To find the value of \( \lambda \) for which the system of equations has infinite solutions, we will analyze the given equations:
1. \( x + 2y + 3z = 6 \) (Equation 1)
2. \( 4x + 5y + 6z = \lambda \) (Equation 2)
3. \( 7x + 8y + 9z = 24 \) (Equation 3)
### Step 1: Write the system in matrix form
We can represent the system of equations in augmented matrix form:
\[
\begin{bmatrix}
1 & 2 & 3 & | & 6 \\
4 & 5 & 6 & | & \lambda \\
7 & 8 & 9 & | & 24
\end{bmatrix}
\]
### Step 2: Apply row operations to find conditions for infinite solutions
For the system to have infinite solutions, the rank of the coefficient matrix must be equal to the rank of the augmented matrix and less than the number of variables (which is 3).
We will perform row operations to simplify the matrix. Let's perform \( R_3 - 7R_1 \) and \( R_2 - 4R_1 \):
1. \( R_2 \) becomes:
\[
R_2 - 4R_1: \quad (4 - 4 \cdot 1, 5 - 4 \cdot 2, 6 - 4 \cdot 3, \lambda - 4 \cdot 6) = (0, -3, -6, \lambda - 24)
\]
2. \( R_3 \) becomes:
\[
R_3 - 7R_1: \quad (7 - 7 \cdot 1, 8 - 7 \cdot 2, 9 - 7 \cdot 3, 24 - 7 \cdot 6) = (0, -6, -12, 24 - 42) = (0, -6, -12, -18)
\]
Now, the augmented matrix looks like this:
\[
\begin{bmatrix}
1 & 2 & 3 & | & 6 \\
0 & -3 & -6 & | & \lambda - 24 \\
0 & -6 & -12 & | & -18
\end{bmatrix}
\]
### Step 3: Simplify further
Next, we can simplify \( R_3 \) by dividing it by -6:
\[
R_3 \rightarrow \frac{1}{-6} R_3: \quad (0, 1, 2, 3)
\]
So the matrix now becomes:
\[
\begin{bmatrix}
1 & 2 & 3 & | & 6 \\
0 & -3 & -6 & | & \lambda - 24 \\
0 & 1 & 2 & | & 3
\end{bmatrix}
\]
### Step 4: Make \( R_2 \) simpler
Next, we can add \( 3R_3 \) to \( R_2 \):
\[
R_2 + 3R_3: \quad (0, -3 + 3 \cdot 1, -6 + 3 \cdot 2, \lambda - 24 + 3 \cdot 3) = (0, 0, 0, \lambda - 24 + 9)
\]
This gives us:
\[
\begin{bmatrix}
1 & 2 & 3 & | & 6 \\
0 & 0 & 0 & | & \lambda - 15 \\
0 & 1 & 2 & | & 3
\end{bmatrix}
\]
### Step 5: Set the condition for infinite solutions
For the system to have infinite solutions, the last row must be consistent, which means:
\[
\lambda - 15 = 0 \implies \lambda = 15
\]
### Conclusion
Thus, the value of \( \lambda \) for which the system has infinite solutions is:
\[
\boxed{15}
\]
