X-Intercepts and Y-Intercepts

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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.

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.

InterceptWhat to doWrite it as
x-interceptSet y = 0 and solve for x(a, 0)
y-interceptSet x = 0 and solve for y(0, b)
  1. To find the x-intercept, replace y with 0.
  2. Solve for x. Call the answer a.
  3. Write the x-intercept as the point (a, 0).
  4. To find the y-intercept, replace x with 0.
  5. Solve for y. Call the answer b.
  6. 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

For the x-intercept, set y = 0.

\begin{aligned} 3x + 2(0) &= 12 \\ 3x &= 12 \\ x &= 4 \end{aligned}

For the y-intercept, set x = 0.

\begin{aligned} 3(0) + 2y &= 12 \\ 2y &= 12 \\ y &= 6 \end{aligned}

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

For the y-intercept, set x = 0: y = 2(0) - 6 = -6. So the y-intercept is (0, -6).

For the x-intercept, set y = 0 and solve.

\begin{aligned} 0 &= 2x - 6 \\ 6 &= 2x \\ 3 &= x \end{aligned}

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

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

For the x-intercept, set y = 0: 5x = 10, so x = 2.

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

\begin{aligned} 5(0) - 4y &= 10 \\ -4y &= 10 \\ y &= \frac{10}{-4} = -\frac{5}{2} \end{aligned}

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

Set y = 0: 2x = 6, so x = 3. Plot (3, 0).

Set x = 0: -3y = 6, so y = -2. Plot (0, -2).

Draw a line through the two points. To check, pick one more point. If x = 6, then 12 - 3y = 6, so y = 2. The point (6, 2) lands on the line, so the graph is right.

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.

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.

For the B-intercept, set c = 0: B = 60 - 4(0) = 60. Before buying any coffee, the card has $60.

For the c-intercept, set B = 0.

\begin{aligned} 0 &= 60 - 4c \\ 4c &= 60 \\ c &= 15 \end{aligned}

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

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

MistakeExample of the errorFix
Setting the wrong variable to 0For 3x + 2y = 12, setting x = 0 and calling 6 the x-interceptFor the x-intercept, set y to 0. That gives x = 4.
Writing the point in the wrong orderWriting the y-intercept 6 as (6, 0)The y-intercept has x = 0, so it is (0, 6).
Losing a negative signFrom -4y = 10, writing y = \tfrac{5}{2}Divide by -4: y = -\tfrac{5}{2}.
Trusting only two pointsPlotting one intercept in the wrong place and drawing the wrong lineFind 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.

  1. x + y = 7
  2. 2x + 5y = 10
  3. 4x - 3y = 24
  4. y = 3x + 9
  5. y = -2x + 5
  6. -6x + 2y = 18
  7. 3x + 4y = 6
  8. y = -4
  9. Use the table to find both intercepts: x: -2, 0, 2, 4 and y: -6, -3, 0, 3.
  10. 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
  1. Set y = 0: x = 7. Set x = 0: y = 7. Intercepts: (7, 0) and (0, 7).
  2. 2x = 10, so x = 5. 5y = 10, so y = 2. Intercepts: (5, 0) and (0, 2).
  3. 4x = 24, so x = 6. -3y = 24, so y = -8. Intercepts: (6, 0) and (0, -8).
  4. 0 = 3x + 9, so x = -3. When x = 0, y = 9. Intercepts: (-3, 0) and (0, 9).
  5. 0 = -2x + 5, so 2x = 5 and x = \tfrac{5}{2}. When x = 0, y = 5. Intercepts: \left(\tfrac{5}{2}, 0\right) and (0, 5).
  6. -6x = 18, so x = -3. 2y = 18, so y = 9. Intercepts: (-3, 0) and (0, 9).
  7. 3x = 6, so x = 2. 4y = 6, so y = \tfrac{3}{2}. Intercepts: (2, 0) and \left(0, \tfrac{3}{2}\right).
  8. This is a horizontal line. The y-intercept is (0, -4). There is no x-intercept, because y is never 0.
  9. The row with x = 0 has y = -3, so the y-intercept is (0, -3). The row with y = 0 has x = 2, so the x-intercept is (2, 0).
  10. Set t = 0: V = 900, so (0, 900) means the pool starts with 900 gallons. Set V = 0: 30t = 900, so t = 30. The point (30, 0) means the pool is empty after 30 minutes.

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

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