The Slope Intercept Form of the Equation of a Straight Line

Consider the straight line l. Let P\left( {x,y} \right) be any point on the given line l. Suppose that \alpha is 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 cuts the Y-axis at the point Q\left( {0,c} \right), i.e. OQ = c = Y-intercept.

From point P draw PM perpendicular to the X-axis, and from Q draw QR perpendicular to the PM.


slope-intercept-form

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 the definition we can use m instead of \tan \alpha , and we get

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


\boxed{y = mx + c}

This is the equation of a straight line having the 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 the Y-intercept is equal to zero i.e. c = 0, so the equation of a straight line becomes y = mx.

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

Now using the formula of straight line having the slope and Y-intercept

y = mx + c

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

y = 3x + 8


3x - y + 8 = 0

This is the required equation of a straight line.