If (3a + 4b)/(3c + 4d) = (3a − 4b)/(3c − 4d) for real numbers a, b, c and d (with denominators non zero), then which of the following relations between a, b, c and d must hold?

Introduction / Context:
This algebra question explores what happens when two rational expressions involving the same four variables are equal. By cross multiplying and comparing coefficients, we can extract a necessary relation among a, b, c and d. Such questions test the ability to handle symmetrical expressions and factor out common terms effectively.

Given Data / Assumptions:

• (3a + 4b)/(3c + 4d) = (3a − 4b)/(3c − 4d).
• 3c + 4d ≠ 0 and 3c − 4d ≠ 0 so the fractions are defined.
• a, b, c and d are real numbers.
• We need to deduce a relation among a, b, c and d that must always hold when the equality is true.

Concept / Approach:
The standard approach is to cross multiply the two fractions and simplify the resulting equality. This yields a polynomial equation in a, b, c and d. By expanding both sides and then moving all terms to one side, we can factor or group the terms to reveal the underlying product relation. Often, expressions of this type simplify nicely to something like ad = bc, ab = cd or similar symmetric conditions.

Step-by-Step Solution:
Given (3a + 4b)/(3c + 4d) = (3a − 4b)/(3c − 4d). Cross multiply: (3a + 4b)(3c − 4d) = (3a − 4b)(3c + 4d). Expand the left side: (3a + 4b)(3c − 4d) = 9ac − 12ad + 12bc − 16bd. Expand the right side: (3a − 4b)(3c + 4d) = 9ac + 12ad − 12bc − 16bd. Set them equal and subtract the right side from the left side: (9ac − 12ad + 12bc − 16bd) − (9ac + 12ad − 12bc − 16bd) = 0. The 9ac and −16bd terms cancel out. We are left with −12ad + 12bc − 12ad + 12bc = 0. This simplifies to −24ad + 24bc = 0. Divide by 24: −ad + bc = 0, so bc = ad.

Verification / Alternative check:
As a check, choose numbers satisfying ad = bc, for example a = 2, b = 3, c = 4, d = 6 (since 2·6 = 3·4 = 12). Then 3a + 4b = 6 + 12 = 18 and 3c + 4d = 12 + 24 = 36, giving (3a + 4b)/(3c + 4d) = 18/36 = 1/2. On the other side, 3a − 4b = 6 − 12 = −6 and 3c − 4d = 12 − 24 = −12, so (3a − 4b)/(3c − 4d) = (−6)/(−12) = 1/2. The fractions are indeed equal, illustrating that ad = bc works.

Why Other Options Are Wrong:
ab = cd, ac = bd, and a = b = c ≠ d would not, in general, guarantee equality of the two fractions when checked with arbitrary numeric examples. The condition a + d = b + c is also unrelated to the cross multiplied expression and fails for many sets of values. Only ad = bc emerges directly from algebraic simplification of the given equality.

Common Pitfalls:
Students sometimes expand incorrectly or forget to move all terms to one side, missing the chance to factor. Another pitfall is cancelling non common terms incorrectly or dividing by expressions that might be zero. Carefully expand, combine like terms and factor out constants before dividing to avoid losing valid solutions.

Final Answer:
The necessary relation is ad = bc.