Solve the system of linear equations: 2x + 3y = 0 3x − 4y = 34 Then find the exact value of x − y.

Introduction / Context:
This problem tests solving simultaneous linear equations and then evaluating a derived expression (x - y). The system has two equations and two unknowns, so it typically has a unique solution. Common methods include substitution and elimination. After finding x and y exactly, computing x - y is straightforward. The main focus is accuracy in algebraic steps and sign handling, especially because one equation has a negative term and the other mixes coefficients.

Given Data / Assumptions:

• 2x + 3y = 0 • 3x - 4y = 34 • Required: x - y

Concept / Approach:
Use substitution from the simpler equation. From 2x + 3y = 0, express x in terms of y (or y in terms of x) and substitute into the second equation. This reduces the system to a single-variable equation. Then back-substitute to find the other variable. Finally compute x - y.

Step-by-Step Solution:
1) From 2x + 3y = 0, solve for x: 2x = -3y x = -3y/2 2) Substitute x = -3y/2 into 3x - 4y = 34: 3(-3y/2) - 4y = 34 3) Simplify: -9y/2 - 4y = 34 4) Write 4y as 8y/2: -9y/2 - 8y/2 = 34 -17y/2 = 34 5) Solve for y: y = 34 * (-2/17) = -4 6) Find x using x = -3y/2: x = -3(-4)/2 = 12/2 = 6 7) Compute x - y: x - y = 6 - (-4) = 10

Verification / Alternative check:
Check in both equations: 2x + 3y = 2*6 + 3*(-4) = 12 - 12 = 0 (correct). 3x - 4y = 3*6 - 4*(-4) = 18 + 16 = 34 (correct). So x = 6 and y = -4 are consistent, and x - y = 10 is confirmed.

Why Other Options Are Wrong:
• -10, 2, -2, 6: these do not match the computed difference from the unique solution of the system.

Common Pitfalls:
• Sign error when substituting x = -3y/2. • Incorrectly combining fractions like -9y/2 - 4y.

Final Answer:
10