Three equations, three unknowns, and one constraint

Question

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.

Answer 1

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.

Answer 2

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 $$2r_a+\frac{r_a}{2}=1$$