# Intercepts

## Definition - Intercepts

An intercept is the point where a line crosses something else. When we're talking about lines there are two that we're interested in: the *y*-intercept is the *y* coordinate of the point where the line crosses the *y*-axis and the *x*-intercept is *x* coordinate of the point where the line crosses the *x*-axis.

So what does an intercept look like? It's really nothing more than a number. If a line crosses the *x*-axis at the point (0, 3) then it's *x*-intercepts is 3. Similarly, if a point crosses the *y*-intercept at (0, -5) then then it's *y*-intercept would be -5. The applet to the right let's you work with these concepts. If you enter numbers for the two intercepts and click on the draw button, the graph will draw a line with the values that you entered.

Once you've drawn some lines, take a look at the points where they cross the axes and think about what the coordinates of those points are. You should see that the coordinates of the *y*-intercept also have a 0 for their *x* part and the coordinates of the *x*-intercept also have a zero for their *y* part. That tells you how you can go about finding the *x*- and *y*-intercepts if you've got an equation:

- To find the
*y*-intercept, substitute 0 for all the*x*'s in your equation and solve for*y*. - To find the
*x*-intercept, substitute 0 for all the*x*'s in your equation and solve for*x*.

# Example 1

**Find the x- and y-intercepts of the line whose equation is y = 3x + 4.**

First, we'll find the *x*-intercpet by substituting 0 for the *y* in the equation:

*y* = 3*x* + 4

0 = 3*x* + 4

-4 = 3*x**x* = -4/3

Then we'll find the *y*-intercpet by substituting 0 for the *x* in the equation:

*y* = 3*x* + 4*y* = 3 · 0 + 4*y* = 4

So we can conclude that the *x*-intercept of *y* = 3*x* + 4 is -4/3 and the *y*-intercept is 4.

# Example 3

**Find the x- and y-intercepts of the line whose equation is y = 14.**

Right away, you should notice that this equation is a little unusual since it doesn't have an *x* in it anywhere. That tells you that *y* is going to equal 14 for every value of *x*, including *x* = 0 so the *y*-intercept must be 14.

By the same argument, there are no *x* values that make *y* anything but 14, including 0 so the line never crosses the *x*-axis and doesn't have an *x*-intercept.

# Example 2

**Find the x- and y-intercepts of the line whose equation is y + x = 2y - 3x + 1.**

This may look a little confusing because there's more than one *x* and more than one *y* in the equation. Don't let that bother you. You can start by simpifying the equation if you're comfortable with that but it isn't necessary. You can still follow the same steps as we did in the previous example. First, we'll find the *x*-intercpet by substituting 0 for the *y* in the equation:

*y* + *x* = 2*y* - 3*x* + 1

0 + *x* = 2 · 0 - 3*x* + 1*x* = 3*x* + 1

-2*x* = 1*x* = -1/2

Then we'll find the *y*-intercpet by substituting 0 for the *x* in the equation:

*y* + *x* = 2*y* - 3*x* + 1*y* + 0 = 2*y* - 3 · 0 + 1*y* = 2*y* + 1

-*y* = 1*y* = -1

So we can conclude that the *x*-intercept of *y* + *x* = 2*y* - 3*x* + 1 is -1/2 and the *y*-intercept is -1.

# Example 4

**Find the x- and y-intercepts of the line whose equation is x = -3.**

This is the other side of Example 3. Because the line's coordinates all have -3 as their *x* value, including the one where *y* = 0, the *x*-intercept must be -3. On the other hand, since the line has no coordinates where *x* = 0, it has no *y*-intercept.