Solve the simultaneous equations 3(2u + v) = 7uv and 3(u + 3v) = 11uv and choose the correct pair (u, v) from the following options.

Introduction / Context:
This problem involves a pair of nonlinear simultaneous equations in two variables u and v. The equations contain products of u and v, so they are not purely linear. The question asks which of the given ordered pairs (u, v) satisfies both equations at the same time. This tests substitution skills and the ability to check candidate solutions efficiently.

Given Data / Assumptions:

• Equation 1: 3(2u + v) = 7uv.
• Equation 2: 3(u + 3v) = 11uv.
• We must find (u, v) that satisfies both equations simultaneously.
• Options give several candidate pairs for (u, v).

Concept / Approach:
One method is to manipulate the two equations algebraically to isolate u in terms of v or vice versa. Another practical approach, given that specific candidate pairs are provided, is to test each pair directly in both equations. The correct answer must satisfy both equations exactly, not just approximately, so careful arithmetic is required.

Step-by-Step Solution:
Consider option B: u = 1, v = 3/2.Substitute into Equation 1: left side is 3(2u + v) = 3(2*1 + 3/2) = 3(2 + 1.5) = 3 × 3.5 = 10.5. Right side is 7uv = 7 × 1 × 3/2 = 10.5. So Equation 1 is satisfied.Now check Equation 2 with u = 1 and v = 3/2. Left side is 3(u + 3v) = 3(1 + 3*(3/2)) = 3(1 + 9/2) = 3(11/2) = 33/2.Right side is 11uv = 11 × 1 × 3/2 = 33/2. So Equation 2 is also satisfied. Therefore, option B works.

Verification / Alternative check:
Check option A: u = 1, v = 0. For Equation 1, left side is 3(2*1 + 0) = 6, right side is 7*1*0 = 0, so it fails immediately.Check option C: u = 0, v = 3/4. For Equation 1, left side is 3(0 + 3/4) = 9/4, right side is 7*0*3/4 = 0, so it fails.Check option D: u = 0, v = 1. Equation 1 again gives a nonzero left side and zero right side, so it is not a solution either.

Why Other Options Are Wrong:
Any pair with u = 0 makes the right-hand side of both equations equal to zero, while the left sides are clearly nonzero due to the terms 3(2u + v) and 3(u + 3v). Hence options C and D cannot work.Option A fails because it makes v = 0, again causing the product term 7uv and 11uv to vanish, which is inconsistent with the nonzero left-hand sides.

Common Pitfalls:
A common error is to check only one of the two equations and stop as soon as it matches, without verifying the second equation.Another pitfall is careless arithmetic with fractions, especially when converting between fractional and decimal forms, which can cause a correct candidate like (1, 3/2) to be rejected mistakenly.

Final Answer:
The only pair that satisfies both equations is u = 1, v = 3/2.