Formal proof for why subtracting two equations in a system of linear equations maintains equivalence with the original system of equations

Question

While solving a system of linear equations, we add/subtract two equations to get a simpler equation which we can solve for a variable. I see why we can do so intuitively, because when we add one equation to the other, we are just adding equal quantities on both sides of the first equation, thus unchanging the equation and since we are finding solution which satisfies both the equations, this solution satisfies the new equation also. Graphically, adding subtracting equations will create new lines (or 2-d planes and higher dimensional planes), but these will intersect at the same point(s) as the original system. But I am not able to show a formal proof for why subtracting two equations while solving a system of equations doesn't change the original system of equations. Could I please get some help with this?

Answer 1

Consider a system of two equations $$ (S_1) \quad l_1 = r_1 \quad\text{and}\quad l_2 = r_2. $$ Then $S_1$ is equivalent to the system $$ (S_2) \quad l_1 + l_2 = r_1 + r_2 \quad\text{and}\quad l_2 = r_2. $$ and to the system $$ (S_3) \quad l_1 + l_2 = r_1 + r_2 \quad\text{and}\quad l_1 = r_1. $$ and also to the system $$ (S_4) \quad l_1 + l_2 = r_1 + r_2 \quad\text{and}\quad l_1 = r_1 \quad\text{and}\quad l_2 = r_2. $$ Note that when adding the two initial equations, you need to keep at least one of the original equations to keep an equivalent system. For instance, you could prove that $S_2$ implies $S_1$ by substracting: $l_1 + l_2 - l_2 = r_1 + r_2 - r_2$, which gives $l_1 = r_1$.