Does anybody know how I could go about solving the following system of equations? I have as many variables as I have equations so it should be possible to find a single solution, however it is complicated by variables being multiplied by other variables and also modular arithmetic.
I need to solve for $a,b,c,x,y,z$. Constraints are that all these variables are positive integers $0$-$9$, so they each equal one of the numbers from the set $0,1,2,3,4,5,6,7,8,9$. Is there any way to solve this by hand without simply trying every combination?
$$\begin{align}7c + z = 4 \quad \textrm{(mod 10)}\\ 7b + zc + y = 0 \quad \textrm{(mod 10)}\\ 7a + zb + yc + x = 8 \quad \textrm{(mod 10)}\\ za + yb + xc = 7 \quad \textrm{(mod 10)}\\ ya + xb = 0 \quad \textrm{(mod 10)}\\ xa = 5 \quad \textrm{(mod 10)}\end{align}$$