The Slope Formula

We’ve seen that we can find the slope of a line on a graph by measuring the rise and the run. We can also find the slope of a line without its graph if we know the coordinates of any two points on that line. Every point has a set of coordinates: an [latex]x[/latex]-value and a [latex]y[/latex]-value, written as an ordered pair [latex](x, y)[/latex]. The [latex]x[/latex] value tells yus where a point is horizontally. The [latex]y[/latex] value tells us 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  [latex]y[/latex]-coordinates. That makes the rise [latex]\left(y_{2}-y_{1}\right)[/latex]. The run between these two points is the difference in the [latex]x[/latex]-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].

To see how the rise and run relate to the coordinates of the two points, let’s take a look at the slope of the line between the points [latex]\left(2,3\right)[/latex] and [latex]\left(7,6\right)[/latex].

Since we have two points, we will use subscript notation.

[latex]\stackrel{{x}_{1},{y}_{1}}{\left(2,3\right)}\stackrel{{x}_{2},{y}_{2}}{\left(7,6\right)}[/latex]

On the graph, we can count the rise of [latex]3[/latex]. The rise can also be found by subtracting the [latex]y\text{-coordinates}[/latex] of the points.

[latex]\begin{array}{c}{y}_{2}-{y}_{1}\\ 6 - 3\\ 3\end{array}[/latex]

We can count a run of [latex]5[/latex] from the graph. The run can also be found by subtracting the [latex]x\text{-coordinates}[/latex].

[latex]\begin{array}{c}{x}_{2}-{x}_{1}\\ 7 - 2\\ 5\end{array}[/latex]

We’ve shown that [latex]m={\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}}[/latex] is another version of [latex]m={\frac{\text{rise}}{\text{run}}}[/latex]. We can use this formula to find the slope of a line when we kmow two points on the line.

It is important to remember that the slope is the same no matter which order we select the points.  Previously, whenever we found the slope by looking at the graph, we always selected our points from left to right so that our run was always a positive value.  Now, let’s take a look at an example in which we select our points from right to left.

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.

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

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

We do not need the graph to find the slope. We 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:

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].

Remember, it doesn’t matter which point is designated as Point 1 and which is Point 2. We 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 we subtract: we must always subtract in the same order [latex]\left(y_{2}-y_{1}\right)[/latex] and [latex]\left(x_{2}-x_{1}\right)[/latex].