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All courses Math I · A-REI.5 11 of 59
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: enter 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

Check the first original equation

\[3~+~2~=~5\]

The point (3,2) satisfies x + y = 5.

02

Check the second original equation

\[3~-~2~=~1\]

The point (3,2) also satisfies x - y = 1.

03

Check the replacement equation

\[2(3)~=~6\]

Substituting x = 3 into 2x = 6 shows the new equation is true.

04

State the verification

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

So (3,2) is still a solution after replacing one equation with 2x = 6.

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Another way

  1. You can notice that adding the original equations gives 2x = 6, so any solution of the original system must satisfy the replacement equation too.

!

Common mistake

A common mistake is to check only the new equation and forget to confirm the point still matches the remaining original equation.