Three-way proportional equality: If (x + y − 8)/2 = (x + 2y − 14)/3 = (3x + y − 12)/11, find the values of x and y respectively.

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:

• (x + y − 8)/2 = (x + 2y − 14)/3 = (3x + y − 12)/11
• All variables are real and expressions are defined.

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:

Verification / 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