Question

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

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

Answer 1

`k=24`

Explanation:

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`.