Justify elimination: replacing one equation in a two-variable system with a linear combination preserves solutions.
Verify that replacing an equation in a system preserves a solution
Problem
The original system is \(x+y=5\) and \(x-y=1\). Adding the equations and retaining \(x-y=1\) gives the transformed system \(x-y=1\) and \(2x=6\). Given that \((3,2)\) solves the original system, test it in both transformed equations: identify each left/right value and still-a-solution yes/no.
Big Picture
What this problem is really about
A solution of a system must satisfy every equation that remains after a transformation. We’ll preserve the ordered-pair roles, substitute into the retained equation and the replacement equation separately, and compare each left side with its right side. Then we’ll trace the replacement back to a linear combination of the originals to explain why retaining the other equation keeps the transformation reversible.
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