Using Gaussian elimination to solve 3x3 matrix with unknown.

Question

I'm trying to find values for k, such that the set of equations(Ax=b) have i)one solution, ii)infinite solutions and iii) no solutions.

All I can think of is dividing rows 1 and 3 by the number 2, and then perhaps subtracting row 1 from row 3.

I've been stuck for a while and have been making no progress.

Any help will be greatly appreciated :)

Answer 1

from the first equation we get $$x_1=1-x_2$$ plugging this in the third equation we get $$kx_3=1$$ if $k=0$ then we have no Solutions, if $$k\neq0$$ then we get $$x_3=\frac{1}{k_3}$$ Can you finish?

Answer 2

Subtract half of the first equation from the third to get a zero bottom left.

Swap the second and third equations (to get a constant in the middle).

Subtract $k$ times the second from the third.

Now you have an echelon form, which is easy to solve.