Question

If the system of linear equations:
$x_1+2x_2+3x_3=6$
$x_1+3x_2+5x_3=9$
$2x_1+5x_2+a{x}_3=b$
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
$A=\left(\begin{array}{ccc}1& 2& 3\\ 1& 3& 5\\ 2& 5& a\end{array}\right)$ and $B=\left(\begin{array}{c}6\\ 9\\ b\end{array}\right)$
Since, system is consistent and has infinitely many solutions
$\therefore \left(adj.A\right)B=0$
$\Rightarrow \left(\begin{array}{ccc}3a-25& 15-2a& 1\\ 10-a& a-6& -2\\ -1& -1& 1\end{array}\right)\left(\begin{array}{c}6\\ 9\\ b\end{array}\right)=\left(\begin{array}{c}0\\ 0\\ 0\end{array}\right)$
$\Rightarrow -6-9+b=0\Rightarrow b=15$
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, $a=8,b=15.$