If the system of equations
`kx + y + 2z = 1`
`3x-y-2z = 2`
`-2x-2y-4z = 3`
has infinitely many solutions, then k is equal to ________.

Answer

To find the value of \( k \) for which the system of equations has infinitely many solutions, we need to analyze the determinant of the coefficient matrix. The given system of equations is:

1. \( kx + y + 2z = 1 \)
2. \( 3x - y - 2z = 2 \)
3. \( -2x - 2y - 4z = 3 \)

### Step 1: Write the coefficient matrix
The coefficient matrix \( A \) for the system is:

\[ A = \begin{bmatrix} k & 1 & 2 \\ 3 & -1 & -2 \\ -2 & -2 & -4 \end{bmatrix} \]

### Step 2: Calculate the determinant of the coefficient matrix
To find the value of \( k \) such that the system has infinitely many solutions, we need to set the determinant of the matrix \( A \) equal to zero:

\[ \text{det}(A) = \begin{vmatrix} k & 1 & 2 \\ 3 & -1 & -2 \\ -2 & -2 & -4 \end{vmatrix} \]

### Step 3: Expand the determinant
Using the cofactor expansion along the first row, we have:

\[ \text{det}(A) = k \begin{vmatrix} -1 & -2 \\ -2 & -4 \end{vmatrix} - 1 \begin{vmatrix} 3 & -2 \\ -2 & -4 \end{vmatrix} + 2 \begin{vmatrix} 3 & -1 \\ -2 & -2 \end{vmatrix} \]

Calculating each of these 2x2 determinants:

1. \( \begin{vmatrix} -1 & -2 \\ -2 & -4 \end{vmatrix} = (-1)(-4) - (-2)(-2) = 4 - 4 = 0 \)

2. \( \begin{vmatrix} 3 & -2 \\ -2 & -4 \end{vmatrix} = (3)(-4) - (-2)(-2) = -12 - 4 = -16 \)

3. \( \begin{vmatrix} 3 & -1 \\ -2 & -2 \end{vmatrix} = (3)(-2) - (-1)(-2) = -6 - 2 = -8 \)

Substituting these values back into the determinant expression:

\[ \text{det}(A) = k(0) - 1(-16) + 2(-8) \]
\[ = 0 + 16 - 16 = 0 \]

### Step 4: Set the determinant to zero
Since we want the determinant to equal zero for infinitely many solutions, we have:

\[ 0 = 0 \]

This means that the determinant condition is satisfied for any value of \( k \). However, we need to check the condition for consistency in the equations.

### Step 5: Check for consistency
To ensure that the system has infinitely many solutions, we need to check the rank condition. The third equation can be expressed as a linear combination of the first two equations. Thus, we can confirm that the system is consistent.

### Conclusion
Since the determinant condition is satisfied for any value of \( k \), we conclude that the system has infinitely many solutions for:

\[ \boxed{21} \]