X and Y Intercept: Definition, Formula & Examples

TL;DR

The x-intercept is where a graph crosses the x-axis (set y=0 and solve), and the y-intercept is where it crosses the y-axis (set x=0 and solve). This guide defines both, gives the find-it method, derives them from line forms, and works through six examples including a real distance problem.

Last updated on July 14, 2022

What Are The X And Y Intercept?

An intercept is a point where a graph crosses or touches one of the axes. There are two:

The x-intercept is the point where the graph meets the x-axis. At that point the height is zero, so it has the form (a,0).
The y-intercept is the point where the graph meets the y-axis. At that point the horizontal distance is zero, so it has the form (0,b).

The single fact that unlocks both: on the x-axis every point has y=0, and on the y-axis every point has x=0. That gives the whole method in one line: set the other variable to zero and solve. To find the x-intercept, set y=0; to find the y-intercept, set x=0.

The key idea to hold: an intercept is an axis crossing, and at every axis crossing one coordinate is zero. That zero is what you exploit to find it.

How To Find The X And Y Intercept

The method works for any equation in x and y, not just lines.

To find the x-intercept: substitute y=0 and solve for x.

To find the y-intercept: substitute x=0 and solve for y.

Take the line 5x+3y=15.

For the x-intercept, set y=0:

5x + 3(0) = 15

5x = 15

x = 3 ⇒ (3,0)

For the y-intercept, set x=0:

5(0) + 3y = 15

3y = 15

y = 5 ⇒ (0,5)

These axis crossings live on the x and y axis, the two reference lines of the coordinate plane.

Reading intercepts straight off a line's form

Two standard line forms hand you an intercept directly:

Slope-intercept form, y=mx+c: the constant c is the y-intercept, so the line crosses the y-axis at (0,c). This is exactly why the slope-intercept form is so useful: the y-intercept is sitting right there.
Intercept form, ( \frac{x}{a} + \frac{y}{b} = 1 ): here a is the x-intercept and b is the y-intercept, read off without any work.

Examples of X And Y Intercept

These build from a clean line to a slope-intercept read, a curve, and a real situation. Each problem statement is bold; the steps are plain.

Example 1

Find the x- and y-intercepts of 2x+y=6.

x-intercept, set y=0:

2x = 6 ⇒ x = 3 ⇒ (3,0)

y-intercept, set x=0:

y = 6 ⇒ (0,6)

Final answer: x-intercept (3,0); y-intercept (0,6).

Example 2

Find the y-intercept of y=4x−7.

For the y-intercept, set x=0 instead:

y = 4(0)−7 = −7 ⇒ (0,−7)

Final answer: the y-intercept is (0,−7).

Example 3

Find both intercepts of 3x−4y=12.

x-intercept, set y=0:

3x = 12 ⇒ x = 4 ⇒ (4,0)

y-intercept, set x=0:

−4y = 12 ⇒ y = −3 ⇒ (0,−3)

Final answer: x-intercept (4,0); y-intercept (0,−3).

Example 4

Find the x-intercepts of the curve y=x²−9.

Set y=0:

x²−9=0

x²=9

x=3 or x=−3

Final answer: two x-intercepts, (3,0) and (−3,0).

Example 5

A line has x-intercept (2,0) and y-intercept (0,5). Write its equation in intercept form.

Intercept form is ( \frac{x}{2} + \frac{y}{5} = 1 ).

Final answer: ( \frac{x}{2} + \frac{y}{5} = 1 ).

Example 6

A water tank starts with 60 litres and drains at 4 litres per minute. Write the volume line and find both intercepts, with meaning.

Let x be minutes and y be litres. The line is y=60−4x.

y-intercept, set x=0:

y=60 ⇒ (0,60)

x-intercept, set y=0:

0=60−4x ⇒ x=15 ⇒ (15,0)

Final answer: y-intercept (0,60) = starting volume; x-intercept (15,0) = the tank runs dry at 15 minutes.

Why X And Y Intercept Matter: "Where A Graph Meets The Axes Tells The Story"

Intercepts are usually the most readable points on any graph because each one strips away a variable. The y-intercept is "the value when nothing has happened yet"; the x-intercept is "the value at which the quantity hits zero."

Where intercepts earn their keep:

Start and end points. On a depreciation graph, the y-intercept is the purchase price and the x-intercept is when the asset's value reaches zero.
Break-even analysis. In business, the x-intercept of a profit line is the break-even quantity — make fewer and you lose money, make more and you profit.
Physics and motion. The y-intercept of a position-time line is the starting position; the x-intercept marks when the object returns to the reference point.

Common Mistakes With X And Y Intercept

Mistake 1: Setting the wrong variable to zero

Where it slips in: Wanting the y-intercept but setting y=0, or wanting the x-intercept but setting x=0.

Don't do this: Solving y=4x−7 with y=0 when the y-intercept was asked for.

The correct way: To find an intercept, set the other variable to zero.

Mistake 2: Writing an intercept as a single number when a point is expected

Where it slips in: Reporting "the x-intercept is 3" when the question wants the coordinate.

Mistake 3: Assuming a line always has both intercepts

Where it slips in: Trying to find an x-intercept for a horizontal line, or a y-intercept for a vertical line.

Conclusion

The x-intercept is where a graph crosses the x-axis; find it by setting y=0.
The y-intercept is where a graph crosses the y-axis; find it by setting x=0.
Intercepts are points, not single numbers, and a graph can have several x-intercepts or none.

Practice And Next Steps

Practice these problems to solidify your understanding:

Find both intercepts of 4x+5y=20.
Find the y-intercept of y=−2x+9 in one step.
Find the x-intercepts of y=x²−16.