#3**+1 **

Here is the answer you have provided.

It lost me after the first sentence ...

Melody Feb 27, 2022

#4**+1 **

I don't understand that answer, but algebraically it works out easily enough.

We need

\(\displaystyle \frac{a(ax+b)/(cx+d)+b}{c(ax+b)/(cx+d)+d}\equiv x.\)

Clearing the fraction within the fraction on the lhs,

\(\displaystyle \frac{a(ax+b)+b(cx+d)}{c(ax+b)+d(cx+d)}\equiv x. \)

For this to work we need three things to happen.

(1) the constant term on the top line should be zero, ie. ab + bd = b(a + d) = 0,

(2) the x coefficient on the bottom line should be zero, ie. ca + dc = c(a + d) = 0,

(3) The resulting fraction should cancel down to leave just x.

Applying (1) and (2) reduces the lhs to

\(\displaystyle \frac{x(a^{2}+bc)}{cb+d^{2}}\)

and this cancels down if

\(\displaystyle a^{2}=d^{2}.\)

\(\displaystyle a+d=0,( \text{ so }a=-d),\) meets all of the requirements.

Guest Feb 28, 2022