Justify elimination: replacing one equation in a two-variable system with a linear combination preserves solutions.
Find a multiplier to eliminate a variable
Problem
What multiplier could you use on one equation to eliminate \(y\) in the system \(2x~+~3y~=~12\), \(5x~-~y~=~7\)?
Big Picture
What this problem is really about
The useful multiplier is the one that makes a chosen pair of variable coefficients opposites. We’ll compare the coefficients, set their sum after scaling equal to zero, and apply the resulting multiplier to every term of the affected equation, including its constant. Adding the scaled rows then verifies whether the targeted variable truly disappears.
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