For jasonsteakums69

Start by assuming $$ad = bc.$$ Our goal is to show

$$\dfrac{ax + b}{cx + d} = \dfrac{\dfrac{x}{d} + \dfrac{1}{c}}{\dfrac{x}{b} + \dfrac{1}{a}}$$ Notice that in the first term of the numerator of our goal, we have $$x/d.$$ But the first term of the numerator of our current expression is $$ax.$$ This tells me that multiplying the numerator and denominator of our current expression by $$1/(ad)$$ is likely a step in the right direction. So now we have

$$\dfrac{(ax + b)\dfrac{1}{ad}}{(cx + d)\dfrac{1}{ad}}$$

Distributing gives us

$$\dfrac{\dfrac{ax}{ad} + \dfrac{b}{ad}}{\dfrac{cx}{ad} + \dfrac{d}{ad}}$$

But remember, we started by assuming $$ad = bc,$$ so we can substitute to obtain

$$\dfrac{\dfrac{ax}{ad} + \dfrac{b}{bc}}{\dfrac{cx}{bc} + \dfrac{d}{ad}}$$

Now we just cancel

$$\dfrac{\dfrac{x}{d} + \dfrac{1}{c}}{\dfrac{x}{b} + \dfrac{1}{a}}$$