General Form to Other Forms

Every general linear equation $Ax+By+C=0$ (with $A,B$ not both zero) can be rewritten in any of the other named forms, by comparing coefficients:

1. Slope-intercept form (needs $B\ne0$): dividing by $B$, $y=-\dfrac{A}{B}x-\dfrac{C}{B}$, so

   $\text{slope}=-\frac{A}{B}, \qquad y\text{-intercept}=-\frac{C}{B}.$

2. Intercept form (needs $A,B,C$ all nonzero): rewriting as $\dfrac{x}{-C/A}+\dfrac{y}{-C/B}=1$ gives

   $x\text{-intercept}=-\frac{C}{A}, \qquad y\text{-intercept}=-\frac{C}{B}.$

3. Normal form (needs $A,B\ne0$): comparing $Ax+By+C=0$ with $x\cos\alpha+y\sin\alpha=p$ (i.e. $x\cos\alpha+y\sin\alpha-p=0$), the coefficients must be proportional:

   $\frac{\cos\alpha}{A}=\frac{\sin\alpha}{B}=\frac{-p}{C}=\pm\frac{\sqrt{\cos^2\alpha+\sin^2\alpha}}{\sqrt{A^2+B^2}}=\pm\frac{1}{\sqrt{A^2+B^2}}.$

   The sign ($+$ or $-$) is chosen so that the resulting $p$ comes out positive (since a normal length is always taken positive); with that choice, $\cos\alpha=\frac{\mp A}{\sqrt{A^2+B^2}}, \qquad \sin\alpha=\frac{\mp B}{\sqrt{A^2+B^2}}, \qquad p=\frac{|C|}{\sqrt{A^2+B^2}}.$ …