and then solve for the corresponding #math.equation(block: false, alt: "y -")[$y "-"$] 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 #math.equation(block: false, alt: "x -")[$x "-"$] and #math.equation(block: false, alt: "y -")[$y "-"$] intercepts and then a third point.
]

#examplebox("Example 7")[][
Identify the most convenient method to graph each line:

+ ⓐ #math.equation(block: false, alt: "y equals −3")[$#h(0.2em) y = −3 #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "4 x −6 y equals 12")[$#h(0.2em) 4 x −6 y = 12 #h(0.2em)$]
+ ⓒ #math.equation(block: false, alt: "x equals 2")[$#h(0.2em) x = 2 #h(0.2em)$]
+ ⓓ #math.equation(block: false, alt: "y equals the fraction 2 over 5 x −1")[$#h(0.2em) y = frac(2, 5) x −1$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "y equals −3")[$#h(0.2em) y = −3$]

This equation has only one variable, #math.equation(block: false, alt: "y .")[$y .$] Its graph is a horizontal line crossing the #math.equation(block: false, alt: "y -axis")[$y "-axis"$] at #math.equation(block: false, alt: "−3 .")[$−3 .$]

ⓑ #math.equation(block: false, alt: "4 x −6 y equals 12")[$#h(0.2em) 4 x −6 y = 12$]

This 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 and one more point.

ⓒ #math.equation(block: false, alt: "x equals 2")[$#h(0.2em) x = 2$]

There is only one variable, #math.equation(block: false, alt: "x .")[$x .$] The graph is a vertical line crossing the #math.equation(block: false, alt: "x -axis")[$x "-axis"$] at #math.equation(block: false, alt: "2 .")[$2 .$]

ⓓ #math.equation(block: false, alt: "y equals the fraction 2 over 5 x −1")[$#h(0.2em) y = frac(2, 5) x −1$]

Since #math.equation(block: false, alt: "y")[$y$] is isolated on the left side of the equation, it will be easiest to graph this line by plotting three points.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Identify the most convenient method to graph each line:

+ ⓐ #math.equation(block: false, alt: "3 x plus 2 y equals 12")[$#h(0.2em) 3 x + 2 y = 12$]
+ ⓑ #math.equation(block: false, alt: "y equals 4")[$#h(0.2em) y = 4$]
+ ⓒ #math.equation(block: false, alt: "y equals the fraction 1 over 5 x −4")[$#h(0.2em) y = frac(1, 5) x −4$]
+ ⓓ #math.equation(block: false, alt: "x equals −7")[$#h(0.2em) x = −7$]

#solutionbox[
+ ⓐ intercepts
+ ⓑ horizontal line
+ ⓒ plotting points
+ ⓓ vertical line
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Identify the most convenient method to graph each line:

+ ⓐ #math.equation(block: false, alt: "x equals 6")[$#h(0.2em) x = 6$]
+ ⓑ #math.equation(block: false, alt: "y equals minus the fraction 3 over 4 x plus 1")[$#h(0.2em) y = − frac(3, 4) x + 1$]
+ ⓒ #math.equation(block: false, alt: "y equals −8")[$#h(0.2em) y = −8$]
+ ⓓ #math.equation(block: false, alt: "4 x −3 y equals −1")[$#h(0.2em) 4 x −3 y = −1$]

#solutionbox[
+ ⓐ vertical line
+ ⓑ plotting points
+ ⓒ horizontal line
+ ⓓ intercepts
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[ACCESS ADDITIONAL ONLINE RESOURCES]

- #link("https://www.openstax.org/l/24findinter")[Graph by Finding Intercepts]
- #link("https://www.openstax.org/l/24useintercept")[Use Intercepts to Graph]
- #link("https://www.openstax.org/l/24statintercept")[State the Intercepts from a Graph]
]


=== Key Concepts

- #emph[Intercepts]- The #emph[x-]intercept is the point, #math.equation(block: false, alt: "open parenthesis a , 0 close parenthesis")[$( a , 0 )$], where the graph crosses the #emph[x-]axis. The #emph[x-]intercept occurs when y is zero.
- The #emph[y-]intercept is the point, #math.equation(block: false, alt: "open parenthesis 0 , b close parenthesis")[$( 0 , b )$], where the graph crosses the #emph[y-]axis. The #emph[y-]intercept occurs when x is zero.
- The #emph[x-]intercept occurs when y is zero.
- The #emph[y-]intercept occurs when x is zero.
- #emph[Find the #emph[x] and #emph[y] intercepts from the equation of a 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].
- 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].#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[x]], [#emph[y]]),
  [], [0],
  [0], [],
))
- #emph[Graph a line 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 #emph[x.]
- 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.
- #emph[Choose the most convenient method to graph a line]+ Determine if the equation has only one variable. Then 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 #emph[a]. #linebreak() #math.equation(block: false, alt: "y equals b")[$y = b$] is a horizontal line passing through the #emph[y-]axis at #emph[b].
+ Determine if #emph[y] is isolated on one side of the equation. The graph by plotting points. #linebreak()  Choose any three values for #emph[x] and then solve for the corresponding #emph[y-] values.
+ Determine 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 and then a third point.


==== Practice Makes Perfect

#strong[Identify the Intercepts on a Graph]

In the following exercises, find the #math.equation(block: false, alt: "x -")[$x "-"$] and #math.equation(block: false, alt: "y -")[$y "-"$] intercepts.

#figure(figph[The graph shows the x y-coordinate plane. The axes run from -10 to 10.  A line passes through the points “ordered pair 0, 3” and “ordered pair 3, 0”.], alt: "The graph shows the x y-coordinate plane. The axes run from -10 to 10. A line passes through the points “ordered pair 0, 3” and “ordered pair 3, 0”.", caption: none)

#solutionbox[
(3,0),(0,3)
]

#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10.  A line passes through the points “ordered pair 0, 2” and “ordered pair 2, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. A line passes through the points “ordered pair 0, 2” and “ordered pair 2, 0”.", caption: none)

#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10.  A line passes through the points “ordered pair 0, -5” and “ordered pair 5, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. A line passes through the points “ordered pair 0, -5” and “ordered pair 5, 0”.", caption: none)

#solutionbox[
(5,0),(0,−5)
]

#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10.  A line passes through the points “ordered pair 0, -1” and “ordered pair 1, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. A line passes through the points “ordered pair 0, -1” and “ordered pair 1, 0”.", caption: none)

#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10.  A line passes through the points “ordered pair 0, -2” and “ordered pair -2, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. A line passes through the points “ordered pair 0, -2” and “ordered pair -2, 0”.", caption: none)

#solutionbox[
(−2,0),(0,−2)
]

#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10.  A line passes through the points “ordered pair 0, -3” and “ordered pair -3, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. A line passes through the points “ordered pair 0, -3” and “ordered pair -3, 0”.", caption: none)

#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10.  A line passes through the points “ordered pair 0, 1” and “ordered pair -1, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. A line passes through the points “ordered pair 0, 1” and “ordered pair -1, 0”.", caption: none)

#solutionbox[
(−1,0),(0,1)
]

#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10.  A line passes through the points “ordered pair 0, 5” and “ordered pair -5, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. A line passes through the points “ordered pair 0, 5” and “ordered pair -5, 0”.", caption: none)

#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7.  A line passes through the points “ordered pair 0, 0” and “ordered pair 4, 2”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points “ordered pair 0, 0” and “ordered pair 4, 2”.", caption: none)

#solutionbox[
(0,0)
]

#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10.  A line passes through the points “ordered pair 0, 0” and “ordered pair 1, 1”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. A line passes through the points “ordered pair 0, 0” and “ordered pair 1, 1”.", caption: none)

#strong[Find the #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$] Intercepts from an Equation of a Line]

In the following exercises, find the intercepts.

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

#solutionbox[
(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[
(−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[
(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[
(−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[
(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[
(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 −3 y equals 12")[$x −3 y = 12$]

#solutionbox[
(12,0),(0,−4)
]

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

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

#solutionbox[
(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[
(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 −2 y equals 12")[$3 x −2 y = 12$]

#solutionbox[
(4,0),(0,−6)
]

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

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

#solutionbox[
(3,0),(0,−1)
]

#math.equation(block: true, alt: "y equals the fraction 1 over 4 x −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[
(−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[
(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[
(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 graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12.  A line passes through the points “ordered pair 0, 2” and “ordered pair -10, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 2” and “ordered pair -10, 0”.", 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 graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12.  A line passes through the points “ordered pair 0, 2” and “ordered pair 4, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 2” and “ordered pair 4, 0”.", 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 graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12.  A line passes through the points “ordered pair 0, 2” and “ordered pair 2, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 2” and “ordered pair 2, 0”.", caption: none)
]

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

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

#solutionbox[
#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12.  A line passes through the points “ordered pair 0, 3” and “ordered pair 3, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 3” and “ordered pair 3, 0”.", caption: none)
]

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

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

#solutionbox[
#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12.  A line passes through the points “ordered pair 0, -1” and “ordered pair 1, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, -1” and “ordered pair 1, 0”.", 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 graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12.  A line passes through the points “ordered pair 0,  4” and “ordered pair -4, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 4” and “ordered pair -4, 0”.", 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 graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12.  A line passes through the points “ordered pair 0,  4” and “ordered pair 1, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 4” and “ordered pair 1, 0”.", caption: none)
]

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

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

#solutionbox[
#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12.  A line passes through the points “ordered pair 0,  6” and “ordered pair -2, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 6” and “ordered pair -2, 0”.", caption: none)
]

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

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

#solutionbox[
#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12.  A line passes through the points “ordered pair 0,  3” and “ordered pair 6, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 3” and “ordered pair 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 −2 y equals 6")[$3 x −2 y = 6$]

#solutionbox[
#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12.  A line passes through the points “ordered pair 0,  -3” and “ordered pair 2, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, -3” and “ordered pair 2, 0”.", caption: none)
]

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

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

#solutionbox[
#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12.  A line passes through the points “ordered pair 0,  4” and “ordered pair -10, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 4” and “ordered pair -10, 0”.", caption: none)
]

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

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

#solutionbox[
#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12.  A line passes through the points “ordered pair 0,  0” and “ordered pair 2, -4”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 0” and “ordered pair 2, -4”.", caption: none)
]

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

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

#solutionbox[
#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12.  A line passes through the points “ordered pair 0,  0” and “ordered pair 2, 2”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair 0, 0” and “ordered pair 2, 2”.", caption: none)
]

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

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

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

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

#solutionbox[
vertical line
]

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

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

#solutionbox[
horizontal line
]

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

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

#solutionbox[
plotting points
]

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

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

#solutionbox[
intercepts
]

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

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

#solutionbox[
plotting points
]

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

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

#solutionbox[
horizontal line
]

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

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

#solutionbox[
intercepts
]

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

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

#solutionbox[
plotting points
]

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


==== Everyday Math

#strong[Road trip] Damien is driving from Chicago to Denver, a distance of #math.equation(block: false, alt: "1,000")[$1,000$] miles. The #math.equation(block: false, alt: "x -axis")[$x "-axis"$] on the graph below shows the time in hours since Damien left Chicago. The #math.equation(block: false, alt: "y -axis")[$y "-axis"$] represents the distance he has left to drive.

#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from - to .  A line passes through the labeled points “ordered pair 0, 1000” and “ordered pair 15, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from - to . A line passes through the labeled points “ordered pair 0, 1000” and “ordered pair 15, 0”.", caption: none)

ⓐ Find the #math.equation(block: false, alt: "x -")[$x "-"$] and #math.equation(block: false, alt: "y -")[$y "-"$] intercepts

ⓑ Explain what the #math.equation(block: false, alt: "x -")[$x "-"$] and #math.equation(block: false, alt: "y -")[$y "-"$] intercepts mean for Damien.

#solutionbox[
ⓐ (0,1,000),(15,0). ⓑ At (0,1,000) he left Chicago 0 hours ago and has 1,000 miles left to drive. At (15,0) he left Chicago 15 hours ago and has 0 miles left to drive.
]

#strong[Road trip] Ozzie filled up the gas tank of his truck and went on a road trip. The #math.equation(block: false, alt: "x -axis")[$x "-axis"$] on the graph shows the number of miles Ozzie drove since filling up. The #math.equation(block: false, alt: "y -axis")[$y "-axis"$] represents the number of gallons of gas in the truck’s gas tank.

#figure(figph[The graph shows the x y-coordinate plane. The x and y-axis each run from - to .  A line passes through labeled points “ordered pair 0, 16” and “ordered pair 300, 0”.], alt: "The graph shows the x y-coordinate plane. The x and y-axis each run from - to . A line passes through labeled points “ordered pair 0, 16” and “ordered pair 300, 0”.", caption: none)

ⓐ Find the #math.equation(block: false, alt: "x -")[$x "-"$] and #math.equation(block: false, alt: "y -")[$y "-"$] intercepts.

ⓑ Explain what the #math.equation(block: false, alt: "x -")[$x "-"$] and #math.equation(block: false, alt: "y -")[$y "-"$] intercepts mean for Ozzie.


==== Writing Exercises

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

#solutionbox[
Answers will vary.
]

How do you find the #math.equation(block: false, alt: "y -intercept")[$y "-intercept"$] of the graph of #math.equation(block: false, alt: "5 x minus y equals 10 ?")[$5 x − y = 10 ?$]

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

#solutionbox[
Answers will vary.
]

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


==== Self Check

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

#figure(figph[A student self-assessment rubric on identifying, finding, and graphing line intercepts and selecting appropriate graphing methods.], alt: "A student self-assessment rubric on identifying, finding, and graphing line intercepts and selecting appropriate graphing methods.", caption: none)

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