# Understanding Line Slopes and the Slope Formula

The *slope* of a line on the coordinate plane basically tells you how steep the line is. If you know the rise and run of a line, you can calculate its slope using the slope formula.

Slope formula: The slope of a line containing two points, (x1, y1) and (x2, y2), is given by the following formula (a line’s slope is often represented by the letter m):

** Note:** It doesn’t matter which points you designate as (

*x*

_{1},

*y*

_{1}) and (

*x*

_{2},

*y*

_{2}); the math works out the same either way. Just make sure that you plug your numbers into the right places in the formula.

The *rise* is the “up distance,” and the *run* is the “across distance” shown in the above figure. To remember this, note that you *rise up* but you *run across,* and also that “rise” rhymes with “*y*‘s.”

Compare the following list with the second figure, which shows you that the slope of a line increases as the line gets steeper and steeper:

- A horizontal line has no steepness at all, so its slope is zero. A good way to remember this is to think about driving on a horizontal, flat road—the road has zero steepness or slope.
- A slightly inclined line might have a slope of, say, 1/5.
- A line at a 45-degree angle has a slope of 1.
- A steeper line could have a slope of 5.
- A vertical line (the steepest line of all) sort of has an infinite slope, but math people say that its slope is
*undefined.*(It’s undefined because with a vertical line, you don’t go across at all, and thus the*run*inwould be zero, and you can’t divide by zero). Think about driving up a vertical road: You can’t do it—it’s impossible. And it’s impossible to compute the slope of a vertical line.

The lines you see in the second figure have *positive* slopes (except for the horizontal and vertical lines). So what about lines with negative slopes? Actually, there are a couple of ways to distinguish the two types of slopes:

**Lines that go up to the right have a positive slope.**Going from left to right, lines with positive slopes go uphill.**Lines that go down to the right have a negative slope.**Going from left to right, lines with negative slopes go downhill.