Question

The values of $\lambda$ and $\mu$ for which the system of linear equation, $x+y+z=2,x+2y+3z=5$ and $x+3y+\lambda z=\mu$ has infinitely many solution. The values respectively are,

• A. $6$ and $8$
• B. $5$ and $8$
• C. $5$ and $7$
• D. $4$ and $9$

Answer 1

The correct option is B

$5$ and $8$

Evaluate the values of the given variable based on system of linear equations

Given, $x+y+z=2,x+2y+3z=5$ and $x+3y+\lambda z=\mu$

As it is known that these lines have infinitely many solutions, so we can say that, $\left|\begin{array}{ccc}{x}_1& y_1& z_1\\ x_2& y_2& z_2\\ x_3& y_3& z_3\end{array}\right|=0$

So, according to the equations, $\left|\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 3& \lambda \end{array}\right|=0$

Applying transformation along the rows,

$$\begin{gathered} R_2\to R_2-R_1 \\ {R}_3\to R_3-R_1 \end{gathered}$$

$\Rightarrow \left|\begin{array}{ccc}1& 1& 1\\ 0& 1& 2\\ 0& 2& \lambda -1\end{array}\right|=0$

$$\begin{gathered} \Rightarrow 1(\lambda -1-4)-1(0-0)+1(0-0)=0 \\ \Rightarrow \lambda -1-4=0 \\ \Rightarrow \lambda =5 \end{gathered}$$

Also, it is given that, $\left|\begin{array}{ccc}{x}_1& y_1& a\\ x_2& y_2& b\\ x_3& y_3& c\end{array}\right|=0$

Again, applying transformation along the rows,

$$\begin{gathered} R_2\to R_2-R_1 \\ {R}_3\to R_3-R_1 \end{gathered}$$

$\Rightarrow \left|\begin{array}{ccc}1& 1& 2\\ 0& 1& 3\\ 0& 2& \mu -2\end{array}\right|=0$

$$\begin{gathered} \Rightarrow 1(\mu -2-6)-1(0-0)+2(0-0)=0 \\ \Rightarrow \mu -2-6=0 \\ \Rightarrow \mu =8 \end{gathered}$$

Therefore, $\lambda =5,\mu =8$.

Thus, option (B) is the correct answer.