I'm a bit puzzled as to why you would be asking. If you are dealing with systems of equations, you must have years of experience with solving individual equations and what you ask is just what you learned long ago:

If you multiply both sides of a true equation by a number, the equation is still true. Any equation is of the form A(x, y, z, ...)= B(x, y, z, ...), where x, y, z, ... are the variables, which is equivalent to A(x, y, z, ...)- B(x, y, z, ...)= 0. For any number, u, u times 0, u(0)= 0 so specifically, u(A(x, y, z, ...)- B(x, y, z, ...))= uA(x, y, z, ...)- uB(x, y, z, ...)= 0. That is the same as u(A(x,y,z,...)= uB(x, y, z, ...)).

Similarly, if A(x, y, z, ...)= B(x, y, z,...) and C(x, y, z, ...)= D(x, y, z, ...), then A(x, y, z, ...)- B(x, y, z, ...)= 0 and C(x, y, z, ...)- D(x, y, z, ...)= 0.

0+ 0= 0 so (A(x, y, z, ...)- B(x, y, z, ...))+ (C(x, y, z, ...)- D(x, y, z, ...))= 0. Then (A(x, y, z, ...)+ C(x, y, z, ...))- (B(x, y, z, ...)+ D(x, y, z, ...))= 0 so A(x, y, z, ...)+ C(x, y, z, ...)= B(x, y, z, ...)+ D(x, y, z, ...).