# How to Find the Slope of a Line Given the Two Coordinates of its Points

## Where to Begin

First, let us define what the slope of a line is. The slope of a line is the change in y divided by the change in x or rise over run. You can also think of slope as the "steepness" of the line. Finding the slope of a line is simple if you're given the coordinates of any two points. Let's start with an example:

Let's say we have two points (X_{1},Y_{1}), (X_{2},Y_{2}) such that, X_{1}= 1, Y_{1 }= 4 X_{2 }= 3 and Y_{2 }= 8. Next, we simply apply the equation for the slope of a line which is_{}

m = Y_{2}-Y_{1} ⁄ X_{2}-X_{1.}

Then we plug in the values for X_{2}_{, }X_{1,}Y_{1, }and Y_{2}, and we obtain the following:

m = (8-4) ⁄ (3-1) = 4 ⁄ 2 = 2. Therefore the slope of this line is 2.

Here are some things to remember:

1) When you are given two pairs of coordinates (X_{2},Y_{2}) and (X_{1},Y_{1}), it does not matter which numbers replace which coordinates as long as you are consistent. For example, if we take the example above, I could have written Y_{2}=4, Y_{1=}8, X_{2}=1 and X_{1}=3. The resulting equation would look like this:

m = (4-8) ⁄ (1-3) = -4 ⁄ -2 = 2.

Notice that the slope is the same as before.

2) If X_{2 = }X_{1, }the denominator evaluates to 0. Therefore, the line will always be vertical because the X coordinate is always the same. Also, a vertical line is said to have no slope because division by zero is undefined. _{}

For example,

m = (17-5) ⁄ ( 5 - 5) = 12 ⁄ 0 = undefined slope.

3) If Y_{2} = Y_{1}, that means the slope = 0.That does not mean that your line has no slope; It simply means that your graph contains the slope of a horizontal line where all points along the Y coordinate are the same.

For example,

m = (13-13) ⁄ (4-2) = 0 ⁄ 2 = 0.

## Slope of a Line and its Formula

## Graph of Horizontal and Vertical Lines

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