How to solve this Linear Algebra problem involving a system of linear equations?

Question

The following is what I have so far.

I'm not sure how to use my echelon matrix to find out which values for the variables can provide an answer to the question or how to prove it. I was thinking of plugging in arbitrary numbers for $x_3\ y_1\ y_2$ but not sure if this is the way to approach this.

Answer 1

If q=1, then you are left with only two equations and three unknowns. This gives you and infinite number of possible solutions for (y1,y3). To find the solutions, solve $5x_2-x_3=y_3$ and $x_1-2x_2+x_3=y_1$.

If $q$ does not equal on3, then you have only one solution which is the right most column of your reduced row echelon matrix.

Can you take it from here?

Answer 2

Expanded coefficients matrix and its reduction:

$$\begin{pmatrix}1&\!\!-2&1&y_1\\ 2&1&q&y_2\\ 0&5&\!\!-1&y_3\end{pmatrix}\stackrel{R_2-2R_1}\rightarrow\begin{pmatrix}1&\!\!-2&1&y_1\\ 0&5&q-2&y_2-2y_1\\ 0&5&\!\!-1&y_3\end{pmatrix}\stackrel{R_3-R_2}\rightarrow$$$${}$$

$$\rightarrow\begin{pmatrix}1&\!\!-2&1&y_1\\ 0&5&q-2&y_2-2y_1\\ 0&0&1-q&y_3-y_2+2y_1\end{pmatrix}$$

Thus, for example:

$$\text{If}\;\;q=1\implies y_3-y_2+2y_1=0$$

and the system has solution (but not unique! Why?) . Try to take it from here.