Definition: The elimination method solves a system by adding or subtracting the equations, after scaling them if needed, so that one variable cancels.
What is the idea?
Equations can be added because equal quantities added to equal quantities stay equal. If the coefficients of a variable are opposites, adding the equations removes that variable.
How do you set it up?
Line up the variables in standard form. Multiply one or both equations so a pair of coefficients become opposites. Add, solve, back-substitute, and check.
Example. Solve 3x + 4y = 18 and 5x – 6y = 4. Multiply the first by 3 and the second by 2: 9x + 12y = 54 and 10x – 12y = 8. Adding gives 19x = 62, so x = 62/19. Then 3(62/19) + 4y = 18 gives 4y = 342/19 – 186/19 = 156/19, so y = 39/19.
Which slips are typical?
Multiplying only one side of an equation, or only part of it. Another is a sign error when subtracting an equation: change every sign in the second equation. Elimination is often cleaner than substitution when both equations are in standard form with no variable alone.