Evaluate an expression after solving a 2×2 system: If x + y − 7 = 0 and 3x + y − 13 = 0, then compute the value of 4x^2 + y^2 + 4xy.

Introduction / Context:This problem couples a small linear system with an algebraic expression. Note that 4x^2 + y^2 + 4xy equals (2x + y)^2, which suggests a quick path once 2x + y is found from the system.

Given Data / Assumptions:

• x + y = 7
• 3x + y = 13

Concept / Approach:Subtract equations to get x, find y, then compute 2x + y and square it. This avoids expanding 4x^2 + y^2 + 4xy directly.

Step-by-Step Solution:(3x + y) − (x + y) = 13 − 7 → 2x = 6 → x = 3Then y = 7 − x = 4Compute 2x + y = 2*3 + 4 = 10Value = (2x + y)^2 = 10^2 = 100

Verification / Alternative check:Direct expansion: 4x^2 + y^2 + 4xy with x = 3, y = 4 gives 36 + 16 + 48 = 100, confirming.

Why Other Options Are Wrong:75, 85, 91, 110 do not match the computed value from the uniquely determined solution.

Common Pitfalls:Expanding incorrectly or missing the identity (2x + y)^2, which simplifies the calculation.

Final Answer:100