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. $a\,\in\,R-\left\{8\right\}$ and $b\,\in \,R-\left\{15\right\}$
• 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
$\therefore \left(adj. A\right)B=0$
$\Rightarrow \begin{pmatrix}3a-25&15-2a&1\\ 10-a&a-6&-2\\ -1&-1&1\end{pmatrix}\begin{pmatrix}6\\ 9\\ b\end{pmatrix}=\begin{pmatrix}0\\ 0\\ 0\end{pmatrix}$
$\Rightarrow -6-9 + b = 0 \Rightarrow b= 15$
and $6\left(10 - a\right)+ 9\left(a-6\right)-2\left(6\right) = 0$
$\Rightarrow 60 - 6a + 9a - 54 - 30 = 0$
$\Rightarrow 3a = 24 \Rightarrow a=8$
Hence, $a = 8, b =15.$