Justify elimination: replacing one equation in a two-variable system with a linear combination preserves solutions.
Compare system transformations and decide which preserves solutions
Problem
The system is \(x~+~y~=~5\) and \(x~-~y~=~1\). Transformation A replaces the first equation with \(2x~=~6\) and keeps the second. Transformation B replaces both equations with \(2x~=~6\). Which complete response correctly evaluates the transformations?
Big Picture
What this problem is really about
A combined equation is a consequence of a system, but it may not contain all of the system’s information by itself. We’ll compare how many independent conditions each transformation retains, solve or test the resulting systems, and ask whether the original rows can be recovered. A transformation preserves the solution set only when its equations retain enough independent information to make that recovery possible.
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