Justify elimination: replacing one equation in a two-variable system with a linear combination preserves solutions.
Decide whether an elimination action loses system information
Problem
A transformation multiplies one equation of a system by \(0\) before adding. Classify whether the transformation preserves or loses information, and explain why.
Big Picture
What this problem is really about
Information is preserved only when an equation transformation can be reversed and still imposes the original constraint. We’ll carry the proposed multiplier through every coefficient and constant, inspect the equation it produces, and test whether that statement restricts any ordered pairs. Comparing this with reversible nonzero scaling reveals whether the original equation could be recovered or whether a condition has disappeared.
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