Question

The system of linear equations \[ \begin{cases} x + y + z = 6 \\ 2x + 5y + az = 36 \\ x + 2y + 3z = b \end{cases} \] has:

• A. Infinitely many solutions for \( a = 8 \) and \( b = 16 \)
• B. Unique solution for \( a = 8 \) and \( b = 16 \)
• C. Unique solution for \( a = 8 \) and \( b = 14 \)
• D. Infinitely many solutions for \( a = 8 \) and \( b = 14 \)

Hint:

For linear systems, infinitely many solutions occur when equations are dependent but consistent.

Answer 1

The Correct Option is D

Step 1: Write the augmented matrix.
\[ \left[ \begin{array}{ccc|c} 1 & 1 & 1 & 6 \\ 2 & 5 & a & 36 \\ 1 & 2 & 3 & b \end{array} \right] \]
Step 2: Find the condition for infinitely many solutions.
For infinitely many solutions, \[ \text{Rank of coefficient matrix} = \text{Rank of augmented matrix}<3 \]
Step 3: Substitute \( a = 8 \).
The determinant of the coefficient matrix becomes zero, making the system dependent.
Step 4: Find the value of \( b \).
Consistency of equations gives \[ b = 14 \]
Step 5: Conclusion.
Hence, the system has infinitely many solutions for \( a = 8 \) and \( b = 14 \).