Converting Linear Equations in Standard Form to Slope Point Form

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

ax + by + c = 0

Where a and b are constants and either a \ne 0 or b \ne 0.

Convert the standard equation of line ax + by + c = 0 into the slope point form

y - {y_1} = m\left( {x - {x_1}} \right)

If the standard form of the line passes through the point \left( {{x_1},{y_1}} \right), then this point must satisfy the standard equation, i.e.

\begin{gathered} a{x_1} + b{y_1} + c = 0 \\ \Rightarrow b{y_1} = - a{x_1} - c \\ \Rightarrow {y_1} = - \frac{{\left( {a{x_1} + c} \right)}}{b} \\ \end{gathered}

Comparing the equation with the slope intercept form, the slope is m = - \frac{a}{b}. The slope point form of the line is y - {y_1} = m\left( {x - {x_1}} \right).

Now we put the values of m and {y_1} in the slope point form y - {y_1} = m\left( {x - {x_1}} \right).

\begin{gathered} \Rightarrow y - \left( { - \frac{{a{x_1} + c}}{b}} \right) = - \frac{a}{b}\left( {x - {x_1}} \right) \\ \Rightarrow y + \frac{{a{x_1} + c}}{b} = - \frac{a}{b}x + \frac{a}{b}{x_1} \\ \Rightarrow y + \frac{a}{b}{x_1} + \frac{c}{b} = - \frac{a}{b}x + \frac{a}{b}{x_1} \\ \Rightarrow y = - \frac{a}{b}x + \frac{a}{b}{x_1} - \frac{a}{b}{x_1} - \frac{c}{b} \\ \Rightarrow y = - \frac{a}{b}x - \frac{c}{b} \\ \Rightarrow y = - \frac{a}{b}x - \frac{{ac}}{{ab}} \\ \Rightarrow y = - \frac{a}{b}\left( {x + \frac{c}{a}} \right) \\ \end{gathered}

This is the equation of a line in point-slope form transferred from its general form.