[Math] If and when a linear system has exactly three solutions

Question

Does the following system have exactly three solutions?
$$\left\{ \begin{array}{l}
2x – y +3z = 1 \\
x + 4y – 2z = -7 \\
3x + y -z = 4 \\
\end{array} \right.$$

I marked this answer as True.
I proceeded to row reduce it and obtained the resultant matrix as –

$$\left[ \begin{array}{ccc|c}
1 & -1/2 & 3/2 & 4 \\
0 & 1 & 9 & 13/9 \\
0 & 0 & 1 & 10/28 \\
\end{array} \right]$$

I'm pretty sure I made some mistake while row-reducing it. I answered this question in an exam setting. Then I gave the explanation as follows –

Form row reduction, we know that the system is consistent and the rank of the matrix is 3 which is equal to the number of variables. So, the system of equations has 3 distinct solutions.

Can someone point out where I went wrong?

Answer 1

HINT: The system of linear equations can have only 0, 1 or infinitely many solutions. Hence no calculations are needed.