Problem

Let \(E_{1}\) be \(x~+~y~=~7\) and \(E_{2}\) be \(2x~-~y~=~5\). Keep \(E_{1}\) and replace \(E_{2}\) by \(E_{2}~+~E_{1}\), giving \(3x~=~12\). Describe the replacement, determine whether it preserves every solution, and justify the decision.

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What this problem is really about

A row combination preserves a system only when the replacement keeps enough information to reverse the operation. We’ll form the proposed sum term by term, retain the untouched equation, and then try to recover the removed equation by subtracting that retained row. This two-way argument, rather than merely finding one shared candidate, establishes whether the complete solution set stays the same.

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Keep E2 and replace E1 with E2 + E1. This is equivalent because the written system keeps E1.

Keep E1 and replace E2 with E2 + E1. This is not equivalent because equations may never be added.

Replace both equations with E2 + E1. This is equivalent because 3x = 12 contains the solution.

Keep E1 and replace E2 with E2 + E1. This is equivalent because E2 can be recovered by subtracting E1, so no constraint is lost.

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Let \(E_{1}\) be \(x~+~y~=~7\) and \(E_{2}\) be \(2x~-~y~=~5\). Keep \(E_{1}\) and replace \(E_{2}\) by \(E_{2}~+~E_{1}\), giving \(3x~=~12\). Describe the replacement, determine whether it preserves every solution, and justify the decision.

What this problem is really about

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