Scale, Then Add

Usually nothing cancels by itself; a multiplication sets it up. In x + 2y = 7; 3x - y = 7 doubling the second equation makes the y terms opposites:

6x - 2y = 14

7x = 21

x = 3

and y follows: 3 + 2y = 7 gives y = 2. Multiplying a whole equation changes nothing about its line, so the scaled system is the same system dressed for cancelling.

Aim the scaling: pick the variable to kill, then choose multipliers that make its coefficients opposite. Sometimes both equations need scaling, say by 2 and by 3, to meet at a common multiple.