Given that the augmented matrix in row-reduced form is equivalent to the augmented matrix of a system of linear equations, do the following. (Use $x, y$, and $z$ as your variables, each representing the columns in turn.)
\[
\left[\begin{array}{lll|l}
1 & 0 & 0 & 1 \\
0 & 1 & 0 & 4 \\
0 & 0 & 0 & 0
\end{array}\right]
\]
(a) Determine whether the system has a solution.
The system has one solution.
The system has infinitely many solutions.
The system has no solution.
(b) Find the solution or solutions to the system, if they exist. (If there is no solution, enter NO sOLUTION. If there are infinitely many solutions, express your answer in terms of the parameter $t$. Use $s$ if a secontuparameter is needed.)
\[
(x, y, z)=(\square)
\]
Answer
The solutions to the system can be expressed as \[(x, y, z) = (1, 4, t)\], where t is any real number.
Step 1 ：Given the augmented matrix in row-reduced form is equivalent to the augmented matrix of a system of linear equations, we have the following matrix: \[\left[\begin{array}{lll|l} 1 & 0 & 0 & 1 \ 0 & 1 & 0 & 4 \ 0 & 0 & 0 & 0 \end{array}\right]\]
Step 2 ：From the matrix, we can see that the system has infinitely many solutions because the third row of the matrix is all zeros, which means that the third variable, z, is free to take any value.
Step 3 ：The solutions to the system can be expressed as \[(x, y, z) = (1, 4, t)\], where t is any real number.
