Let be the inclination of the straight line as shown in the given diagram. Let be any point on the given line . Consider another point , since the line passes through the point .
From draw perpendicular to the X-axis and from point draw also perpendicular to the X-axis. Also from draw perpendicular to on .
Now from the given diagram, consider the triangle . By the definition of a slope we take
Now by the definition we can use instead of , and we get
This is the equation of a straight line having slope and passing through the point .
NOTE: There is an alternate way to prove the slope point form of the equation of a line. Let be any point on the line.
Considering the slope intercept form of the equation of a line, we have
Since the line passing through the point above equation (i) becomes
Now subtracting equation (ii) from equation (i), we get
Example: Find the equation of a straight line having the slope and passing through the point
Here we have slope and point
Now the slope point form of the equation of a straight line is
Substitute the above values in the formula to get the equation of a straight line
This is the required equation of a straight line.