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}.