Where x, y, and z are unkown variables, find all values for a and b for which

[skyturnred]

Homework Statement

Where x, y, and z are unknown variables and a and b are parameters, find all values a and b for which this system has (a) no solution (b) infinitely many solutions, and find the solutions (c) a unique solution, and find the solution when a=b=1.

matrix=[2 -1 1 a; 1 1 2 1; b 3 3 a] (where each ; denotes a new line in the matrix, so the matrix is 3 rows by 4 columns. And btw it is an augmented matrix).

Homework Equations

The Attempt at a Solution

I don't really know where to start! I know I have to somehow get the system into REF first, but it seems impossible because of the variables. I know that for (a) I have to somehow get a line that is something like [0 0 0 a] or [0 0 0 b] where a/b is not zero.

 

[HallsofIvy]

It's not "impossible", just requires a bit of algebra.

$$\begin{bmatrix}2 & -1 & 1 & a \\ 1 & 1 & 2 & 1 \\ b & 3 & 3 & a \end{bmatrix}$$
1a) swap the first two rows
1b) Subtract 2 times the new first row from the second row
1c) Subtract b times the new first row from the third row
That gives
$$\begin{bmatrix}1 & 1 & 2 & 1 \\ 0 & -3 & -3 & a- 2 \\ 0 & 3+ b & 1 & a- ab\end{bmatrix}$$
Can you continue? The algebra gets complicated but not "impossible".

Alternatively, you could start by determining the values of a and b that make the determinant 0 or not 0.