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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

the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is
anas Reply
?
Choli
a rock is thrown directly upward with an initial velocity of 96feet per second from a cliff 190 feet above a beach. The hight of tha rock above the beach after t second is given by the equation h=_16t^2+96t+190
Usman
Stella bought a dinette set on sale for $725. The original price was $1,299. To the nearest tenth of a percent, what was the rate of discount?
Manhwa Reply
44.19%
Scott
40.22%
Terence
44.2%
Orlando
p mulripied-5 and add 30 to it
Tausif Reply
p mulripied-5 and add30
Tausif
p mulripied-5 and addto30
Tausif
Can you explain further
Monica Reply
p mulripied-5 and add to 30
Tausif
How do you find divisible numbers without a calculator?
Jacob Reply
TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13
BAINAMA
When she graduates college, Linda will owe $43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was $1,585. What is the amount of each loan?
Ariana Reply
Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7 2/3 hours, what was the speed of the bus?
Kirisma Reply
66miles/hour
snigdha
How did you work it out?
Esther
s=mi/hr 2/3~0.67 s=506mi/7.67hr = ~66 mi/hr
Orlando
hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused.
Alicia Reply
what do you need help in?
Felix
subtracting a negative....is adding!!
Heather
look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number...
Felix
for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer...
Felix
—12
Niazmohammad
Thanks Felix.l also get confused with signs.
Esther
Thank you for this
Shatey
ty
Graham
think about it like you lost $19 (-19), then found $7(+7). Totally you lost just $12 (-12)
Annushka
I used to struggle a lot with negative numbers and math in general what I typically do is look at it in terms of money I have -$5 in my account I then take out 5 more dollars how much do I have in my account well-$10 ... I also for a long time would draw it out on a number line to visualize it
Meg
practicing with smaller numbers to understand then working with larger numbers helps too and the song/rhyme same sign add and keep opposite signs subtract keep the sign of the bigger # then you'll be exact
Meg
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles
John Reply
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
achol Reply
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce
where's the answers?
Ed Reply
I don't see where the answers are.
Ed
Cindy and Richard leave their dorm in Charleston at the same time. Cindy rides her bicycle north at a speed of 18 miles per hour. Richard rides his bicycle south at a speed of 14 miles per hour. How long will it take them to be 96 miles apart?
Maddy Reply
3
Christopher
18t+14t=96 32t=96 32/96 3
Christopher
show that a^n-b^2n is divisible by a-b
Florence Reply
What does 3 times your weight right now
Cherokee Reply
Use algebra to combine 39×5 and the half sum of travel of 59+30
Cherokee
What is the segment of 13? Explain
Cherokee
my weight is 49. So 3 times is 147
Cherokee
kg to lbs you goin to convert 2.2 or one if the same unit your going to time your body weight by 3. example if my body weight is 210lb. what would be my weight if I was 3 times as much in kg. that's you do 210 x3 = 630lb. then 630 x 2.2= .... hope this helps
tyler
How to convert grams to pounds?
paul
What is the lcm of 340
Kendra Reply
Yes
Cherokee
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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