Question

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

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

24

Explanation:

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 :

• `R_2 leftarrowR_2+R_1,quad R_3 leftarrowR_3+2R_1` :

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

• `R_3 leftarrowR_3-4 R_1` :

`((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`