# Converting Linear Equations in Standard Form to the Intercepts Form

The general equation or standard equation of a straight line is:

\[ax + by + c = 0\]

In which, $$a$$ and $$b$$ are constants and either $$a \ne 0$$ or $$b \ne 0$$.

Now to convert this linear equation in standard form to the intercepts form, i.e. $$X$$-intercept and $$Y$$-intercept, by definition the intercepts form is written as \[\frac{x}{a} + \frac{y}{b} = 1\]

To convert an equation from standard form to intercepts form, take the constant value $$c$$ and move it to the left hand side. Then divide both sides of the equation by $$c$$ and $$1$$ on the right hand side as follows:

\[\begin{gathered} ax + by + c = 0 \\ \Rightarrow ax + by = – c \\ \end{gathered} \]

Divide both sides of the above equation by $$ – c$$:

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

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

__Example__**: **Convert the equation $$2x + 5y – 6 = 0$$ into the intercepts form.

We have the equation of a line in standard form as $$2x + 5y – 6 = 0$$

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

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