If (5x − y) / (5x + y) = 3/7 for real numbers x and y with non zero denominator, then what is the value of the expression (4x^2 + y^2 − 4xy) / (9x^2 + 16y^2 + 24xy)?

Introduction / Context:
This question checks algebraic simplification skills and the ability to recognize expressions as perfect squares. By rewriting the numerator and denominator in a suitable form, the value of the big fraction can often be obtained without explicitly solving for x and y, although solving is also possible in this case.

Given Data / Assumptions:

(5x − y) / (5x + y) = 3/7.
x and y are real numbers and 5x + y is not zero.
The required value is (4x^2 + y^2 − 4xy) / (9x^2 + 16y^2 + 24xy).

Concept / Approach:
The numerator and denominator can be recognized as squares of linear expressions in x and y. In particular, 4x^2 + y^2 − 4xy equals (2x − y)^2, and 9x^2 + 16y^2 + 24xy equals (3x + 4y)^2. If we can relate 2x − y and 3x + 4y to the given ratio involving 5x − y and 5x + y, we may evaluate the expression. An even simpler approach here is to use the given ratio to find y in terms of x and then substitute directly.

Step-by-Step Solution:
Start with (5x − y) / (5x + y) = 3/7.Cross multiply: 7(5x − y) = 3(5x + y).This gives 35x − 7y = 15x + 3y.Rearrange: 35x − 15x = 3y + 7y, so 20x = 10y.Hence y = 2x.Now compute the numerator: 4x^2 + y^2 − 4xy = 4x^2 + (2x)^2 − 4x(2x) = 4x^2 + 4x^2 − 8x^2 = 0.Therefore the numerator is zero for every non zero x, while the denominator is positive.Thus (4x^2 + y^2 − 4xy) / (9x^2 + 16y^2 + 24xy) = 0 / (positive) = 0.

Verification / Alternative check:
We can verify that the denominator is not zero for y = 2x. Substitute y = 2x: 9x^2 + 16(4x^2) + 24x(2x) = 9x^2 + 64x^2 + 48x^2 = 121x^2, which is non zero when x is non zero. Hence the fraction is well defined. Also, as noted earlier, 4x^2 + y^2 − 4xy = (2x − y)^2, which becomes (2x − 2x)^2 = 0, confirming the result in a more conceptual way.

Why Other Options Are Wrong:
The options 3/7, 18/49, 1/6, and 1/3 might tempt students who try to connect the result directly to the given ratio, but none of these values is consistent with the exact algebraic substitution y = 2x. Since the numerator reduces to zero for all valid values of x and y that satisfy the given ratio, the only correct value is 0.

Common Pitfalls:
One common mistake is to assume (4x^2 + y^2 − 4xy) is equal to (2x + y)^2 or another incorrect square, which leads to algebraic errors. Another pitfall is trying to manipulate the ratio (5x − y) / (5x + y) into the desired expression without first solving for y. Being comfortable with recognizing perfect squares and systematically solving the ratio equation helps avoid these issues.

Final Answer:
The value of the expression (4x^2 + y^2 − 4xy) / (9x^2 + 16y^2 + 24xy) is 0.