Converting Linear Equations in Standard Form to Intercepts 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.
Now to convert this linear equation in standard form to intercepts form i.e. X-intercept and Y-intercept, by definition intercepts form is written as

\frac{x}{a}  + \frac{y}{b} = 1


The procedure of converting standard form to intercepts form, take the constant value c move on left hand side then dividing by c on both sides of equation to 1 on the right hand side as follows

\begin{gathered} ax + by + c = 0 \\ \Rightarrow ax + by =  - c \\ \end{gathered}


Dividing  - c on both sides of the above equation

\begin{gathered} \Rightarrow \frac{{ax}}{{ - c}} +  \frac{{by}}{{ - c}} = 1 \\ \Rightarrow \frac{x}{{ - \frac{c}{a}}} +  \frac{y}{{ - \frac{c}{b}}} = 1 \\ \end{gathered}


This is equation of line intercepts form with X-intercept  -  \frac{c}{a} and Y-intercept  -  \frac{c}{b}.

Example: Convert the equation 2x + 5y - 6 = 0 into intercepts form.
We have equation of line in standard form is 2x + 5y - 6 = 0

\begin{gathered} \Rightarrow 2x + 5y = 6 \\ \Rightarrow \frac{{2x}}{6} + \frac{{5y}}{6}  = 1 \\ \Rightarrow \frac{x}{3} +  \frac{{5y}}{{\frac{6}{5}}} = 1 \\ \end{gathered}


Compare with intercepts form \frac{x}{a} + \frac{y}{b} = 1, where X-intercept is 3 and Y-intercept \frac{6}{5}.

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