Find function that verifies the following relationships for any $x \in R$

Question

Find all the functions $f:\mathbb R\to\mathbb R $, $f(x) = ax + b$ for which the following relationship holds:

$$f(2x+3) - 2f(x-2) = 26$$

I have tried substituting $x - 2$ with $x$, substituting $2x-3$ with $x$, inputting $2$ into the relationship and solving for $f(0)$ (which I assumed to be equal to $b$), same for $-3/2$, and also simply inputting the arguments in the relationship and solving for a and b which gave me $7a - b = 26$

I have no idea how to proceed from now on, and I feel stupid. Help!

Answer 1

$f(x) = ax + b$ and$f(2x+3) - 2f(x-2) = 26$ means

$a(2x+3)+b-2(a(x-2)+b)=2ax+3a+b-2ax +4a-2b=7a-b=26$.

So the desired equations hold whenever $7a-b=26$.

One example would be when $a=3$ and $b=-5$, but there are many others.

Answer 2

We rewrite $f(2x+3)-2f(x-2)=26$ as $a(2x+3)+b-2(a(x-2)+b)=26$. The left side can be simplified by writing $a(2x+3)+b-2(a(x-2)+b)=2ax+3a+b-2ax+4a-2b=7a-b$. Thus $7a-b=26$ or equivalently $b=7a-26$ which gives the solutions $f(x)=ax+7a-26$.

We can also check to be sure of our answer $f(2x+3)-2f(x-2)=a(2x+3)+7a-26-2(a(x-2)+7a-26)=2ax+3a+7a-26-2ax+4a-14a+2\cdot 26=26$ after everything cancels out, which is what we expected.