Converting Linear Equations in Standard Form to Slope Intercept 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 slope intercept form, by definition slope-intercept is written as y = mx + c.
The procedure of converting standard form to slope intercept, 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 slope intercept form y = mx + c where slope is  - \frac{a}{b} and y-intercept is  - \frac{c}{b}.

Example: Convert the equation 2x + 5y - 6 = 0 into slope intercept form.
We have equation of 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 with equation y  =  - \frac{2}{5}x + \frac{6}{5} with slope intercept form y = mx + c. Where slope is  - \frac{2}{5} and y-intercept \frac{6}{5}.

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