Consider the following system of linear equations in variables $x, y$, and $z$.
\[
\left\{\begin{array}{r}
x+2 y+2 z=0 \\
-x-y+2 z=0 \\
2 x+y+k z=0
\end{array}\right.
\]
where $k$ is a constant.
Which of the following statements is true? (Read them carefully!)
The system has a nontrivial solution for any value of $k$
The system is inconsistent for any value of $k$.
The system has only the trivial solution for any value of $k$.
The system has a nontrivial solution for exactly two values of $k$.
The system has a nontrivial solution for exactly one value of $k$.

Step 1 ：Consider the following system of linear equations in variables \(x, y\), and \(z\):

Step 2 ：\[\begin{cases} x+2y+2z=0 \\ -x-y+2z=0 \\ 2x+y+kz=0 \end{cases}\]

Step 3 ：where \(k\) is a constant. The system of equations is homogeneous, meaning that the right-hand side of each equation is zero. This implies that the system always has at least the trivial solution (where \(x = y = z = 0\)). The question is whether there are nontrivial solutions, and if so, for what values of \(k\).

Step 4 ：To find out, we can write the system of equations in matrix form and then calculate its determinant. If the determinant is nonzero for a certain value of \(k\), then the system has a unique solution (the trivial one). If the determinant is zero, then the system has either no solutions or an infinite number of solutions (including nontrivial ones).

Step 5 ：Let's calculate the determinant of the matrix formed by the coefficients of the variables in the system of equations.

Step 6 ：\[A = \begin{bmatrix} 1 & 2 & 2 \\ -1 & -1 & 2 \\ 2 & 1 & k \end{bmatrix}\]

Step 7 ：The determinant of the matrix is \(k + 8\). This means that the system of equations has a nontrivial solution if and only if \(k + 8 = 0\), or equivalently, \(k = -8\).

Step 8 ：\[\boxed{\text{The system has a nontrivial solution for exactly one value of } k}\]