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 consider $\alpha$ be the inclination of the line $l$ as shown in the given diagram. From point $P$ draw $PM$ perpendicular on X-axis also from $Q$ draw $QN$ perpendicular on X-axis.

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

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

Example: Find the slope of 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 slope passing through two given points of the straight line

Substitute the above points in the formula we the slope of line as

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