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 $– \frac{c}{b}$.