The values of lambda and mu for which the sys | vedclass

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(C) The given system of equations is:$x+y+z=6$$x+2y+3z=10$$x+2y+\lambda z=\mu$For the system to have infinitely many solutions,the augmented matrix $[

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$\lambda=3, \mu=10$

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(C) The given system of equations is:$x+y+z=6$$x+2y+3z=10$$x+2y+\lambda z=\mu$For the system to have infinitely many solutions,the augmented matrix $[A|B]$ must have a rank less than the number of variables $(3)$.Writing the augmented matrix:$\begin{bmatrix} 1 & 1 & 1 & | & 6 \ 1 & 2 & 3 & | & 10 \ 1 & 2 & \lambda & | & \mu \end{bmatrix}$Applying row operations:$R_2 	o R_2 - R_1$ and $R_3 	o R_3 - R_1$:$\begin{bmatrix} 1 & 1 & 1 & | & 6 \ 0 & 1 & 2 & | & 4 \ 0 & 1 & \lambda-1 & | & \mu-6 \end{bmatrix}$Applying $R_3 	o R_3 - R_2$:$\begin{bmatrix} 1 & 1 & 1 & | & 6 \ 0 & 1 & 2 & | & 4 \ 0 & 0 & \lambda-3 & | & \mu-10 \end{bmatrix}$For infinitely many solutions,the last row must be a zero row,meaning $\lambda-3=0$ and $\mu-10=0$.Thus,$\lambda=3$ and $\mu=10$.

The values of $\lambda$ and $\mu$ for which the system of equations $x+y+z=6, x+2y+3z=10, x+2y+\lambda z=\mu$ has infinitely many solutions are

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