Replace One Variable with an Equal Expression

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Solve y=2x+1 and x+y=7. Candidate answers: (A) (2,5) (B) (2,6) (C) (3,5) (D) (5,2).
• Independent check inputs: Solve y=2x+1 and x+y=10. Candidate answers: (A) (3,7) (B) (3,8) (C) (4,7) (D) (7,3).

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. Substitution uses one equation to express a variable, then replaces that variable in the other equation. The resulting one-variable equation preserves the need to satisfy both original relationships.

Here is the method on the supplied teaching case. Substitute 2x+1 for y: x+2x+1=7. Then 3x=6, x=2; y=2(2)+1=5. The ordered pair satisfies both equations.

Use parentheses around a substituted expression, particularly when it is multiplied or subtracted. Verify both coordinates in both original equations; satisfying only one equation is insufficient for a system.

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 y=2x+1 and x+y=28. Candidate answers: (A) (10,19) (B) (19,9) (C) (9,19) (D) (9,20). Show the mathematical reason, not only the selected answer. Expected result: (9,19). Substitute 2x+1 for y: x+2x+1=28. Then 3x=27, x=9; y=2(9)+1=19. The ordered pair satisfies both equations.

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