The augmented matrix can be used to determine the number of solutions:

$$\left[\begin{array}{ccc|c}a & b & c & d\\0 & e & f & g\\0 & 0 & h & i\end{array}\right]$$

If \( h \neq 0\) there is a unique solution.

If \( h = 0 \text{ and } i \neq 0 \) there is no solution

If \( h = 0 \text{ and } i = 0 \) there are infinitely many solutions (let \(z=t\)).

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