Justify elimination: replacing one equation in a two-variable system with a linear combination preserves solutions.
Show that an original system and transformed system share the same ordered-pair solution
Problem
Solve the original system \(x~+~y~=~5\) and \(x~-~y~=~1\) and the transformed system \(2x~=~6\) and \(x~-~y~=~1\). Give the shared ordered-pair solution and verify it in both systems.
Big Picture
What this problem is really about
Sharing one derived equation is not enough to prove two systems have the same ordered-pair solution. We’ll solve the transformed system from its simplest equation, substitute to recover the second coordinate, and then test that full pair in every equation of both systems. Comparing each left side with its right side shows whether the candidate satisfies every original and transformed constraint.
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