When solving a system of equations using the elimination method, the equations are either added or subtracted. When the sides of two equations are added with each other, why is then the sum of the two left-hand sides equal to the sum of the right-hand sides?

In equations it is permitted to add or subtract any constant or variable as long as the same is added or subtracted on both sides. For example, in the first equation ${\text{LHS}}_2$ can be added to both sides.

$$\{\begin{array}{l}{\text{LHS}}_1+{\text{LHS}}_2={\text{RHS}}_1+{\text{LHS}}_2\\ {\text{LHS}}_2={\text{RHS}}_2\end{array}$$

However, according to the second equation, ${\text{LHS}}_2$ is equal to ${\text{RHS}}_2.$ This can be substituted into the right-hand side of the first equation.

$$\{\begin{array}{l}{\text{LHS}}_1+{\text{LHS}}_2={\text{RHS}}_1+{\text{RHS}}_2\\ {\text{LHS}}_2={\text{RHS}}_2\end{array}$$

The first equation is now exactly the result when the two equations are added together column by column. In other words, the sum of the left-hand sides is equal to the sum of the right-hand sides. That the elimination method also works when the equations are subtracted can be shown in a similar way.