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Find the x - and y - intercepts from the equation of a line

Use the equation of the line. To find:

  • the x - intercept of the line, let y = 0 and solve for x .
  • the y - intercept of the line, let x = 0 and solve for y .

 

Find the intercepts of 2 x + y = 6 .

Solution

We will let y = 0 to find the x - intercept, and let x = 0 to find the y - intercept. We will fill in the table, which reminds us of what we need to find.

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

To find the x - intercept, let y = 0 .

.
Let y = 0. .
Simplify. .
.
The x -intercept is (3, 0)
To find the y -intercept, let x = 0.
.
Let x = 0. .
Simplify. .
.
The y -intercept is (0, 6)

The intercepts are the points ( 3 , 0 ) and ( 0 , 6 ) as shown in [link] .

2 x + y = 6
x y
3 0
0 6
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Find the intercepts of 3 x + y = 12 .

x - intercept: ( 4 , 0 ) , y - intercept: ( 0 , 12 )

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Find the intercepts of x + 4 y = 8 .

x - intercept: ( 8 , 0 ) , y - intercept: ( 0 , 2 )

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Find the intercepts of 4 x 3 y = 12 .

Solution

To find the x -intercept, let y = 0.
.
Let y = 0. .
Simplify. .
.
.
The x -intercept is (3, 0)
To find the y -intercept, let x = 0.
.
Let x = 0. .
Simplify. .
.
.
The y -intercept is (0, −4)

The intercepts are the points (3, 0) and (0, −4) as shown in [link] .

4 x 3 y = 12
x y
3 0
0 −4
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Find the intercepts of 3 x 4 y = 12 .

x - intercept: ( 4 , 0 ) , y - intercept: ( 0 , −3 )

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Find the intercepts of 2 x 4 y = 8 .

x - intercept: ( 4 , 0 ) , y - intercept: ( 0 , −2 )

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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 x - and 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.

How to graph a line using intercepts

Graph x + 2 y = 6 using the intercepts.

Solution

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)”. 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.” 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 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).
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Graph x 2 y = 4 using the intercepts.

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).

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Graph x + 3 y = 6 using the intercepts.

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).

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

Graph a linear equation using the intercepts.

  1. Find the x - and y - intercepts of the line.
    • Let y = 0 and solve for x
    • Let x = 0 and solve for y .
  2. Find a third solution to the equation.
  3. Plot the three points and check that they line up.
  4. Draw the line.

Graph 4 x 3 y = 12 using the intercepts.

Solution

Find the intercepts and a third point.

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.

We list the points in [link] and show the graph below.

4 x 3 y = 12
x y ( x , y )
3 0 ( 3 , 0 )
0 −4 ( 0 , −4 )
6 4 ( 6 , 4 )
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).
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Graph 5 x 2 y = 10 using the intercepts.

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).

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Graph 3 x 4 y = 12 using the intercepts.

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).

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Graph y = 5 x using the intercepts.

Solution

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).

This line has only one intercept. It is the point ( 0 , 0 ) .

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

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.

See [link] .

y = 5 x
x y ( x , y )
0 0 ( 0 , 0 )
1 5 ( 1 , 5 )
−1 −5 ( −1 , −5 )

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

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).
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Graph y = 4 x using the intercepts.

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).

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Graph y = x the intercepts.

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).

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Key concepts

  • Find the x - and y - Intercepts from the Equation of a Line
    • Use the equation of the line to find the x - intercept of the line, let y = 0 and solve for x .
    • Use the equation of the line to find the y - intercept of the line, let x = 0 and solve for y .
  • Graph a Linear Equation using the Intercepts
    1. Find the x - and y - intercepts of the line.
      Let y = 0 and solve for x .
      Let x = 0 and solve for y .
    2. Find a third solution to the equation.
    3. Plot the three points and then check that they line up.
    4. Draw the line.



  • 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.
      x = a is a vertical line passing through the x - axis at a
      y = b is a horizontal line passing through the y - axis at b .
    • If y is isolated on one side of the equation, graph by plotting points.
    • Choose any three values for x and then solve for the corresponding y - values.
    • If the equation is of the form a x + b y = c , find the intercepts. Find the x - and y - intercepts and then a third point.

Practice makes perfect

Identify the x - and y - Intercepts on a Graph

In the following exercises, find the x - and y - intercepts on each graph.

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).

( 5 , 0 ) , ( 0 , −5 )

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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).

( −2 , 0 ) , ( 0 , −2 )

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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).

( −1 , 0 ) , ( 0 , 1 )

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Find the x - and y - Intercepts from an Equation of a Line

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

x + y = 4

( 4 , 0 ) , ( 0 , 4 )

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x + y = −2

( −2 , 0 ) , ( 0 , −2 )

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x y = 5

( 5 , 0 ) , ( 0 , −5 )

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x y = −3

( −3 , 0 ) , ( 0 , 3 )

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x + 2 y = 8

( 8 , 0 ) , ( 0 , 4 )

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3 x + y = 6

( 2 , 0 ) , ( 0 , 6 )

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x 3 y = 12

( 12 , 0 ) , ( 0 , −4 )

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4 x y = 8

( 2 , 0 ) , ( 0 , −8 )

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2 x + 5 y = 10

( 5 , 0 ) , ( 0 , 2 )

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3 x 2 y = 12

( 4 , 0 ) , ( 0 , −6 )

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y = 1 3 x + 1

( 3 , 0 ) , ( 0 , −1 )

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y = 1 5 x + 2

( −10 , 0 ) , ( 0 , 2 )

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Graph a Line Using the Intercepts

In the following exercises, graph using the intercepts.

Everyday math

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

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.
  1. Find the x - and y - intercepts.
  2. Explain what the x - and y - intercepts mean for Damien.

( 0 , 1000 ) , ( 15 , 0 )
At ( 0 , 1000 ) , he has been gone 0 hours and has 1000 miles left. At ( 15 , 0 ) , he has been gone 15 hours and has 0 miles left to go.

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

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
  1. Find the x - and y - intercepts.
  2. Explain what the x - and y - intercepts mean for Ozzie.
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Writing exercises

How do you find the x - intercept of the graph of 3 x 2 y = 6 ?

Answers will vary.

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Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation 4 x + y = −4 ? Why?

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Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation y = 2 3 x 2 ? Why?

Answers will vary.

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Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation y = 6 ? Why?

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Self check

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

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

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

Questions & Answers

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Elizabeth Reply
ggfcc
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1,75hrs
Mike
I'm going to guess. Divide Levi's time by 2. Then divide 1 hour by 2. 1.25 + 0.5 = 1.3?
John
Oops I mean 1.75
John
I'm guessing this because since I have divide 1 hour by 2, I have to do the same for the 2.5 hours it takes Levi by himself.
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Debra
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Parker Reply
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800 sq. ft @ $1.50 & 200 sq. ft @ $9.00
Marco
Geneva treated her parents to dinner at their favorite restaurant. The bill was $74.25. Geneva wants to leave 16 % of the total - bill as a tip. How much should the tip be?
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Practice Key Terms 3

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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