make x the subject of the relation $y = \frac{ax^3 - b}{3z}$

Analyze the question: We need to rearrange y = (ax³ − b)/(3z) so that x stands alone on one side.

Concept: To change the subject, undo the operations around x in reverse order: first the division, then the subtraction, then the multiplication by a, and finally the cube.

Step-by-step solution:

1. Start with: y = (ax³ − b)/(3z)
2. Multiply both sides by 3z: 3yz = ax³ − b
3. Add b to both sides: 3yz + b = ax³
4. Divide both sides by a: x³ = (3yz + b)/a
5. Take the cube root of both sides: x = ∛[(3yz + b)/a]

Checking the options:

• A: Still has x on the right-hand side, so it is not a valid answer.
• B: Has −b, but we added b, so it is wrong.
• C: x = ∛[(3yz + b)/a], which matches our result.
• D: Multiplies by b instead of adding, so it is wrong.

Tip: Whenever you move a term across the equals sign, reverse its operation: subtraction becomes addition, and multiplication becomes division. Always do the addition or subtraction before dividing by the coefficient. Also, a quick check is that the answer should never contain the subject (x) on both sides.