# 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](/content/math/geometry/x-and-y-axis/index.html), 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](/content/math/geometry/slope-intercept-form-of-a-line/index.html) is so useful: the [y-intercept](/content/math/geometry/y-intercept/index.html) 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:

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