# How can one understand if a system is consistent from RREF?

We row reduce the corresponding augmented matrix to RREF.

$$\left[\begin{array}{rrr|r}2&3&3&-4\\-1&3&-6&2\\3&5&4&-6\end{array}\right] \begin{array}{c}R_1+R_2\\\\R_3+3R_2\end{array}\sim \left[\begin{array}{rrr|r}1&6&-3&-2\\-1&3&-6&2\\0&14&-14&0\end{array}\right] \begin{array}{c}R_2+R_2\end{array}\sim \left[\begin{array}{rrr|r}1&6&-3&-2\\0&9&-9&0\\0&14&-14&0\end{array}\right] \begin{array}{c}\frac19R_2\end{array}\sim \left[\begin{array}{rrr|r}1&6&-3&-2\\0&1&-1&0\\0&14&-14&0\end{array}\right] \begin{array}{c}R_1-6R_2\\\\R_3-14R_2\end{array}\sim \left[\begin{array}{rrr|r}1&0&3&-2\\0&1&-1&0\\0&0&0&0\end{array}\right]$$

Thus, we can see the system if consistent and contains $1$ free variable. Thus, the system is consistent with infinitely many solutions. Therefore, the answer is (b).

I was looking at my Linear Algebra quiz solutions and I saw the following:

Thus from RREF, we can see the system if consistent and contains 1 free variable. Thus, the system is consistent with infinitely many solutions.

1. How can one understand if a system is consistent from RREF (that it has at least one solution). Is this about the free variable?

2. Which theorem does the answer use to conclude "Thus, the system is consistent with infinitely many solutions"?