General Form of Equation of a Line

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

ax  + by + c = 0


Where a and b are any constants and also 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 of the form

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


Which is the equation of straight line parallel to Y-axis. Thus, the line is parallel to 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 of the form

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


Which is the equation of straight line parallel to X-axis. Thus, the line is parallel to 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 of the form

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


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

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