Justify elimination: replacing one equation in a two-variable system with a linear combination preserves solutions.
Decide whether an elimination step is valid
Problem
For \(2x~+~y~=~7\) and \(x~-~y~=~2\), a proposed step adds the equations to get \(3x~=~9\). Evaluate the step and state the correct result.
Big Picture
What this problem is really about
A proposed elimination step must use one consistent equation operation and simplify every column correctly. We’ll align like terms, check whether one pair of coefficients are opposites, and add the variable terms and constants using the same operation on both sides. If a variable cancels and the combined equality is computed term by term, the result can be checked against the original rows.
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