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?

Consider the system:
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:

Given:

`3x + 5y + 5z = −2`
`−3x + −1y + 0z = 7`
`−6x + 6y + 10z = k`

The augmented matrix is:

#[ (3, 5, 5, |, −2), (−3, −1, 0,|, 7), (−6, 6, 10,|, k) ]#

We can easily make the first column become a 3 in the top left and the rest 0s:

#[ (3, 5, 5, |, −2), (0, 4, 5,|, 5), (0, 16, 20,|, k-4) ]#

Divide the third row by 4:

#[ (3, 5, 5, |, −2), (0, 4, 5,|, 5), (0, 4, 5,|, k/4-1) ]#

To be consistent (In this case, "consistent" means have a linear equation that describes an infinite number of solutions) row 2 must be identical to row 3, which implies:

`k/4 - 1 = 5`

`k/4 = 6`

`k = 24`