Converting Linear Equations in Standard Form to Two Points Form

The general equation or standard equation of a straight line is:
\[ax + by + c = 0\]

Where $$a$$ and $$b$$ are constants and either $$a \ne 0$$ or $$b \ne 0$$.

Putting $$y = 0$$ in the above standard equation of a line, we have
\[\begin{gathered} ax + b\left( 0 \right) + c = 0 \\ \Rightarrow ax = – c \\ \Rightarrow x = – \frac{c}{a} \\ \end{gathered} \]

This shows that the line is passing through the point $$\left( { – \frac{c}{a},0} \right)$$.

Now, similarly, by putting $$x = 0$$ in the same equation of a straight line, we have
\[\begin{gathered} a\left( 0 \right) + by + c = 0 \\ \Rightarrow by = – c \\ \Rightarrow y = – \frac{c}{b} \\ \end{gathered} \]

This shows that the line is passing through the point $$\left( {0, – \frac{c}{b}} \right)$$.

Now the equation of a straight line passing through two points $$\left( { – \frac{c}{a},0} \right)$$ and $$\left( {0, – \frac{c}{b}} \right)$$ is
\[\begin{gathered} \frac{{y – 0}}{{ – \frac{c}{b} – 0}} = \frac{{x – \left( { – \frac{c}{a}} \right)}}{{0 – \left( { – \frac{c}{a}} \right)}} \\ \Rightarrow \frac{y}{{ – \frac{c}{b}}} = \frac{{x + \frac{c}{a}}}{{\frac{c}{a}}} \\ \Rightarrow \frac{y}{{ – \frac{1}{b}}} = \frac{{x + \frac{c}{a}}}{{\frac{1}{a}}} \\ \Rightarrow – by = a\left( {x + \frac{c}{a}} \right) \\ \Rightarrow y = – \frac{a}{b}\left( {x + \frac{c}{a}} \right) \\ \end{gathered} \]

This is the equation of a line in two-point form transformed from its general form or standard form. It is noted that the transformation of the equation of a line from its general or standard form to a point slope form and a two points form is the same.