make u the subject in x =$\frac{2u-3}{3u + 2}$

Analyze the question: We need to rearrange x = (2u − 3)/(3u + 2) so that u stands alone on one side. This is "changing the subject of a formula" where the variable appears in both numerator and denominator.

Concept: Cross-multiply to clear the fraction, group all u-terms on one side, factorise out u, then divide.

Solution:

1. Cross-multiply: x(3u + 2) = 2u − 3

2. Expand: 3xu + 2x = 2u − 3

3. Collect u-terms on the left and the other terms on the right: 3xu − 2u = −3 − 2x

4. Factorise u: u(3x − 2) = −(2x + 3)

5. Divide: u = −(2x + 3)/(3x − 2)

6. Multiply the numerator and denominator by −1 to get a positive-looking denominator: u = (2x + 3)/(2 − 3x)

Check with a number: Let u = 1, then x = (2 − 3)/(3 + 2) = −1/5. Using option C: (2(−1/5) + 3)/(2 − 3(−1/5)) = (2.6)/(2.6) = 1 ✓

Answer: C

Tip: In "change of subject" questions where the subject appears twice, always remember: cross-multiply, group, factorise, divide. Watch your signs when moving terms. A quick substitution check with a simple number can confirm your answer in seconds.