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

\begin{gathered} \tan \alpha   = \frac{{QR}}{{PR}} = \frac{{QN - RN}}{{MN}} \\ \Rightarrow \tan \alpha  = \frac{{QN - RN}}{{ON - OM}} \\ \Rightarrow \tan \alpha  = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} \\ \end{gathered}

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

\boxed{m  = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}}

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

m =  \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}

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

m =  \frac{{9 - 4}}{{7 - 3}} = \frac{5}{4}

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