We now have two algebraic strategies for solving systems of equations: by substitution and by elimination. In some systems, the equations may give us a clue as to which strategy to use. For example:

In other systems, which strategy to use is less straightforward, either because no variables are isolated, or because no variables have equal or opposite coefficients. For example:

$\begin{cases} \begin{aligned} 2x+3y&=15 \quad&\text{Equation A}\\ 3x-9y&=18 \quad&\text{Equation B} \\ \end{aligned}\end{cases}$

To solve this system by elimination, we first need to rewrite one or both equations so that one variable can be eliminated. To do that, we can multiply both sides of an equation by the same factor. Remember that doing this doesn't change the equality of the two sides of the equation, so the $x$- and $y$-values that make the first equation true also make the new equation true.

There are different ways to eliminate a variable with this approach. For instance, we could:

Each multiple of an original equation is equivalent to the original equation. So each new pair of equations is equivalent to the original system and has the same solution.

When we solve a system by elimination, we are essentially writing a series of equivalent systems, or systems with the same solution. Each equivalent system gets us closer and closer to the solution of the original system.