# Converting Linear Equations in Standard Form to Normal 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 normal form \[x\cos \alpha + y\sin \alpha = p\]

The procedure of converting the equation into the normal form is as follows:

\[\begin{gathered} ax + by + c = 0 \\ \Rightarrow – ax – by = c \\ \end{gathered} \]

Dividing both sides of the equation by $$ \pm \sqrt {{a^2} + {b^2}} $$ we have

\[ \Rightarrow – \frac{a}{{ \pm \sqrt {{a^2} + {b^2}} }}x – \frac{b}{{ \pm \sqrt {{a^2} + {b^2}} }}y = \frac{c}{{ \pm \sqrt {{a^2} + {b^2}} }}\]

This is the equation of the line in normal form, where $$\frac{c}{{ \pm \sqrt {{a^2} + {b^2}} }}$$ is the length of the normal form origin of the line. Since the length of the normal form origin to the line must be positive, so:

(i) $$ – \frac{a}{{\sqrt {{a^2} + {b^2}} }}x – \frac{b}{{\sqrt {{a^2} + {b^2}} }}y = \frac{c}{{\sqrt {{a^2} + {b^2}} }}$$ is the normal form of the line if $$c > 0$$.

(ii) $$\frac{a}{{\sqrt {{a^2} + {b^2}} }}x + \frac{b}{{\sqrt {{a^2} + {b^2}} }}y = \frac{c}{{ – \sqrt {{a^2} + {b^2}} }}$$ is the normal form of line if $$c < 0$$.

Example: Convert the equation $$2x + 5y – 6 = 0$$ into normal form.

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

\[ \Rightarrow 2x + 5y = 6\]

Dividing both sides of the equation by the normalizing factor $$\sqrt {{{\left( 2 \right)}^2} + {{\left( 5 \right)}^2}} = \sqrt {29} $$, we have

\[\Rightarrow \frac{{2x}}{{\sqrt {29} }} + \frac{{5y}}{{\sqrt {29} }} = \frac{6}{{\sqrt {29} }}\]

Compare with the normal form of a line $$x\cos \alpha + y\sin \alpha = p$$, where $$\cos \alpha = \frac{2}{{\sqrt {29} }}$$, $$\sin \alpha = \frac{5}{{\sqrt {29} }}$$and $$p = \frac{6}{{\sqrt {29} }}$$.