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All courses Algebra I · A-REI.5 11 of 76
Justify elimination: replacing one equation in a two-variable system with a linear combination preserves solutions.

Verify that replacing an equation in a system preserves a solution

Problem
The original system is \(x+y=5\) and \(x-y=1\). Adding the equations and retaining \(x-y=1\) gives the transformed system \(x-y=1\) and \(2x=6\). Given that \((3,2)\) solves the original system, test it in both transformed equations: identify each left/right value and still-a-solution yes/no.
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Hint

Substitute (3,2) into each original equation first, then into the replacement equation 2x = 6.

If a point satisfies the original equations and the new replacement equation, it is still a solution of the transformed system.

Solution walkthrough

01

Identify the point and both transformed equations

\[x~=~3;~y~=~2;~\text{equation}~1:~x~-~y~=~1;~\text{equation}~2:~2x~=~6\]

The given point supplies x = 3 and y = 2. The transformed system retains x - y = 1 and replaces the other condition with the sum 2x = 6.

02

Check the retained equation

\[\text{left}~=~3~-~2~=~1;~\text{right}~=~1\]

Substituting the point into x - y = 1 makes the left and right sides equal, so the point satisfies equation 1.

03

Check the combined equation

\[\text{left}~=~2(3)~=~6;~\text{right}~=~6\]

Substituting x = 3 into 2x = 6 also makes the two sides equal, so the point satisfies equation 2.

04

Explain why the transformation is valid

\[(x~+~y)~+~(x~-~y)~=~5~+~1~->~2x~=~6\]

The replacement equation comes from adding equal quantities in the original equations. Because the retained equation remains available, the original first equation can be recovered by subtracting it from 2x = 6.

05

Report the requested audit

\[\text{equation}~1:~\text{left}~=~1,~\text{right}~=~1;~\text{equation}~2:~\text{left}~=~6,~\text{right}~=~6;~\text{still}~a~\text{solution}~=~\text{yes}\]

Both substitutions produce true equations, so (3,2) remains a solution of the transformed system.

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Another way

  1. Solve the transformed system: 2x = 6 gives x = 3, and x - y = 1 then gives y = 2, reproducing the given point.

!

Common mistake

Do not merely check 2x = 6. A point solves a system only when it satisfies every equation in that system.

Solution walkthrough video