X and Y Intercepts: How to Find and Graph

The x and y intercepts are the points where a graph crosses the axes. To find the x-intercept, set $y = 0$ and then solve for $x$ . To find the y-intercept, set $x = 0$ and then solve for $y$ . For $3x + 2y = 12$ , the intercepts are $(4, 0)$ and $(0, 6)$ .

Before this lesson: You should know how to plot points on the coordinate plane, solve two-step equations, and find solutions of two-variable linear equations.

Extra practice: The Algebra 1 Unit 4 Practice Pack (Part 1) has printable problems, answer keys and a cheat sheet for this unit.

What Are X-Intercepts and Y-Intercepts?

Think of a line as a road on a map. The two axes are like two big streets. An intercept is a spot where the road crosses one of those streets.

The x-intercept is where a graph crosses the x-axis. Every point on the x-axis has a height of 0. So at the x-intercept, $y = 0$ .

The y-intercept is where a graph crosses the y-axis. Every point on the y-axis is 0 steps left or right. So at the y-intercept, $x = 0$ .

The line 3x + 2y = 12 on a grid with x and y from −2 to 7. It crosses the x-axis at the point (4, 0), labeled x-intercept, and the y-axis at the point (0, 6), labeled y-intercept. A note beside the graph says: at the x-intercept, y = 0; at the y-intercept, x = 0. — The x-intercept is where the line crosses the x-axis. The y-intercept is where it crosses the y-axis.

How to Find X and Y Intercepts

Use the same idea for every equation. Put a zero in for one variable, then solve for the other.

Intercept	What to do	Write it as
x-intercept	Set $y = 0$ and solve for $x$	$(a, 0)$
y-intercept	Set $x = 0$ and solve for $y$	$(0, b)$

To find the x-intercept, replace $y$ with 0.
Solve for $x$ . Call the answer $a$ .
Write the x-intercept as the point $(a, 0)$ .
To find the y-intercept, replace $x$ with 0.
Solve for $y$ . Call the answer $b$ .
Write the y-intercept as the point $(0, b)$ .

A quick way to remember: the x-intercept has a zero for $y$ , and the y-intercept has a zero for $x$ . The zero always goes in the other spot.

Worked Examples

Example 1: Find the intercepts of $3x + 2y = 12$

Find the $x$ -intercept and the $y$ -intercept of $3x + 2y = 12$ .

3x + 2(0) = 12

Step 1. For the x-intercept, set $y = 0$ .

3x = 12

Step 2. Multiply. The term 2(0) is 0.

x = 4

Step 3. Divide both sides by 3.

3(0) + 2y = 12

Step 4. For the y-intercept, set $x = 0$ .

2y = 12

Step 5. Multiply. The term 3(0) is 0.

y = 6

Step 6. Divide both sides by 2.

Answer: The x-intercept is $(4, 0)$ . The y-intercept is $(0, 6)$ . This is the line in the graph above.

Example 2: Find the intercepts of $y = 2x - 6$

Find both intercepts of $y = 2x - 6$ .

y = 2(0) - 6 = 0 - 6 = -6

Step 1. For the y-intercept, set $x = 0$ .

0 = 2x - 6

Step 2. For the x-intercept, set $y = 0$ .

6 = 2x

Step 3. Add 6 to both sides, as in two-step equations.

3 = x

Step 4. Divide both sides by 2.

Answer: The x-intercept is $(3, 0)$ . The y-intercept is $(0, -6)$ .

Check: Notice that the y-intercept, $-6$ , is the number at the end of $y = 2x - 6$ . In slope-intercept form, $y = mx + b$ , the y-intercept is always $(0, b)$ .

Example 3: Intercepts with a fraction, $5x - 4y = 10$

Find both intercepts of $5x - 4y = 10$ .

5x - 4(0) = 10

Step 1. For the x-intercept, set $y = 0$ .

5x = 10

Step 2. Multiply.

x = 2

Step 3. Divide both sides by 5.

5(0) - 4y = 10

Step 4. For the y-intercept, set $x = 0$ . Watch the negative sign.

-4y = 10

Step 5. Multiply.

y = \frac{10}{-4}

Step 6. Divide both sides by $-4$ .

y = -\frac{5}{2}

Step 7. Simplify the fraction. Divide the top and bottom by 2, and a positive divided by a negative is negative (see fractions).

Answer: The x-intercept is $(2, 0)$ . The y-intercept is $\left(0, -\tfrac{5}{2}\right)$ . On a graph, that is halfway between $-2$ and $-3$ on the y-axis.

Example 4: Graph $2x - 3y = 6$ using intercepts

Graph $2x - 3y = 6$ using its intercepts.

2x - 3(0) = 6 \text{, so } 2x = 6

Step 1. Set $y = 0$ and simplify.

x = 3

Step 2. Divide both sides by 2. Plot $(3, 0)$ .

2(0) - 3y = 6 \text{, so } -3y = 6

Step 3. Set $x = 0$ and simplify.

y = -2

Step 4. Divide both sides by $-3$ (see multiplying and dividing integers). Plot $(0, -2)$ .

Step 5. Draw a line through the two points.

2(6) - 3y = 6

Step 6. Check with a third point. Pick $x = 6$ and substitute.

12 - 3y = 6

Step 7. Multiply.

-3y = -6

Step 8. Subtract 12 from both sides.

y = 2

Step 9. Divide both sides by $-3$ .

The line 2x − 3y = 6 on a grid with x and y from −3 to 7. The x-intercept (3, 0) and the y-intercept (0, −2) are marked with closed dots. A third check point, (6, 2), is marked with an open circle on the same line. Steps beside the graph: 1. Let y = 0, so x = 3. 2. Let x = 0, so y = −2. 3. Plot both points and draw the line. — Find both intercepts, plot them, and draw the line. A third point checks your work.

Answer: The line goes through $(3, 0)$ and $(0, -2)$ . The check point $(6, 2)$ lands on the same line, so the graph is right.

Example 5: Word problem with a gift card

Maya has a $60 gift card for a coffee shop. Each coffee costs $4. The balance after $c$ coffees is $B = 60 - 4c$ . Find both intercepts and explain what they mean.

B = 60 - 4(0) = 60 - 0 = 60

Step 1. For the $B$ -intercept, set $c = 0$ .

Step 2. Explain it. Before buying any coffee, the card has $60.

0 = 60 - 4c

Step 3. For the $c$ -intercept, set $B = 0$ .

4c = 60

Step 4. Add $4c$ to both sides.

c = 15

Step 5. Divide both sides by 4.

Step 6. Explain it. After 15 coffees, the balance is $0.

Answer: The intercepts are $(0, 60)$ and $(15, 0)$ . The card starts with $60, and it is empty after 15 coffees.

Check: In word problems, the intercept on the vertical axis is often the starting amount. The intercept on the horizontal axis is often when something runs out.

Common Mistakes

Mistake	Example of the error	Fix
Setting the wrong variable to 0	For $3x + 2y = 12$ , setting $x = 0$ and calling 6 the x-intercept	For the x-intercept, set $y$ to 0. That gives $x = 4$ .
Writing the point in the wrong order	Writing the y-intercept 6 as $(6, 0)$	The y-intercept has $x = 0$ , so it is $(0, 6)$ .
Losing a negative sign	From $-4y = 10$ , writing $y = \tfrac{5}{2}$	Divide by $-4$ : $y = -\tfrac{5}{2}$ .
Trusting only two points	Plotting one intercept in the wrong place and drawing the wrong line	Find a third point and check that it is on your line.

Intercepts from a Graph or a Table

You don’t always need an equation. You can read intercepts from a graph or a table too.

From a graph: find where the line touches each axis. Read the point, like $(4, 0)$ or $(0, 6)$ .
From a table: look for the row where $x = 0$ . Its $y$ -value gives the y-intercept. The row where $y = 0$ gives the x-intercept.
Special lines: a horizontal line like $y = 3$ has a y-intercept but no x-intercept. A vertical line like $x = -2$ has an x-intercept but no y-intercept. See horizontal and vertical lines.
Lines through the origin: $y = 4x$ has both intercepts at $(0, 0)$ . To graph it, you need one more point, such as $(1, 4)$ .

Intercepts are also key features of curved graphs. The x-intercepts of a curve are its zeros. You saw these in key features of functions.

Practice Problems

Find the x-intercept and the y-intercept. Write each one as a point.

$x + y = 7$
$2x + 5y = 10$
$4x - 3y = 24$
$y = 3x + 9$
$y = -2x + 5$
$-6x + 2y = 18$
$3x + 4y = 6$
$y = -4$
Use the table to find both intercepts: $x$ : $-2$ , 0, 2, 4 and $y$ : $-6$ , $-3$ , 0, 3.
A pool holds 900 gallons. It drains at 30 gallons per minute, so $V = 900 - 30t$ . Find both intercepts and say what they mean.

Show answers

x + 0 = 7, \text{ so } x = 7

Set y = 0.

0 + y = 7, \text{ so } y = 7

Set x = 0.

Intercepts: $(7, 0)$ and $(0, 7)$ .

2x + 5(0) = 10

Set y = 0.

2x = 10

Simplify.

x = 5

Divide both sides by 2.

2(0) + 5y = 10

Set x = 0.

5y = 10

Simplify.

y = 2

Divide both sides by 5.

Intercepts: $(5, 0)$ and $(0, 2)$ .

4x - 3(0) = 24

Set y = 0.

4x = 24

Simplify.

x = 6

Divide both sides by 4.

4(0) - 3y = 24

Set x = 0.

-3y = 24

Simplify.

y = -8

Divide both sides by −3.

Intercepts: $(6, 0)$ and $(0, -8)$ .

0 = 3x + 9

Set y = 0.

-9 = 3x

Subtract 9 from both sides.

x = -3

Divide both sides by 3.

y = 3(0) + 9 = 9

Set x = 0.

Intercepts: $(-3, 0)$ and $(0, 9)$ .

0 = -2x + 5

Set y = 0.

2x = 5

Add 2x to both sides.

x = \tfrac{5}{2}

Divide both sides by 2.

y = -2(0) + 5 = 5

Set x = 0.

Intercepts: $\left(\tfrac{5}{2}, 0\right)$ and $(0, 5)$ .

-6x + 2(0) = 18

Set y = 0.

-6x = 18

Simplify.

x = -3

Divide both sides by −6.

-6(0) + 2y = 18

Set x = 0.

2y = 18

Simplify.

y = 9

Divide both sides by 2.

Intercepts: $(-3, 0)$ and $(0, 9)$ .

3x + 4(0) = 6

Set y = 0.

3x = 6

Simplify.

x = 2

Divide both sides by 3.

3(0) + 4y = 6

Set x = 0.

4y = 6

Simplify.

y = \tfrac{6}{4}

Divide both sides by 4.

y = \tfrac{3}{2}

Simplify the fraction.

Intercepts: $(2, 0)$ and $\left(0, \tfrac{3}{2}\right)$ .

This is a horizontal line. Every point has y = −4
At x = 0, y = −4, so the y-intercept is $(0, -4)$
Since y is never 0, the line never crosses the x-axis
There is no x-intercept.

Find the row with x = 0. Its y-value is −3
Find the row with y = 0. Its x-value is 2
The y-intercept is $(0, -3)$ and the x-intercept is $(2, 0)$ .

10.

V = 900 - 30(0) = 900

Set t = 0.

This point, (0, 900), means the pool starts with 900 gallons

0 = 900 - 30t

Set V = 0.

30t = 900

Add 30t to both sides.

t = 30

Divide both sides by 30.

The point $(30, 0)$ means the pool is empty after 30 minutes.

Want more practice? The Algebra 1 Unit 4 Practice Pack (Part 1) has 39 pages of problems on this and the other lessons in the unit, with step-by-step answers and a cheat sheet.

X and Y Intercepts FAQ

How do you find the x-intercept?

Set $y = 0$ in the equation and solve for $x$ . Then write the answer as a point, $(x, 0)$ .

How do you find the y-intercept?

Set $x = 0$ and solve for $y$ . If the equation is in the form $y = mx + b$ , the y-intercept is just $(0, b)$ .

Can a line have no x-intercept?

Yes. A horizontal line like $y = 5$ never crosses the x-axis, so it has no x-intercept. The x-axis itself, $y = 0$ , touches it everywhere.

Is an intercept a number or a point?

Both are used. Many books say “the x-intercept is 4” and mean the point $(4, 0)$ . Writing the point is clearer, so we use points in this lesson.

Summary

The x-intercept is where a graph crosses the x-axis. The y-intercept is where it crosses the y-axis.
To find the x-intercept, set $y = 0$ . To find the y-intercept, set $x = 0$ .
Write intercepts as points: $(a, 0)$ and $(0, b)$ .
Two intercepts are enough to graph a line. A third point is a good check.
In word problems, intercepts often show a starting amount and when something runs out.

Previous lesson: Two-Variable Linear Equations

Next lesson: Slope

Back to the Algebra 1 Index

What Are X-Intercepts and Y-Intercepts?

How to Find X and Y Intercepts

Worked Examples

Example 1: Find the intercepts of 3x + 2y = 12

Example 2: Find the intercepts of y = 2x - 6

Example 3: Intercepts with a fraction, 5x - 4y = 10

Example 4: Graph 2x - 3y = 6 using intercepts

Example 5: Word problem with a gift card

Common Mistakes

Intercepts from a Graph or a Table

Practice Problems

X and Y Intercepts FAQ

How do you find the x-intercept?

How do you find the y-intercept?

Can a line have no x-intercept?

Is an intercept a number or a point?

Summary

Example 1: Find the intercepts of $3x + 2y = 12$

Example 2: Find the intercepts of $y = 2x - 6$

Example 3: Intercepts with a fraction, $5x - 4y = 10$

Example 4: Graph $2x - 3y = 6$ using intercepts