If the simultaneous linear equations 3x + 2y = 7 and 4x − y = 24 hold for real numbers x and y, what is the value of (x − y)?

Introduction / Context:
This question involves solving a system of two linear equations in two variables and then computing a simple expression involving x and y. Such problems are central in algebra and appear frequently in aptitude exams. The aim is to find x and y and then evaluate x − y. You can use substitution or elimination; elimination works quickly here.

Given Data / Assumptions:

• The first equation is 3x + 2y = 7.
• The second equation is 4x − y = 24.
• x and y are real numbers.
• We must find x − y once the system is solved.

Concept / Approach:
To solve two linear equations, we can express one variable in terms of the other and substitute, or we can eliminate one variable by adding or subtracting equations after scaling them. Here, solving the second equation for y and substituting into the first equation gives a simple single variable equation. Once x is found, y is obtained easily and then x − y can be computed.

Step-by-Step Solution:
Start with the second equation: 4x − y = 24.Solve for y: y = 4x − 24.Substitute this expression for y into the first equation 3x + 2y = 7.This gives 3x + 2(4x − 24) = 7.Expand: 3x + 8x − 48 = 7, so 11x − 48 = 7.Add 48 to both sides: 11x = 55, so x = 55 / 11 = 5.Now find y from y = 4x − 24: y = 4 * 5 − 24 = 20 − 24 = −4.Compute x − y: 5 − (−4) = 5 + 4 = 9.

Verification / Alternative check:
Substitute x = 5 and y = −4 back into both original equations. For 3x + 2y: 3 * 5 + 2 * (−4) = 15 − 8 = 7, which is correct. For 4x − y: 4 * 5 − (−4) = 20 + 4 = 24, which also matches. Thus (x, y) = (5, −4) satisfies both equations. The value x − y = 9 is therefore reliable.

Why Other Options Are Wrong:

• Values 1 and −1 would require different values of x and y that do not satisfy both equations simultaneously.
• −9 is the negative of the correct result and might arise from reversing the expression as y − x instead of x − y.
• 0 would correspond to x = y, which is not true here.
• Only 9 is consistent with the correct solution pair (5, −4).

Common Pitfalls:

• Mistakes in solving for y from the second equation, for example getting y = 24 − 4x instead of 4x − 24.
• Errors in distributing 2 when substituting 2(4x − 24).
• Forgetting that x − y is x minus y, not y minus x, so the sign matters.

Final Answer:
9