# How to use Intercepts

An intercept in algebra refers to the point at which a line or curve intersects with a coordinate axis. The $$x$$-intercept is the point at which the line or curve crosses the $$x$$-axis, and the $$y$$-intercept is the point at which it crosses the $$y$$-axis.

## Step-by-step to use intercepts

To find out how to use intercepts, follow the step-by-step guide below:

1. To find the $$x$$-intercept of a linear equation in slope-intercept form $$(y=mx+b)$$, set y equal to zero and solve for $$x$$. This will give you the $$x$$-coordinate of the $$x$$-intercept.
2. To find the $$y$$-intercept, set $$x$$ equal to zero and solve for $$y$$. This will give you the $$y$$-coordinate of the y-intercept.
3. For example, the equation $$y=2x+1$$ has an $$x$$-intercept at $$(-0.5, 0)$$ and a $$y$$-intercept at $$(0, 1)$$.
4. To find the intercepts of a non-linear equation, you can set the equation equal to zero and solve for $$x$$ or $$y$$, depending on what type of intercept you are looking for. This may require factoring, completing the square, or using other algebraic techniques.

### Using Intercepts – Example 1:

Find the $$x$$-intercept and $$y$$-intercept of the equation $$y=2x+1$$.

Solution:

To find the $$x$$-intercept, we set $$y$$ equal to zero and solve for $$x$$:

$$y = 0 = 2x + 1$$ →$$x = -0.5$$

So, the $$x$$-intercept is $$(-0.5, 0)$$.

To find the $$y$$-intercept, we set $$x$$ equal to zero and solve for $$y$$:

$$y=2(0)+1=1$$

So the $$y$$-intercept is $$(0, 1)$$.

### Using Intercepts – Example 2:

Find the $$x$$-intercept and $$y$$-intercept of the equation $$y=-3x^2+4x -5$$.

Solution:

To find the $$x$$-intercepts we set $$y$$ equal to zero and solve for $$x$$:

$$y = -3x^2 + 4x – 5 = 0$$→ $$-3x^2 + 4x – 5 = 0$$ →$$x^2 – (\frac{4}{3})x + (\frac{5}{3}) = 0$$

To solve this equation we can factor it or use the quadratic formula. Factoring gives $$(x – (\frac{5}{3}))(x – (\frac{4}{3}))=0$$ so $$x=\frac{5}{3}$$ or $$\frac{4}{3}$$

so the $$x$$ intercepts are $$(\frac{5}{3}, 0)$$ and $$(\frac{4}{3}, 0)$$

since it’s a parabola, it doesn’t have any $$y$$-intercept.

## Exercises for Using Intercept

1. Find the $$x$$-intercept and $$y$$-intercept of the equation $$3y=6x-9$$.
2. Find the $$x$$-intercept and $$y$$-intercept of the equation $$y=2x^2 + 7x – 8$$.
1. $$\color{blue}{x-intercept=(\frac{3}{2}, 0), y-intercept= (0, -3)}$$
2. $$\color{blue}{x-intercepts=(-4, 0) and (1, 0), y-intercept= (0, -8)}$$

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