Slope Intercept Form of Equation of Straight Line

Consider the straight line l. Let P\left(  {x,y} \right) be any point on the given line l. Suppose that \alpha be the inclination of the line l as shown in the given diagram, i.e. \tan \alpha   = m
Take c as a Y-intercept of the straight line because it cutting the Y-axis at the point Q\left( {0,c} \right), i.e. OQ = c = Y-intercept.
From point P draw PM perpendicular on X-axis also from Q draw QR perpendicular on PM.


Now from the given diagram, consider the triangle \Delta PQR, i.e. m\angle RQP = \alpha , by the definition of slope we take

\begin{gathered} \tan \alpha   = \frac{{PR}}{{QR}} = \frac{{PM - RM}}{{OM}} \\ \Rightarrow \tan \alpha  = \frac{{PM - OQ}}{{OM}} \\ \Rightarrow \tan \alpha  = \frac{{y - c}}{x} \\ \end{gathered}

Now by definition we can use m instead of \tan \alpha , we get

\begin{gathered} \Rightarrow m = \frac{{y - c}}{x} \\ \Rightarrow mx = y - c \\ \end{gathered}

\boxed{y  = mx + c}

Which is the equation of straight line having slope m and Y-intercept c.

NOTE: It may be noted that if the line passes through the origin \left( {0,0} \right), then take Y-intercept is equal to zero i.e. c = 0, so the equation of straight line becomes y = mx.

Example: Find the equation of straight line having slope 3 and Y-intercept is equal to 8.
Here we have slope m =  3 and Y-intercept c = 8
Now using the formula of straight line having slope and Y-intercept

y =  mx + c

Substitute the above values in the formula to get the equation of straight line

y =  3x + 8

3x  - y + 8 = 0

Which is the required equation of straight line.