for what value of $y$ does $\sum_{k= 0}^n a_0 x^k = \sum_{k=0}^n y^k$

Question

for what value of $y$ does $$\sum_{k= 0}^n a_0 x^k = \sum_{k=0}^n y^k$$

This was just an idea I was playing around with. I tried solving

$$\frac{a_0(x^{n+1}-1)}{x-1} = \frac{y^{n+1}-1}{y-1} $$

This does not seem easy to solve algebraically.

It was easy enough for me to do as $n \to \infty$, as we get:

$$\frac{a_0}{1-x} = \frac1{1-y} \implies y = \frac{a_0 + x -1}{a_0}$$ However, I've had no luck in the finite case.