Find all real solutions to the following system of equations:

$$
\left\{\begin{aligned}
x+y+z+t & =5 \\
x y+y z+z t+t x & =4 \\
x y z+y z t+z t x+t x y & =3 \\
x y z t & =-1
\end{aligned}\right.
$$
Answer: $x=\frac{1 \pm \sqrt{2}}{2}, y=2, z=\frac{1 \mp \sqrt{2}}{2}, t=2$ or $x=2, y=\frac{1 \pm \sqrt{2}}{2}$, $z=2, t=\frac{1 \mp \sqrt{2}}{2}$.

Let $A=x+z$ and $B=y+t$. Then the system of equations is equivalent to

$$
\left\{\begin{aligned}
A+B & =5 \\
A B & =4 \\
B x z+A y t & =3 \\
(B x z) \cdot(A y t) & =-4 .
\end{aligned}\right.
$$

The first two of these equations imply $\{A, B\}=\{1,4\}$ and the last two give $\{B x z, A y t\}=\{-1,4\}$. Once $A=x+z, B=y+t, B x z$ and Ayt are known, it is easy to find the corresponding values of $x, y, z$ and $t$. The solutions are shown in the following table.

| $A$ | $B$ | Bxz | Ayt | $x, z$ | $y, t$ |
| :---: | :---: | :---: | :---: | :---: | :---: |
| 1 | 4 | -1 | 4 | $\frac{1 \pm \sqrt{2}}{2}$ | 2 |
| 1 | 4 | 4 | -1 | - | - |
| 4 | 1 | -1 | 4 | - | - |
| 4 | 1 | 4 | -1 | 2 | $\frac{1 \pm \sqrt{2}}{2}$ |