The real numbers $x, y$ and $z$ satisfy both of the equations below:

$$
\begin{aligned}
& 4 x+7 y+z=11 \\
& 3 x+y+5 z=15
\end{aligned}
$$

Given that $x+y+z=\frac{p}{q}$ where $p$ and $q$ are positive integers and the fraction $\frac{p}{q}$ is in lowest terms, what is the value of $p-q$ ?
Solution 1

Multiplying $4 x+7 y+z=11$ by 2 gives $8 x+14 y+2 z=22$ and multiplying $3 x+y+5 z=15$ by 3 gives $9 x+3 y+15 z=45$.

Adding $8 x+14 y+2 z=22$ and $9 x+3 y+15 z=45$ gives $17 x+17 y+17 z=67$, and after dividing through by 17 , we get $x+y+z=\frac{67}{17}$.

The fraction $\frac{67}{17}$ is in lowest terms, which means $p=67$ and $q=17$, so $p-q=67-17=50$.

## Solution 2

We will try to solve the system of equations

$$
\begin{aligned}
4 x+7 y+z & =11 \\
3 x+y+5 z & =15 \\
x+y+z & =\frac{p}{q}
\end{aligned}
$$

for $x, y$, and $z$ in terms of $p$ and $q$ and see what happens. For this particular system, it is impossible to express $x, y$, and $z$ in terms of only $p$ and $q$. However, a relationship between $p$ and $q$ will be revealed by the process of attempting to solve the system.

Subtracting the second equation from the first, we get $x+6 y-4 z=-4$, and subtracting $x+y+z=\frac{p}{q}$ from this equation gives $5 y-5 z=-4-\frac{p}{q}$.

After dividing by 5 , we get $y-z=-\frac{4}{5}-\frac{p}{5 q}$.

Next, we subtract 4 times the second equation from 3 times the first equation to get

$$
\begin{aligned}
3(4 x+7 y+z)-4(3 x+y+5 z) & =3(11)-4(15) \\
12 x+21 y+3 z-12 x-4 y-20 z & =33-60 \\
17 y-17 z & =-27 \\
y-z & =-\frac{27}{17}
\end{aligned}
$$

We now have two expressions for $y-z$, and if we set them equal, we get an equation involving $p$ and $q$. We can manipulate this equation as follows.

$$
\begin{aligned}
-\frac{4}{5}-\frac{p}{5 q} & =-\frac{27}{17} \\
-\frac{4}{5}+\frac{27}{17} & =\frac{p}{5 q} \\
-4+\frac{5 \times 27}{17} & =\frac{p}{q} \\
\frac{-68+135}{17} & =\frac{p}{q} \\
\frac{67}{17} & =\frac{p}{q}
\end{aligned}
$$

and since $\frac{67}{17}$ is in lowest terms, $p=67$ and $q=17$, so $p-q=67-17=50$.

ANSWER: 50