Solve a 2×2 linear system, then sum: If 3x + 7y = 75 and 5x − 5y = 25, find the value of x + y.

Introduction / Context:We are given a pair of linear equations and asked for x + y directly. Solving the system gives x and y, which we can then add. Alternatively, manipulation can sometimes find x + y without individual values.

Given Data / Assumptions:

• 3x + 7y = 75
• 5x − 5y = 25

Concept / Approach:First simplify the second equation to x − y = 5. Substitute y = x − 5 into the first and solve for x. Then compute x + y.

Step-by-Step Solution:From 5x − 5y = 25 → x − y = 5 → y = x − 53x + 7(x − 5) = 75 → 3x + 7x − 35 = 7510x = 110 → x = 11 → y = 6x + y = 11 + 6 = 17

Verification / Alternative check:Plug back: 3*11 + 7*6 = 33 + 42 = 75 and 5*11 − 5*6 = 55 − 30 = 25, both hold.

Why Other Options Are Wrong:14, 15, 16, and 18 are not equal to the verified sum 17 for the unique solution.

Common Pitfalls:Missing the simple reduction of the second equation by dividing by 5, or sign errors when substituting y.

Final Answer:17