Mathematics » Introducing Graphs » Graphing with Intercepts

# Finding the Intercepts from an Equation of a Line

## Finding the Intercepts from an Equation of a Line

Recognizing that the $$x\text{-intercept}$$ occurs when $$y$$ is zero and that the $$y\text{-intercept}$$ occurs when $$x$$ is zero gives us a method to find the intercepts of a line from its equation. To find the $$x\text{-intercept,}$$ let $$y=0$$ and solve for $$x.$$ To find the $$y\text{-intercept},$$ let $$x=0$$ and solve for $$y.$$

### Definition: Find the x and y from the Equation of a Line

Use the equation 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.
xy
0
0

## Example

Find the intercepts of $$2x+y=6$$

We’ll fill in the figure below.

To find the x- intercept, let $$y=0$$:

 Substitute 0 for y. Add. Divide by 2. The x-intercept is (3, 0).

To find the y- intercept, let $$x=0$$:

 Substitute 0 for x. Multiply. Add. The y-intercept is (0, 6).

The intercepts are the points $$\left(3,0\right)$$ and $$\left(0,6\right)$$.

## Example

Find the intercepts of $$4x-3y=12.$$

### Solution

To find the $$x\text{-intercept,}$$ let $$y=0.$$

 $$4x-3y=12$$ Substitute 0 for $$y.$$ $$4x-3·0=12$$ Multiply. $$4x-0=12$$ Subtract. $$4x=12$$ Divide by 4. $$x=3$$

The $$x\text{-intercept}$$ is $$\left(3,0\right).$$

To find the $$y\text{-intercept},$$ let $$x=0.$$

 $$4x-3y=12$$ Substitute 0 for $$x.$$ $$4·0-3y=12$$ Multiply. $$0-3y=12$$ Simplify. $$-3y=12$$ Divide by −3. $$y=-4$$

The $$y\text{-intercept}$$ is $$\left(0,-4\right).$$

The intercepts are the points $$\left(-3,0\right)$$ and $$\left(0,-4\right).$$

$$4x-3y=12$$
xy
$$3$$$$0$$
$$0$$$$-4$$

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