# Slope of a Line Through Two Points

Let $P\left( {{x_1},{y_1}} \right)$ and $Q\left( {{x_2},{y_2}} \right)$ be any two points on the given line $l$. Also let $\alpha$ be the inclination of the line $l$ as shown in the given diagram. From point $P$ draw $PM$ perpendicular to the X-axis, and from $Q$ draw $QN$ perpendicular to the X-axis.

Now from the given diagram, consider the triangle $\Delta PQR$. From the definition of a slope we take

Now by the definition we can use $m$ instead of $\tan \alpha$, and we get the slope of a line through two points:

Example: Find the slope of a straight line passing through the pair of points $\left( {3,4} \right)$ and $\left( {7,9} \right)$.

Here we have two points. Suppose that $P\left( {3,4} \right) = \left( {{x_1},{y_1}} \right)$ and $Q\left( {7,9} \right) = \left( {{x_2},{y_2}} \right)$. Now using the formula of a slope passing through two given points of the straight line:

By substituting the above points in the formula we get the slope of the line as

Here $m = \frac{5}{4}$ is the required slope of the line.