Question

Consider the system of linear equations: \[ x + 2y + z = -3, \] \[ 3x + 3y - 2z = -1, \] \[ 2x + 7y + 7z = -4. \] Determine the nature of its solutions.

• A. Infinite number of solutions
• B. No solution
• C. Unique solution
• D. Finite number of solutions

Hint:

To determine whether a system has a solution, convert it to row echelon form. If a row of the form \( 0 = c \) (where \( c \neq 0 \)) appears, the system is inconsistent and has no solution.

Answer 1

The Correct Option is B

Step 1: Representing the system in augmented matrix form
The given system of equations can be written in matrix form as: \[ \begin{bmatrix} 1 & 2 & 1 & | -3
3 & 3 & -2 & | -1
2 & 7 & 7 & | -4 \end{bmatrix} \] Step 2: Row reduction
Performing row operations to convert the augmented matrix into row echelon form: 1. Make the first pivot 1 (it is already 1 in the first row). 2. Subtract \( 3 \times \) (Row 1) from Row 2. 3. Subtract \( 2 \times \) (Row 1) from Row 3. After performing these operations, we get: \[ \begin{bmatrix} 1 & 2 & 1 & | -3
0 & -3 & -5 & | 8
0 & 3 & 5 & | -2 \end{bmatrix} \] Adding Row 2 and Row 3 to eliminate the second column entry in Row 3: \[ \begin{bmatrix} 1 & 2 & 1 & | -3
0 & -3 & -5 & | 8
0 & 0 & 0 & | 6 \end{bmatrix} \] Step 3: Checking for inconsistency
The last row corresponds to the equation: \[ 0x + 0y + 0z = 6, \] which is a contradiction (since \( 0 \neq 6 \)). This indicates that the system has no solution. Step 4: Conclusion
Since the system is inconsistent, it has no solution.