Question

If the system of linear equation
$x_1+2x_2+3x_3=6$
$x_1+3x_2+5x_3=9$
$2x_1+5x_2+a{x}_3=b$
is consistent and has infinite number of solution, then :-

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

Answer 1

Answer: (B)

$a=8,b$ can be any real number

Converting the equations into matrix form we get

$\left(\begin{array}{cccc}1& 2& 3& 6\\ 1& 3& 5& 9\\ 2& 5& a& b\end{array}\right)$

$R_2\to R_2-R_1$

$R_3\to R_3-2R_1$

$\left(\begin{array}{cccc}1& 2& 3& 6\\ 0& 1& -1& -3\\ 0& 1& a-6& b-12\end{array}\right)$

$R_3\to R_3-2R_1$

$\left(\begin{array}{cccc}1& 2& 3& 6\\ 0& 1& -1& -3\\ 0& 0& a-5& b-9\end{array}\right)$

Hence through the echolon form

we get $a-5\ne 0\Rightarrow a\ne 5$

b can be any real number