How do I solve this system of equations for $s_0$ and $s_1$?

I am trying to solve this system of equations for $s_0$ and $s_1$.

$$s_0 = s_0 (1 - a) + s_1 b \\ s_1 = s_0 a + s_1 (1 - b)$$

I have tried to solve each equation one at a time:

$$s_0 = s_0 (1 - a) + s_1 b \\ s_0 - s_0 (1 - a) = s_1 b\\ s_0 a = s_1 b \\$$

$$s_1 = s_0 a + s_1 (1 - b) \\ s_1 - s_1 (1 - b) = s_0 a \\ s_1 b = s_0 a$$

Unfortunately, I am only going in circles... eventually, I guessed correctly that $s_0 = \frac{b}{a + b}$ and $s_1 = \frac{a}{a + b}$. How do I actually solve it?

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You're quite correct that both equations are equivalent to $s_0a=s_1b$, and that isn't enough information to get a single solution. In fact, there are infinitely many solutions to this system.
If $a=b=0$, then both $s_0$ and $s_1$ can be anything. If $b=0$ and $a\neq 0$, then we must have $s_0=0$, but $s_1$ can be anything. If $b\neq 0$, then we must have $s_1=\frac{a}{b}s_0$, but $s_0$ can be anything.
I suspect you may have copied one or both of them down incorrectly, that the problem is mistaken, or that the problem mentioned that $a$ and $b$ aren't both $0$ and asked only for some solution.