Linear system with a ratio condition: Given 6x − 10y = 10 and x / (x + y) = 5/7, find the value of (x − y).

Introduction / Context:
Here we must solve a linear equation coupled with a ratio constraint. Converting the ratio x/(x + y) = 5/7 into a linear relation between x and y allows combining it with 6x − 10y = 10 to determine x and y exactly, and then compute x − y.

Given Data / Assumptions:

• 6x − 10y = 10
• x / (x + y) = 5/7 (x + y ≠ 0)

Concept / Approach:
From x/(x + y) = 5/7, cross-multiply to relate x and y directly. Substitute into the linear equation to solve. Finish by evaluating x − y. Keep arithmetic exact to avoid rounding errors.

Step-by-Step Solution:

Verification / Alternative check:
Check ratio: x/(x + y) = 5/(5 + 2) = 5/7 ✔; check equation: 6*5 − 10*2 = 30 − 20 = 10 ✔.

Why Other Options Are Wrong:
6, 8, 12, 5 do not match the computed difference from the consistent pair (x, y) = (5, 2).

Common Pitfalls:
Mis-cross-multiplying the ratio or simplifying 7x = 5x + 5y incorrectly; another pitfall is substituting into 6x − 10y with sign errors.

Final Answer:
3