Scale, Then Add

Often no pair of terms cancels as the equations stand. Then you multiply one equation by a number first, so that one variable gets opposite coefficients. In $x+2y = 7;\quad 3x-y = 7$, multiply the second equation by 2. Now the $y$ terms are $2y$ and $-2y$, which are opposites:

$$6x-2y = 14$$

Add this to the first equation. The $y$ terms cancel:

$$7x = 21$$

$$x = 3$$

Then the first equation gives $3+2y = 7$, so $y = 2$. Multiplying every term of an equation by the same number does not change its solutions, so the scaled system has the same solution as the original.

First choose which variable to eliminate. Then choose the multiplier that makes its coefficients opposites. Sometimes you need to scale both equations. For example, if the coefficients are 2 and 3, multiply the first equation by 3 and the second by 2, so that both become 6.