Add Equations to Cancel a Variable

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Solve x+y=5 and x−y=-1. Candidate answers: (A) (2,3) (B) (3,2) (C) (4,2) (D) (5,-1).
• Independent check inputs: Solve x+y=7 and x−y=-1. Candidate answers: (A) (3,4) (B) (4,3) (C) (5,3) (D) (7,-1).

Learn the idea

A system requires the same ordered pair to satisfy every equation. Substitution and elimination find that shared pair, while parallel or coincident line structure explains whether a system has none or many solutions. Elimination combines equations so one variable’s coefficients sum to zero. The surviving equation finds one coordinate, and substitution into an original equation finds the other.

Here is the method on the supplied teaching case. Add the equations so y cancels: 2x=4, x=2. Substitute in x+y=5 to get y=3.

Add entire equations, including their right sides. A zero coefficient is a purposeful cancellation here; preserve the sign of a subtracted variable and verify the final pair in both relationships.

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 x+y=19 and x−y=-1. Candidate answers: (A) (10,9) (B) (11,9) (C) (19,-1) (D) (9,10). Show the mathematical reason, not only the selected answer. Expected result: (9,10). Add the equations so y cancels: 2x=18, x=9. Substitute in x+y=19 to get y=10.

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