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}.