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 · 2017-09-21T14:32:49

$k = 24$ $k=24$

Let's create a matrix from the system and then perform Gaussian elimination.

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

Let's do $R_2=>R_2+R_3$ and $R_3=>R_3+2R_1$ .

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

Now let's do $R_1=>1/3R_1$ .

$=((1, 5/3, 5/3, -2/3), (0, 4, 5, 5), (0, 16, 20, k-4))$

$R_1=>R_1-5/12R_2$

$=((1, 0, -5/12, -33/12), (0, 4, 5, 5), (0, 16, 20, k-4))$

$R_2=>1/4R_2$

$=((1, 0, -5/12, -33/12), (0, 1, 5/4, 5/4), (0, 16, 20, k-4))$

$R_3=>R_3-16R_2$

$=((1, 0, -5/12, -33/12), (0, 1, 5/4, 5/4), (0, 0, 0, k-24))$

We can stop now: recall that the bottom row translates into $0x+0y+0x=k-24$ , that is, $0=k-24$ . So $k=24$ .