6. [8.58/11.11 Points) DITAILS MY NOTES TANFIN12 2.3.004. Given that the augmented matrix in row-reduced form is equivalent to the augmented matrix of a system as your variables, each representing the columns in turn.) $$ \left[\begin{array}{lll|l} 1 & 0 & 0 & 5 \ 0 & 1 & 0 & 3 \ 0 & 0 & 0 & 0 \end{array}\right] $$ (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 S your answer in terms of the parameters $$ t $$ and/or s.) $$ (x, y, z)=(\square) $$

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To analyze the given augmented matrix and determine the solutions to the corresponding system of equations, we will follow these steps:

### Step 1: Analyze the Augmented Matrix

The given augmented matrix in row-reduced form is:

$$
\left[\begin{array}{lll|l}
1 & 0 & 0 & 5 \
0 & 1 & 0 & 3 \
0 & 0 & 0 & 0
\end{array}\right]
$$

This matrix corresponds to the following system of equations:

1. $$ x = 5 $$
2. $$ y = 3 $$
3. $$ 0 = 0 $$ (which is always true)

### Step 2: Determine if the System has a Solution

- The first two rows indicate that $$ x $$ and $$ y $$ have specific values (5 and 3, respectively).
- The third row does not impose any restrictions on $$ z $$ since it simplifies to $$ 0 = 0 $$.

Since there are no contradictions in the equations, the system has a solution.

### Conclusion for Part (a)

The system has **infinitely many solutions** because $$ z $$ can take any value.

### Step 3: Find the Solution or Solutions

From the equations derived from the augmented matrix, we have:

- $$ x = 5 $$
- $$ y = 3 $$
- $$ z $$ is free (let's denote it as $$ t $$)

Thus, we can express the solution in terms of the parameter $$ t $$:

$$
(x, y, z) = (5, 3, t)
$$

### Conclusion for Part (b)

The solution can be expressed as:

$$
(x, y, z) = (5, 3, t)
$$

where $$ t $$ is any real number.

### Final Answers

(a) The system has **infinitely many solutions**.

(b) The solution is:

$$
(x, y, z) = (5, 3, t)
$$
The system has infinitely many solutions, and the solution is $$ (x, y, z) = (5, 3, t) $$, where $$ t $$ is any real number.

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