# Converting Linear Equations in Standard Form to the Slope Intercept 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 slope intercept form, by definition the slope-intercept is written as $$y = mx + c$$.

To convert the standard form to the slope intercept form, take the standard equation and separate the variable $$y$$ on the left hand side as follows:

\[\begin{gathered} ax + by + c = 0 \\ \Rightarrow by = – ax – c \\ \Rightarrow y = \frac{{ – ax – c}}{b} \\ \Rightarrow y = – \frac{a}{b}x – \frac{c}{b} \\ \end{gathered} \]

Compare this equation with the slope intercept form $$y = mx + c$$ where the slope is $$ – \frac{a}{b}$$ and the y-intercept is $$ – \frac{c}{b}$$.

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

The equation of the line in standard form is $$2x + 5y – 6 = 0$$

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

Compare this with equation $$y = – \frac{2}{5}x + \frac{6}{5}$$ with the slope intercept form $$y = mx + c$$. Here the slope is $$ – \frac{2}{5}$$ and the y-intercept $$\frac{6}{5}$$.