Difficulty: Medium
Correct Answer: 2, 6
Explanation:
Introduction / Context:Equalizing three expressions to a common parameter is a common technique. Assign a parameter t to the shared value, convert each proportional statement into a linear equation, and then solve the resulting system for x and y efficiently.
Given Data / Assumptions:
Concept / Approach:Let each fraction equal t. Then x + y − 8 = 2t, x + 2y − 14 = 3t, and 3x + y − 12 = 11t. Eliminate t by equating suitable pairs, reducing the problem to two linear equations in x and y.
Step-by-Step Solution:
From first two: x + 2y − 14 = (3/2)(x + y − 8) 2x + 4y − 28 = 3x + 3y − 24 ⇒ 0 = x − y + 4 ⇒ x − y = −4 ⇒ x = y − 4 Use the first and third with t: 3x + y − 12 = 11 * [(x + y − 8)/2] 2(3x + y − 12) = 11(x + y − 8) 6x + 2y − 24 = 11x + 11y − 88 ⇒ 0 = 5x + 9y − 64 Substitute x = y − 4: 5(y − 4) + 9y − 64 = 0 ⇒ 14y − 84 = 0 ⇒ y = 6 Then x = 6 − 4 = 2Verification / Alternative check:Compute t: (x + y − 8)/2 = (2 + 6 − 8)/2 = 0; the other two also evaluate to 0. All three equal, confirming correctness.
Why Other Options Are Wrong:Other pairs do not satisfy all three equalities simultaneously when substituted back.
Common Pitfalls:Dropping constants during cross-multiplication or equating only two of the three expressions without checking the third one for consistency.
Final Answer:2, 6
Discussion & Comments