June 2018 Paper 1 Q10

10 Rearrange \(\qquad \dfrac{1}{xy} = 4 - \dfrac{3}{y} \qquad\) to make \(x\) the subject. [3 marks]

Mark scheme

Answer	Mark	Comments
A correct first step using algebra	M1	Here are some of the possible alternatives \(\dfrac{1}{x} = y\left(4 - \dfrac{3}{y}\right)\) multiplying through by \(y\) \(1 = xy\left(4 - \dfrac{3}{y}\right)\) multiplying through by \(xy\) \(1 = 4xy - \dfrac{3xy}{y}\) multiplying through by \(xy\) \(y = 4xy^2 - 3xy\) multiplying through by \(xy^2\) \(\dfrac{1}{xy} = \dfrac{4y - 3}{y}\) making the RHS an algebraic fraction \(\dfrac{1 + 3x}{xy} = 4\) rearranging and making the LHS an algebraic fraction
Further correct algebra which leads to an equation that is one step from the final answer.	M1dep	Following two of the above alternatives … \(y = 4xy^2 - 3xy\) \(y = x(4y^2 - 3y)\) M1dep gained \(\dfrac{1 + 3x}{xy} = 4\) \(1 + 3x = 4xy\) \(1 = 4xy - 3x\) \(1 = x(4y - 3)\) M1dep gained
A correct final answer in any form	A1	\(x = \dfrac{1}{4y - 3} \qquad x = \dfrac{-1}{3 - 4y}\) \(x = \dfrac{y}{4y^2 - 3y} \qquad x = \dfrac{-y}{3y - 4y^2}\) \(x = \dfrac{1}{y\left(4 - \frac{3}{y}\right)} \qquad x = \dfrac{-1}{y\left(\frac{3}{y} - 4\right)}\) \(x = \dfrac{1}{\left(4 - \frac{3}{y}\right)} \div y\)

There are many ways of scoring the first M mark. They do not need to give any reasons but you need to check that what they do is valid.
For the M1dep mark you must check that their algebra is correct and will lead to a result that is one step from the final answer. ‘One step from …’ means that when they divide through, they have a correct version where \(x\) is the subject.
Some of the final answers are more compact than others, but we didn’t ask for any simplification so we have to accept a correct answer in any form.
… and, finally, one to look out for … correct answer from wrong working … 0 marks \(\dfrac{1}{xy} = 4 - \dfrac{3}{y} \ \rightarrow\ xy = \dfrac{1}{4} - \dfrac{y}{3} \ \rightarrow\ x = \dfrac{1}{4y} - \dfrac{1}{3} \ \rightarrow\ x = \dfrac{1}{4y - 3}\) (creative thinking !)