General Form of the Equation of a Line

The general equation or standard equation of a straight line is given by

ax + by + c = 0

In this case, a and b are constants and either a \ne 0 or b \ne 0.

CASE – I: If a \ne 0 or b = 0, then the general equation or standard equation of  straight line ax + by + c = 0 can be written in the following form:

\begin{gathered} ax + \left( 0 \right)y + c = 0 \\ \Rightarrow ax = - c \\ \Rightarrow x = - \frac{c}{a} \\ \end{gathered}

This is the equation of a straight line parallel to the Y-axis. Thus, the line is parallel to the Y-axis if b = 0.

CASE – II: If a = 0 or b \ne 0, then the general equation or standard equation of straight line ax + by + c = 0 can be written in the following form:

\begin{gathered} \left( 0 \right)x + by + c = 0 \\ \Rightarrow by = - c \\ \Rightarrow y = - \frac{c}{b} \\ \end{gathered}

This is the equation of a straight line parallel to the X-axis. Thus, the line is parallel to the X-axis if a = 0.

CASE – III: If a \ne 0 or b \ne 0, then the general equation or standard equation of straight line ax + by + c = 0 can be written in the following form:

\begin{gathered} ax + by + c = 0 \\ \Rightarrow by = - ax - c \\ \Rightarrow y = - \frac{a}{b}x - \frac{c}{b} \\ \end{gathered}

This is the equation of a straight line in slope intercept form with a slope of  - \frac{a}{b} and a y-intercept of  - \frac{c}{b}.