For the following system to be consistent, what must k not be equal to?

Question

$6x - 4y + 4z = 5$

$9x - 6y + kz = -4$

$12x - 8y = -10$

Originally I just multiplied the first row by $\frac{3}{2}$ and subtracted it from the second, which gives you a value of $6$ for the answer. However, this is not the correct answer. Any idea what I am doing wrong?

Answer 1

Hint:

you need to look at the third equation as well! How is that related to the second equation?

Answer 2

Use the first and third equations to find $z$. Then substitute that into the equation you got from the first two equations. There is only one value $k$ can be.

Answer 3

Subtracting the first equation from the third twice yields $-8z=-20$, so $z=\frac52$. Then multiplying the second equation by two and subtracting the first three times yields $$-5k-30=-23.$$ This shows that $k=-\frac75$.