16. The numbers $x, y$ and $z$ satisfy the equations $9 x+3 y-5 z=-4$ and $5 x+2 y-2 z=13$. What is the mean of $x, y$ and $z$ ?
A 10
B 11
C 12
D 13
E 14
Solution
A
Commentary
The mean of $x, y$ and $z$ is $\frac{1}{3}(x+y+z)$. Therefore to answer this question we need to find the value of $x+y+z$. We are given just two equations for the three unknowns $x$, $y$ and $z$. It follows that if these equations have a solution, they will have an infinite number of solutions.

A systematic method for answering the question would be to use the two equations to find expressions for two of the unknowns in terms of the third unknown. For example, we could find $x$ and $y$ in terms of $z$, and thus work out $x+y+z$ in terms of $z$.

However, the wording of the question suggests that $x+y+z$ is independent of $z$. Thus a good starting point is to try to find a way to use the two equations we are given to find a value for $x+y+z$ without the need to find $x$ and $y$ in terms of $z$.

We are given that
$$
9 x+3 y-5 z=-4
$$
and
$$
5 x+2 y-2 z=13
$$

If we multiply equation (2) by 2 , and subtract equation (1) we obtain
$$
2(5 x+2 y-2 z)-(9 x+3 y-5 z)=2(13)-(-4),
$$
that is,
$$
10 x+4 y-4 z-9 x-3 y+5 z=26+4,
$$
that is,
$$
x+y+z=30 \text {. }
$$

We deduce that $\frac{1}{3}(x+y+z)=10$.
Hence the mean of $x, y$ and $z$ is 10 .