The

** ***x*-intercept is the point where the graph intersects the

*x*-axis and the

*y*-intercept is the point where the graph intersects the

*y*-axis. These points have the form (

*x*, 0) and (0,

*y*) respectively.

To find the

*x*- and

*y*-intercepts algebraically, we use the fact that all

*x*-intercepts have a

*y*-value of zero and all

*y*-intercepts have an

*x*-value of zero. For example,

**Graph:** 3*x* − 5*y* = 15

**Tip 1: **To find the

*y*-intercept, set

*x* = 0 and determine the corresponding

*y*-value. Similarly, to find the

*x*-intercept we set

*y* = 0 and determine the corresponding

*x*-value.

Keep in mind that the intercepts are ordered pairs and not numbers. In other words, the

*x*-intercept is not

*x* = 5 but rather (5, 0).

Two points determine a line. If we find the

*x*- and

*y*-intercepts, then we can use them to graph the line. As you can see, they are fairly easy to find. Plot the points and draw a line through them with a straightedge.

Done. Let’s do another one.

**Graph:** *y* = −*x* + 9

We begin by finding the

*x*-intercept.

The

* x*-intercept is (3, 0).

The

*y*-intercept is (0, 9). Now graph the two points.

**Tip 2:** Use

Desmos.com to check your answer – it’s totally free. Just type in the equation.

This is a nice and easy method for determining the two points you need for graphing a line. In fact, we will use this exact technique for finding intercepts when we study the graphs of all the conic sections later in our study of Algebra.

**Graph **−4

*x* + 3

*y* = 12 using the intercepts.

**Graph** −4

*x* + 2

*y* = −6 using the intercepts.

**Graph ***y* = −5x +15 using the intercepts.

**Graph** *y* = −3/4

*x *+ 9 using the intercepts.

This brings us to one of the most popular questions in linear graphing. Do all lines have

*x*- and

*y*-intercepts? The answer is NO. Horizontal lines, of the form

*y* =

*b*, do not necessarily have

*x*-intercepts. Vertical lines, of the form

*x* =

*a*, do not necessarily have

*y*-intercepts.

**Graph** *y* = 3.

**Graph** *x* = −2.

Many students this method, but I will tell you, there is a better way. Even less work... [ Graph Lines using Slope and Intercepts ] Read on!

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