Matrices aren't necessary. Also, guess and check is still easier. Here's to algebra:
Basically, you have 4 variables and several equations, namely:
We only need to find the "smallest" set of a,b,x, and y, which makes this question possible. Otherwise, we would need 4 equations.
Using the first two equations, we can form a relation between a and b:
The smallest integer solution is a=3 and b=4. That means x=1. Now we solve for y.
In conclusion, a=3, b=4, x=1, and y=5. Note this is possible only because we took the smallest integer solution for 4a=3b. Otherwise, at best, we would have the coefficients in terms of one variable.
Thanks to Joe Shmo for pointing out an error.