### Learning Objectives

- Find the Slope from Two Points
- Use the formula for slope to define the slope of a line through two points

## Finding the Slope from Two Points on the Line

You’ve seen that you can find the slope of a line on a graph by measuring the rise and the run. You can also find the slope of a straight line without its graph if you know the coordinates of any two points on that line. Every point has a set of coordinates: an *x*-value and a *y*-value, written as an ordered pair (*x*, *y*). The *x* value tells you where a point is horizontally. The *y* value tells you where the point is vertically.

Consider two points on a line—Point 1 and Point 2. Point 1 has coordinates [latex]\left(x_{1},y_{1}\right)[/latex] and Point 2 has coordinates [latex]\left(x_{2},y_{2}\right)[/latex].

The rise is the vertical distance between the two points, which is the difference between their *y*-coordinates. That makes the rise [latex]\left(y_{2}-y_{1}\right)[/latex]. The run between these two points is the difference in the *x*-coordinates, or [latex]\left(x_{2}-x_{1}\right)[/latex].

So, [latex] \displaystyle \text{Slope}=\frac{\text{rise}}{\text{run}}[/latex] or [latex] \displaystyle m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}[/latex]

In the example below, you’ll see that the line has two points each indicated as an ordered pair. The point [latex](0,2)[/latex] is indicated as Point 1, and [latex](−2,6)[/latex] as Point 2. So you are going to move from Point 1 to Point 2. A triangle is drawn in above the line to help illustrate the rise and run.

You can see from the graph that the rise going from Point 1 to Point 2 is 4, because you are moving 4 units in a positive direction (up). The run is [latex]−2[/latex], because you are then moving in a negative direction (left) 2 units. Using the slope formula,

[latex] \displaystyle \text{Slope}=\frac{\text{rise}}{\text{run}}=\frac{4}{-2}=-2[/latex].

You do not need the graph to find the slope. You can just use the coordinates, keeping careful track of which is Point 1 and which is Point 2. Let’s organize the information about the two points:

Name | Ordered Pair | Coordinates |
---|---|---|

Point 1 | [latex](0,2)[/latex] | [latex]\begin{array}{l}x_{1}=0\\y_{1}=2\end{array}[/latex] |

Point 2 | [latex](−2,6)[/latex] | [latex]\begin{array}{l}x_{2}=-2\\y_{2}=6\end{array}[/latex] |

The slope, [latex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{6-2}{-2-0}=\frac{4}{-2}=-2[/latex]. The slope of the line, *m*, is [latex]−2[/latex].

It doesn’t matter which point is designated as Point 1 and which is Point 2. You could have called [latex](−2,6)[/latex] Point 1, and [latex](0,2)[/latex] Point 2. In that case, putting the coordinates into the slope formula produces the equation [latex]m=\frac{2-6}{0-\left(-2\right)}=\frac{-4}{2}=-2[/latex]. Once again, the slope is [latex]m=-2[/latex]. That’s the same slope as before. The important thing is to be consistent when you subtract: you must always subtract in the same order [latex]\left(y_{2},y_{1}\right)[/latex]_{ }and [latex]\left(x_{2},x_{1}\right)[/latex].

### Example

What is the slope of the line that contains the points [latex](5,5)[/latex] and [latex](4,2)[/latex]?

The example below shows the solution when you reverse the order of the points, calling [latex](5,5)[/latex] Point 1 and [latex](4,2)[/latex] Point 2.

### Example

What is the slope of the line that contains the points [latex](5,5)[/latex] and [latex](4,2)[/latex]?

Notice that regardless of which ordered pair is named Point 1 and which is named Point 2, the slope is still 3.

### Example (Advanced)

What is the slope of the line that contains the points [latex](3,-6.25)[/latex] and [latex](-1,8.5)[/latex]?

Let’s consider a horizontal line on a graph. No matter which two points you choose on the line, they will always have the same *y*-coordinate. The equation for this line is [latex]y=3[/latex]. The equation can also be written as [latex]y=\left(0\right)x+3[/latex].

The following video shows more examples of finding the slope of a line given two points on the line: