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If the system of equations $$x + 4y - z = \lambda$$, $$7x + 9y + \mu z = -3$$, $$5x + y + 2z = -1$$ has infinitely many solutions, then $$(2\mu + 3\lambda)$$ is equal to :
For infinitely many solutions, the system must satisfy:
Determinant condition gives:
$$\mu=0$$
Now enforce consistency. From coefficient comparison:
$$R_3=-2R_1+R_2$$
Apply same to constants:
$$-1=-2\lambda+(-3)$$
$$\Rightarrow-2\lambda-3=-1$$
$$\Rightarrow\lambda=-1$$
Finally:
$$2\mu+3\lambda=2(0)+3(-1)=-3$$
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