If the system of equations [x+y+z=5, x+2y+3z=9, x+3y+ z=] has infinitely many solutions, t…

Question

If the system of equations
$$x+y+z=5,\qquad x+2y+3z=9,\qquad x+3y+\lambda z=\mu$$
has infinitely many solutions, then the value of $\lambda+\mu$ is:

Accepted Answer

Using Cramer's determinants: $D=\lambda-5$, $D_1=\lambda+\mu-18$, $D_2=4\lambda-2\mu+6$, $D_3=\mu-13$. For infinitely many solutions, all of $D,D_1,D_2,D_3$ must vanish.

$D=0\Rightarrow\lambda=5$. $D_3=0\Rightarrow\mu=13$.

Checking: $D_1=\lambda+\mu-18=5+13-18=0$ , and $D_2=4(5)-2(13)+6=20-26+6=0$ . Consistent.

So $\lambda+\mu=5+13=18$.

Solutions · 1