Justify elimination: replacing one equation in a two-variable system with a linear combination preserves solutions.
Form and verify an elimination row combination
Problem
\(E_{1}\): \(x+y=7\); \(E_{2}\): \(2x-y=5\). Use \(E_{1}+E_{2}\).
Big Picture
What this problem is really about
The requested row combination should eliminate one variable while preserving equality. We’ll add corresponding terms and constants, confirm that the opposite coefficients cancel, and solve the resulting one-variable equation. Back-substitution recovers the other coordinate, and checking the ordered pair in both original equations verifies the combination and the complete system solution.
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