Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Mini Lesson Question

Question #3: Find x- and y-intercepts

Select the x- and y- intercepts for the line

2x - 3y = 12

.

x-intercept $(0, 6)$ ; y-intercept $(0, 6)$
x-intercept $(-4, 0)$ ; y-intercept $(-4, 0)$
x-intercept $(0, -4)$ ; y-intercept $(6, 0)$
x-intercept $(6, 0)$ ; y-intercept $(0, -4)$

Finding Intercepts of a Line

Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that when graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing the line might use different sets of three points.

At first glance, their two lines might not appear to be the same, since they would have different points labeled. But if all the work was done correctly, the lines should be exactly the same. One way to recognize that they are indeed the same line is to look at where the line crosses the $x$ -axis and the $y$ -axis. These points are called the intercepts of a line.

Let’s look at the graphs of the lines.

The figure shows four graphs of different equations. In example a the graph of 2 x plus y plus 6 is graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The points (0, 6) and (3, 0) are plotted and labeled. A straight line goes through both points and has arrows on both ends. In example b the graph of 3 x minus 4 y plus 12 is graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The points (0, negative 3) and (4, 0) are plotted and labeled. A straight line goes through both points and has arrows on both ends. In example c the graph of x minus y plus 5 is graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The points (0, negative 5) and (5, 0) are plotted and labeled. A straight line goes through both points and has arrows on both ends. In example d the graph of y plus negative 2 x is graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The point (0, 0) is plotted and labeled. A straight line goes through this point and the points (negative 1, 2) and (1, negative 2) and has arrows on both ends.

First, notice where each of these lines crosses the x-axis.

Now, let’s look at the points where these lines cross the y-axis.

Figure	The line crosses the $x$ -axis at:	Ordered pair for this point	The line crosses the $y$ -axis at:	Ordered pair for this point
Figure (a)	3	(3,0)	6	(0,6)
Figure (b)	4	(4,0)	-3	(0,-3)
Figure (c)	5	(5,0)	-5	(0,-5)
Figure (d)	0	(0,0)	0	(0,0)
General figure	a	(a,0)	b	(0,b)

For each line, the y-coordinate of the point where the line crosses the $x$ -axis is zero. The point where the line crosses the $x$ -axis has the form (a, 0) and is called the $x$ -intercepts of the line. The $x$ -intercepts occurs when y is zero.

In each line, the x-coordinate of the point where the line crosses the $y$ -axis is zero. The point where the line crosses the $y$ -axis has the form $(0, b)$ and is called the $y$ -intercepts of the line. The $y$ -intercepts occurs when x is zero.

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

We can use this method to find the intercepts of the equation $2 x + y = 8$

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

Step 1 - Let $y = 0$ . $2 x + 0 = 8$

Step 2 - Simplify. $2 x = 8$

Step 3 - Simplify. $x = 4$

Step 4 - The $x$ -intercept is: $(4, 0)$

To find the $y$ -intercepts, let $x = 0$ .

Step 1 - Let $y = 0$ . $2 (0) + y = 8$

Step 2 - Simplify. $0 + y = 8$

Step 3 - Simplify. $y = 8$ The $x$ -intercept is: $(0, 8)$

The intercepts are the points (4,0) and (0,8).

Try it

Try It: Finding Intercepts of a Line

Find the intercepts of the equation $3 x + y = 12$ .

Here is how to find each intercept:

To find the $x$ -intercepts, let $y = 0$ .

$3 x + y = 12$

Step 1 - Let $y = 0$ . $3 x + 0 = 12$

Step 2 - Simplify. $3 x = 12$ $x = 4$

Step 3 - The $x$ -intercept is: $(4, 0)$

To find the $y$ -intercepts, let $x = 0$ .

$3 x + y = 12$

Step 1 - Let $x = 0$ . $3 (0) + y = 12$

Step 2 - Simplify. $0 + y = 12$

$y = 12$

Step 3 - The $y$ -intercept is $(0, 12)$

The intercepts are the points $(4, 0)$ and $(0, 12)$ .

Check Your Understanding

Select the x- and y- intercepts for the line $- 5 y = 10 x - 20$ .

Multiple Choice:

$x$ -intercepts $(0, 4)$ ; $y$ -intercepts $(2, 0)$
$x$ -intercepts $(2, 0)$ ; $y$ -intercepts $(0, 4)$
$x$ -intercepts $(0, 2)$ ; $y$ -intercepts $(4, 0)$
$x$ -intercepts $(4, 0)$ ; $y$ -intercepts $(0, 2)$

$x$ -intercepts $(2, 0)$ ; $y$ -intercepts $(0, 4)$ .

Check yourself: To find the $x$ -intercepts, let $y = 0$ and solve for $x$ : $- 5 (0) = 10 x - 20$ ; $x = 2$ . To find the $y$ -intercepts, let $x = 0$ and solve for $y$ : $- 5 y = 10 (0) - 20$ ; $y = 4$ .

Video: Finding Intercepts

Khan Academy: Finding Intercepts

Watch this video to see how to find the x- and $y$ -intercepts of an equation.

Access multimedia content

Find the X- andY-intercepts: Mini-Lesson Review

Question #3: Find x- and y-intercepts

Finding Intercepts of a Line

Try It: Finding Intercepts of a Line

Check Your Understanding

Video: Finding Intercepts