Problem preview

A-REI.5 Core M1-011-R02-V04

Justify elimination: replacing one equation in a two-variable system with a linear combination preserves solutions.

Decide whether a system replacement is equivalent

Problem

Let \(E_{1}\) be \(3x~+~y~=~7\) and \(E_{2}\) be \(x~-~2y~=~0\). Keep \(E_{1}\) and replace \(E_{2}\) by \(0E_{2}\), giving \(0~=~0\). Describe the replacement, determine whether it preserves every solution, and justify the decision.

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Replace E2 with 0E2 and keep E1. This is equivalent because E1 is unchanged.

Replace E2 with 0E2 and keep E1. This is not equivalent because 0 = 0 removes E2, leaving every point on 3x + y = 7 instead of the single original intersection.

Replace E2 with 0E2 and keep E1. This is not equivalent because the new system has no solutions.

Replace E2 with 0E2 and keep E1. This is equivalent because 0 = 0 can be divided by 0.

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Curriculum context

Course: Math I
Standard: A-REI.5
Category: Algebra
Domain: Reasoning with Equations and Inequalities
Objective: Justify elimination: replacing one equation in a two-variable system with a linear combination preserves solutions.
Problem type: Decide whether a system replacement is equivalent