# Solve linear system with variables?

I have a system of equations like the below:

$$x + 3y - z = a \\ x + y + 2z = b \\ 2y - 3z = c$$

And have put it in an augmented matrix:

$$\begin{bmatrix} 1 & 3 & -1 & a \\ 1 & 1 & 2 & b \\ 0 & 2 & -3 & c\end{bmatrix}$$

I need to find the conditions where the system is consistent (values of $a$, $b$, and $c$ whereby the system has a solution).

I've attempted to reduce the matrix to row echelon form, but the last column is getting quite crazy. I have to wonder what I should do (if this is even the correct start) once it is reduced: I am left with things like:

$$\begin{bmatrix} 1 & 0 & 0 & \text{a mess}\\ 0 & 1 & 0 & \text{a mess}\\ 0 & 0 & 1 & \text{a mess}\end{bmatrix}$$

If (or when) I get to the endpoint, and only if this is the right methodology, how do I determine the values of $a$, $b$, and $c$ in relation to $z_1$, $z_2$, and $z_3$?

• Is this an augmented matrix? May 19, 2015 at 7:22
• you have copied the question wrong, check your steps.
– UNM
May 19, 2015 at 7:27
• @UNM I've corrected the typo. May 19, 2015 at 7:28
• your row reduced is not the same after correcting the typo.
– UNM
May 19, 2015 at 7:31
• @UNM, I'm sorry but I don't follow. I did not show the reduced matrix, just a theoretical endpoint I am approaching. I haven't yet got to this stage, I'm just curious if I'm going about it correctly in the first place. May 19, 2015 at 7:33

## 2 Answers

HINT:

How can a system be inconsistent? Try to think about how does this look on the reduced matrix of the system.

• Call me naive, but I don't see how this helps. An inconsistent system is where two of the formulae conflict (such as $4x = 2$ and $3x = 5$). May 19, 2015 at 7:26
• Inconsistency can be seen from the matrix when you have a row such that the variable coefficients are zero but the "free" coefficient is non-zero. Can this happen here? (I'm assuming this is an augmented matrix). May 19, 2015 at 7:28
• This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. May 19, 2015 at 7:44
• @PeterWoolfitt I'm trying to give a subtle hint that hopefully will help. If it doesn't help, I wait for the OP to comment and explain himself. If you request this answer to be deleted, I'll do it without a problem. May 19, 2015 at 7:48
• @SolidSnake I'm sorry, it looks like I reviewed this too hastily. I interpreted your answer as another question posted as an answer (in particular I interpreted it to mean you were asking for a clarification of the OP's definition of inconsistent which should be a comment). Could you perhaps change your answer to make it not look like a short question? Perhaps changing it to "Think about what makes a system inconsistent." Still, it was completely my fault - I should have been clued in by the fact that you said hint. May 19, 2015 at 7:54

After you row-reduce the matrix, observe that you get all zeros in the original part of the matrix and an expression of a,b and c in the augmented part in the last row. So for the consistency of the solution the expression in a,b and c has to be zero in the last row.That's the condition.