## Task 3 - 281213

a) Provide two quadruples $(x, y, z, u)$ of real numbers that satisfy the following system of equations (1) to (4).

b) Determine a quadruple $(x, y, z, u)$ of integers such that one of the variables $x, y, z, u$ has the value 1988 and the system of equations (1) to (4) is satisfied.

$$
\begin{aligned}
1 x+9 y+8 z+8 u & =1 \\
9 x+9 y+24 z+24 u & =9 \\
8 x-13 y+8 z+7 u & =8 \\
8 x-21 y-10 z+8 u & =8
\end{aligned}
$$
a) Suppose there is a quadruple $(x, y, z, u)$ of real numbers that satisfies the system of equations (1) to (4). Then, from (1),

$$
x=1-9 y-8 z-8 u
$$

Substituting this into (2) to (4) yields

$$
\begin{aligned}
3 y+2 z+2 u & =0 \\
85 y+56 z+57 u & =0 \\
93 y+74 z+56 u & =0
\end{aligned}
$$

From (5), we have $z=-\frac{3}{2} y-u$. Substituting this into (6) and (7) results in $y+u=0 \quad$ (8) in both cases.

Now let $y=t$, where $t$ is any real number. From (8), (5), and (1), we get $u=-t, z=-\frac{1}{2} t$, and $x=1+3 t$. Verification confirms that for any $t$, the quadruple

$$
\left(1+3 t, t,-\frac{1}{2} t,-t\right)
$$

is a solution. Thus, the solution to the system of equations (1) to (4) is given by (9), where the parameter $t$ runs through all real numbers.

b) The following table provides a selection of quadruples.

| $t$ | $x$ | $y$ | $z$ | $u$ |
| :---: | :---: | :---: | :---: | :---: |
| 0 | 1 | 0 | 0 | 0 |
| 1988 | 5965 | 1988 | -994 | -1988 |
| -1988 | -5963 | -1988 | 994 | 1988 |

$x=1998$ yields $t=\frac{x-1}{3}$, which does not result in an integer for $y$.

Adapted from $[5]$