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.