Three equations, three unknowns, and one constraint Suppose we have the following three equations:
$$
r_y = \frac{r_y}{2} + \frac{r_a}{2} \\
r_a = \frac{r_y}{2} + r_m \\
r_m = \frac{r_a}{2}
$$

We also have additional constraint for uniqueness:
$$
r_y + r_a + r_m = 1
$$

The solution is:
$$
r_y = r_a = \frac{2}{5}, \quad r_m = \frac{1}{5}
$$

I kindly as for any pointer how to solve such system of equations.
How to solve this can be a bit tricky, 
I will suggest this approach. First, get rid of the fraction form. Instead have this form of equations.

your first equation is $r_{y} - r_{a} = 0$, then your second equation $-r_{y} + 2r_{a} + 2 r_{m} = 0$ and finally your third one $-2r_{a} + 2 r_{m} = 0$. Then put it in the form of a matrix. Then use gauss elimination process to solve. Hope that help.
Your first equation yields, after simplifying, $r_y=r_a$.  With your constraints, along with your identity for $r_m$ in the third equation yields
$$2r_a+\frac{r_a}{2}=1$$