linear over linear

So there is one situation. I hope you can use it sometimes.

Let numbers a b c d, and we want to simplify the expression $\frac{ax+b}{cx+d}$, which is a linear expression over a linear expression.

Follow these steps:

• $\frac{a}{c}$ will be the constant term of the result.
• Let $n$ be the value of x where the denominator is zero.
• $a(n)+b$ will be the numerator of the fraction part of the result. With the same denominator as the original expression.

For example, $\frac{3x+4}{6x+5}$ can be simplified with this method. The constant term of the result is $\frac{3}{6}=\frac{1}{2}$. Find $n=\frac{-5}{6}$. The numerator is $3(\frac{-5}{6})+4=\frac{3}{8}$. The result is $\frac{1}{2}+\frac{-5}{6(6x+5)}$.

For example, $\frac{2-3x}{x-1}$. The constant term is $\frac{-3}{1}=-3$. The numerator is $2-3(1)=-1$. The result is $-3+\frac{-1}{x-1}$

Note that substitution works too. For example, $\frac{3x^2+4}{6x^2+5}$ can be simplified using this shortcut if you just let $k=x^2$ and use this method.