lesson
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Imagine two balancing scales: scale A balances 3x+2y with 16, and scale B balances 5x−2y with 8. What happens if you dump the contents of scale B directly onto scale A?
Because both scales are already balanced, adding equal quantities to both sides keeps the new scale perfectly balanced—and the +2y and −2y wipe each other out entirely.
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The Elimination Method
The elimination method (also called linear combination) solves a system of equations by adding or subtracting equations to cancel out one variable, leaving a single one-step equation.
A coefficient is the constant number multiplying a variable, like the 3 in 3x.
Historical texts show this technique dates back over 2,000 years to China's Han dynasty in The Nine Chapters on the Mathematical Art, later refined in Europe by Carl Friedrich Gauss.
Let's see step-by-step how to solve a system when opposite coefficients already exist.
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What if no coefficients cancel out immediately when you add the equations?
Multiplying One Equation
When coefficients don't match, use the multiplication property of equality to scale an entire equation by a constant so that one variable gets opposite coefficients.
Consider the system: