# 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 $$ – \frac{c}{b}$$.