With the system of equations you posted you only need 2 of them to make it solvable. You don't want to include more than 2 of them at once, otherwise, you'll get a "least-squares" solution that tries to approximate solutions for each of the equations which is rubbish for cryptographic applications.
So basically with Gaussian elimination, you want to reduce the left-hand side to
Which will then give you two equations on the right for your target variables. But it will be less complicated if you already know some variables so that you can convert them to constants.
You'd only need the full 2x2 system if you're trying to find r1 and s1. The rest of the variables can be solved independently with just one equation.
