The Elimination Method

Algebra 1 · Axiom Academy

Scale the equations so a variable cancels when you add them — then solve any system, step by step. 1. When Direct Elimination Fails Here is the system we'll solve. Sometimes you can add two equations as-is and a variable just disappears — but not here. The coefficients of x aren't opposites, and neither are the coefficients of y . Watch what happens if we add them directly. Press play: the columns line up, the rows fall together, and the sum still carries both variables. 2. The Multiplication Property of Equality The key tool is the Multiplication Property of Equality : you may multiply every term in an equation by the same non-zero number, and the equation stays true. This is our legal move to reshape an equation without changing its solution. Think of a balanced scale. Press play: both pans start level, and when we triple what's on each side, the scale stays balanced . Multiplying both sides by the same amount never tips it. 3. Creating Opposite Coefficients Our goal is to make the coefficients of one variable opposites — like +6 and -6 — so they'll cancel when added. Look at the y terms: +3y on top and -2y on the bottom. The smallest common size for 3 and 2 is 6 . So we multiply the top equation by 2 and the bottom equation by 3 . Press play to watch each y -coefficient grow to length 6 , pointing in opposite directions. Now the y coefficients are +6 and -6 — a perfectly matched pair of opposites, ready to cancel.

This is the written version of the interactive lesson above. See the full Algebra 1 course.