Converting Linear Equations in Standard Form to Two Points Form

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

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