For what value of $k$ is this system consistent? $3x + 5y + 5z = −2$ , $−3x + −1y + 0z = 7$ and $−6x + 6y + 10z = k$

Question

Consider the system:
3x + 5y + 5z = −2
−3x + −1y + 0z = 7
−6x + 6y + 10z = k
For what value of
k
is this system consistent?

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Answer 1 · 2018-05-05T17:30:24

24

Consider the adjoint matrix $(A|b)$ given by

$((3,5,5,|,-2),(-3,-1,0,|,7),(-6,6,10,|,k))$

We perform the row reductions :

$((3,5,5,|,-2),(0,4,5,|,5),(0,16,20,|,k-4))$

$((3,5,5,|,-2),(0,4,5,|,5),(0,0,0,|,k-24))$

So, the last row now reads

$0x+0y+0z = k-24$

which can be consistent only for $k=24$