Question

The system of equations \( x + 2y + 3z = 6 \), \( x + 3y + 5z = 9 \), \( 2x + 5y + az = 12 \) has no solution when \( a = \):

• A. 5
• B. 6
• C. 7
• D. 8

Hint:

To find the value of \(a\) that makes a system of equations inconsistent, use row operations to obtain a row echelon form of the augmented matrix and check for a row of the form \(0 \quad 0 \quad 0 \, | \, \text{nonzero}\).

Answer 1

The Correct Option is D

Step 1: Subtract the first row from the second row: \( R_2 \rightarrow R_2 - R_1 \)
\( \begin{pmatrix} 1 & 2 & 3 & | & 6 \\ 0 & 1 & 2 & | & 3 \\ 2 & 5 & a & | & 12 \end{pmatrix} \)

Step 2: Subtract twice the first row from the third row: \( R_3 \rightarrow R_3 - 2R_1 \)
\( \begin{pmatrix} 1 & 2 & 3 & | & 6 \\ 0 & 1 & 2 & | & 3 \\ 0 & 1 & a - 6 & | & 0 \end{pmatrix} \)

Step 3: Subtract the second row from the third row: \( R_3 \rightarrow R_3 - R_2 \)
\( \begin{pmatrix} 1 & 2 & 3 & | & 6 \\ 0 & 1 & 2 & | & 3 \\ 0 & 0 & a - 8 & | & -3 \end{pmatrix} \)

For the system to have no solution, the third row must be inconsistent, meaning we need the last entry in the third row to be nonzero while the coefficient of \(z\) is zero. Thus, for inconsistency:
\( a - 8 = 0 \quad \Rightarrow \quad a = 8 \)
Thus, the value of \(a\) that makes the system inconsistent is \(a = 8\).