Use an Eliminating Linear Combination

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Solve 2x+y=7 and x−y=-1. Candidate answers: (A) (2,3) (B) (3,2) (C) (4,2) (D) (7,-1).
• Independent check inputs: Solve 2x+y=10 and x−y=-1. Candidate answers: (A) (10,-1) (B) (3,4) (C) (4,3) (D) (5,3).

Learn the idea

A mathematical plan can require simultaneous equalities or inequalities. The feasible set contains inputs satisfying all constraints, and a linear objective asks which allowed choice gives the greatest or least modeled value. A system can be solved by adding equations whose opposite variable coefficients cancel. The resulting one-variable equation must still be paired with an original equation to recover the full solution.

Here is the method on the supplied teaching case. Add the equations: 3x=6. Thus x=2, and y=x−(-1)=3. Verification in the first equation gives 7.

The operation applies to right-side constants as well as coefficients. A solved x value alone is incomplete; find y and verify the ordered pair in every original relationship.

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 2x+y=28 and x−y=-1. Candidate answers: (A) (10,9) (B) (11,9) (C) (28,-1) (D) (9,10). Show the mathematical reason, not only the selected answer. Expected result: (9,10). Add the equations: 3x=27. Thus x=9, and y=x−(-1)=10. Verification in the first equation gives 28.

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