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

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