Collect Variables from Both Sides

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Solve 5x+3=2x+9. Candidate answers: (A) 1 (B) 2 (C) 3 (D) 4.
• Independent check inputs: Solve 5x+4=2x+13. Candidate answers: (A) 2 (B) 3 (C) 4 (D) 6.

Learn the idea

An equation’s solution count depends on its entire structure. Balanced simplification may isolate a variable, reduce to an identity, or expose a contradiction; each outcome has a different interpretation. When a variable appears on both sides, subtract a matching variable term from both sides before isolating the remaining coefficient. This keeps the relationship balanced throughout.

Here is the method on the supplied teaching case. Subtract 2x and then 3 from both sides: 3x=6. Divide by 3 to get x=2. Both original sides equal 13.

Moving a term is shorthand for an operation on both sides, not permission to change its sign arbitrarily. Check the resulting value in the original, unsimplified equation.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: Solve 5x+10=2x+37. Candidate answers: (A) 10 (B) 18 (C) 8 (D) 9. Show the mathematical reason, not only the selected answer. Expected result: 9. Subtract 2x and then 10 from both sides: 3x=27. Divide by 3 to get x=9. Both original sides equal 55.

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