Slope measures the steepness of a line. It's measured as the ratio of the amount of horizontal change to the amount of vertical change, or "Rise over Run." You can use a graph to find slope by counting spaces to find the rise and run.
There are 4 different types of slope based on the direction of the line. Check out the intro to slope lesson to learn more about each type.
You can use a graph to find the slope of a line, but it's possible to find the slope without one. If you're given two points that are on the line, there is a formula that you can use to find the slope of the line. You can always pull out graph paper to find the slope, but you may find that the formula for slope can be faster and easier.
Open the next tab to see the formula for the slope of a line:
The Formula for Slope
Remember, slope is "Rise over Run." On a graph when you find the rise, you're finding the vertical change. Subtracting the y-coordinates will give you the Rise without needing a graph. Subtracting the x-coordinates will give you the amount of horizontal change, the Run. When you use the formula, make sure to subtract the y-values in the numerator and subtract the x-values in the denominator. It doesn't matter which point you call Point 1 or Point 2 as long as you subtract in the same order at each step.
When using the formula, follow the steps below: Step 1: Label the points. Step 2: Plug the values into the formula. Step 3: Write the slope as a simplified fraction.
Example 1 Find the slope of the line through the points (2,5) and (4,8).
Step 1: Label the points. It doesn't matter which one you pick as "Point 1" and "Point 2." Remember the x's are listed first in an ordered pair and the y's are listed second.
Step 2: Plug in the values. Subtract the y's on the top, subtract the x's on the bottom. Make sure to subtract in the same order in the numerator and denominator.
Step 3: Make sure your answer is simplified. 3/2 cannot be reduced, so we leave the answer as 3/2.
Example 2 Find the slope of a line through the points (-5,6) and (7,2).
Step 1: Label the points. Remember the x's are always listed first in the ordered pairs and the y's are listed second in each ordered pair.
Step 2: Plug the values into the formula. Make sure to subtract the y's in the numerator and subtract the x's in the same order for the denominator. Be especially careful when dealing with negative numbers. 7 minus a negative 5 is the same as 7 plus a positive 5.
Step 3: Make sure your answer is simplified. In this case, we can divide the numerator and denominator by 4 to simplify the answer.
It does not matter if you put the negative sign out in front of the fraction or if you leave it in the numerator.
Zero Slope
A line with a slope of zero is just a flat, horizontal line. Its rise is 0 because it's not going up or down from left to right. If you pick any two points on a horizontal line, they will have the same y-value. If you plug the points into the slope formula, you'll end up with a zero in the numerator. 0 divided by any non-zero number is just equal to 0.
Example Find the slope of the line through the points (1,2) and (4,2). Step 1: Label the points. It doesn't matter which one you pick as "Point 1" and "Point 2." Remember the x's are listed first in an ordered pair and the y's are listed second.
Step 2: Plug the values into the slope formula. Subtract the y's on the top, subtract the x's on the bottom. Make sure to subtract in the same order in the numerator and denominator.
Step 3: Make sure the answer is simplified. In this case, 0 divided by 3 is just 0. Zero divided by any non-zero number will always be 0.
Take a look at this on a graph to verify the answer. The points (1,2) and (4,2) have the same y-values and form a horizontal line. We got a 0 in the numerator when we used the slope formula because the rise was 0. The line didn't go up or down from left to right. The value for the run doesn't really matter because 0 divided by any non-zero number is still 0.
Undefined Slope
A vertical line that goes straight up and down has an undefined slope. With a vertical line, the run will be 0 because the x-values are not changing. If you use two points on a vertical line and plug them into the formula for slope, you're going to end up with a 0 in the denominator of the fraction. Since you can't divide by 0, we say that the slope is undefined. Think of the word NO to help you remember:
Example Find the slope of the line through the points (3,-1) and (3,2).
Step 1: Label the points.
Step 2: Plug the values into the formula for slope.
Step 3: Make sure the slope is a simplified fraction. There's a problem here! You can't have a 0 in the denominator of a fraction. This means the slope is undefined.
Open the next tab to practice finding using the formula for slope on your own.