Slope Formula

If we have the points $\left( x_1, y_1 \right)$ and $\left( x_2, y_2 \right)$, the slope of the line passing through them can be calculated using the following formula:

$$m = {{{y_2} – {y_1}} \over {{x_2} – {x_1}}}$$

but ${x_1} \ne {x_2}$ since a division of zero is not allowed.

If the slope $m$ is positive, the line is rising.
If the slope $m$ is negative, the line is falling.
If the slope $m$ is zero, it is a horizontal line.
If the slope $m$ is undefined, it is a vertical line.

Example 1: Find the slope of the line through the points $\left( { – 2,9} \right)$ and $\left( {2,5} \right)$.

Let $\left( { – 2,9} \right)$ be the first point while $\left( {2,5} \right)$ be the second.

$$\eqalign{
& \left( {{x_1},{y_1}} \right) = \left( { – 2,9} \right) \cr
& \left( {{x_2},{y_2}} \right) = \left( {2,5} \right) \cr} $$

That means the values are

$$\eqalign{
{x_1} &= – 2 \cr
{y_1} &= 9 \cr
{x_2} &= 2 \cr
{y_2} &= 5 \cr} $$

We substitute the values in the slope formula and then simplify.

$$\eqalign{
m &= {{{y_2} – {y_1}} \over {{x_2} – {x_1}}} \cr
&= {{5 – 9} \over {2 – \left( { – 2} \right)}} \cr
&= {{5 – 9} \over {2 + 2}} \cr
&= {{ – 4} \over 4} \cr
m &= – 1 \cr} $$

The final answer is $m=-1$. Since the slope is negative, the line is decreasing from left to right.

Example 2: Find the slope of the line through the points $\left( { -5,-3} \right)$ and $\left( {4,12} \right)$.

If $\left( { – 5, – 3} \right)$ is the first point, we have

$$\eqalign{
& {x_1} = – 5 \cr
& {y_1} = – 3 \cr} $$

This forces the second point to be $\left( {4,12} \right)$. Thus,

$$\eqalign{
{x_2} &= 4 \cr
{y_2} &= 12 \cr} $$

Plug the values into the formula then simplify.

$$\eqalign{
m &= {{{y_2} – {y_1}} \over {{x_2} – {x_1}}} \cr
&= {{12 – \left( { – 3} \right)} \over {4 – \left( { – 5} \right)}} \cr
&= {{12 + 3} \over {4 + 5}} \cr
&= {{15} \over 9} \cr
m &= {5 \over 3} \cr} $$

The final answer is $m ={\large{ {5 \over 3}}}$. Because the slope is positive, the line is increasing from left to right.