Question

If the system of linear equations :
$x_1 + 2x_2 + 3x_3 = 6$
$x_1 + 3x_2 + 5x_3 = 9$
$2x_1 + 5x_2 + ax_3 = b$
is consistent and has infinite number of solutions, then :

• A. a = 8, b can be any real number
• B. b = 15, a can be any real number
• C. a $\in$ R -{8} and b $\in$ R -{15}
• D. a = 8, b = 15

Answer 1

Answer: (D)

Given system of equations can be written in matrix form as AX = B where
$A = \begin{pmatrix}1&2&3\\ 1&3&5\\ 2&5&a\end{pmatrix} $ and $B = \begin{pmatrix}6\\ 9\\ b\end{pmatrix}$
Since, system is consistent and has infinitely many solutions
$ \therefore \left(\text{adj} . A\right)B=0$
$ \Rightarrow \begin{pmatrix}3a-25&15-2a&1\\ 10-a&a-6&-2\\ -1&-1&1\end{pmatrix} \begin{pmatrix}6\\ 9\\ b\end{pmatrix}= \begin{pmatrix}0\\ 0\\ 0\end{pmatrix} $
$ \Rightarrow - 6 - 9 + b = 0 \ \Rightarrow \ b = 15$
and $6(10 - a) + 9(a - 6) - 2(b) = 0$
$\Rightarrow \ 60 - 6a + 9a - 54 - 30 = 0$
$\Rightarrow \ 3a = 24 \ \Rightarrow \ a = 8$
Hence, $a = 8, b = 15$.