If the system of linear equations [x+y+z=6, x+2y+5z=10, 2x+3y+ z=] has infinitely many sol…

Question

If the system of linear equations
$$x+y+z=6,\qquad x+2y+5z=10,\qquad 2x+3y+\lambda z=\mu$$
has infinitely many solutions, then the value of $\lambda+\mu$ equals:

Accepted Answer

$D=\begin{vmatrix}1&1&1\1&2&5\2&3&\lambda\end{vmatrix}=\lambda-6$. For infinitely many solutions, $D=0\Rightarrow\lambda=6$.

$D_3=\begin{vmatrix}1&1&6\1&2&10\2&3&\mu\end{vmatrix}=\mu-16$, and setting $D_3=0$ gives $\mu=16$.

So $\lambda+\mu=6+16=22$.

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