I'm asked to solve the system of equation and to write the answer in parametric form.
However, I'm not sure what to do if my system has no free variable.
This is the system
$
\left[ \begin{array}{ccc|c}
1 & 2 & 3 &1\\
4 & 5 & 6 &1\\
5 & 8 & 9 &1\end{array} \right]
$
Using gauss jordan I found that
$
X1 = 0\\
X2 = -1\\
X3 = 1\\
$
Since I have no free variable, I'm not sure what it means to write this in parametric form.
This is what I figured out…
$0t – t2 + t3 = 1$
Is it correct ?
Best Answer
To give this problem an answer: as you found, the system of equations defined by the augmented matrix $$ \left[ \begin{array}{ccc|c} 1 & 2 & 3 & 1 \\ 4 & 5 & 6 & 1 \\ 5 & 8 & 9 & 1 \\ \end{array} \right] $$ has a unique solution, namely $(0,-1,1)^T$; we can verify this in various ways, e.g. by checking $$\det\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 5 & 8 & 9 \\ \end{pmatrix} = 6 \neq 0.$$ It doesn't really make sense to ask for this unique solution to be written in parametric form (unless you count $(0,-1,1)^T$ as in parametric form).
Thus, I think the $5$ in the bottom left corner of the augmented matrix should be a $7$. In this case, the system of equations becomes: $$ \left[ \begin{array}{ccc|c} 1 & 2 & 3 & 1 \\ 4 & 5 & 6 & 1 \\ 7 & 8 & 9 & 1 \\ \end{array} \right] $$ Since $(0,-1,1)^T$ is (also) a solution to this system of equations and $$\det\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{pmatrix} = 0,$$ there must be an infinite number of solutions. In this case, it makes sense to talk about writing the solutions in parametric form.