Make x the subject of the formula: y = $\frac {3x - 9c}{4x + 5d}$

Analysis

To make $x$ the subject of the formula, we need to isolate $x$ on one side of the equation. This involves clearing the fraction, expanding the terms, grouping all terms containing $x$ on one side, and then factoring out $x$.

Concept

The equation is a rational function. We use the following algebraic steps:

1. Cross-multiply to eliminate the fraction.
2. Expand the brackets.
3. Rearrange the terms to bring all terms with $x$ to one side and terms without $x$ to the other.
4. Factorize $x$ out.
5. Divide by the coefficient of $x$ to solve for it.

Step-by-Step Solution

Step 1: Cross-multiply Given: $y = \frac{3x - 9c}{4x + 5d}$ Multiply both sides by $(4x + 5d)$: $y(4x + 5d) = 3x - 9c$

Step 2: Expand the left side $4xy + 5dy = 3x - 9c$

Step 3: Group terms with $x$ on one side Move $3x$ to the left and $5dy$ to the right: $4xy - 3x = -9c - 5dy$

Step 4: Factorize $x$ $x(4y - 3) = -(9c + 5dy)$

Step 5: Solve for $x$ Divide both sides by $(4y - 3)$: $x = \frac{-(9c + 5dy)}{4y - 3}$

Comparison with Options

Looking at the result $x = \frac{-(9c + 5dy)}{4y - 3}$, it matches Option D exactly.

Note: If you look at Option A, it is mathematically equivalent to D if you distribute the negative sign into the bracket, but Option D is the direct representation of the algebraic steps taken.

Exam Tip

When dealing with "make $x$ the subject" questions involving fractions, always remember the "cross-multiply first" rule. It immediately turns a complex-looking fraction into a simple linear equation that is much easier to manage. Also, watch your signs carefully—the negative sign in front of the bracket in the final answer is a common place where students make mistakes!