https://file.aiquickdraw.com/mathbot/file/1730339162458fqly1e16.jpg

Let's analyze the matrix in reduced echelon form:

\[
\begin{bmatrix}
1 & 0 & 2 & -1 \\
0 & 1 & 5 & 4 \\
0 & 0 & 0 & 0 \\
\end{bmatrix}
\]

Each row represents an equation. Let's write the system of equations:

### Step 1: Write the Equations
1. **Row 1**: \(1x_1 + 0x_2 + 2x_3 = -1\)
   - This simplifies to: \(x_1 + 2x_3 = -1\).

2. **Row 2**: \(0x_1 + 1x_2 + 5x_3 = 4\)
   - This simplifies to: \(x_2 + 5x_3 = 4\).

3. **Row 3**: \(0 = 0\), which is always true and doesn’t add any new information.

So the system of equations is:
\[
x_1 + 2x_3 = -1
\]
\[
x_2 + 5x_3 = 4
\]

### Step 2: Express in Terms of Free Variable
Since \(x_3\) is not constrained by any leading variable (it has no pivot in any row), it’s a free variable. Let \(x_3 = t\), where \(t\) is any real number.

Now we can express \(x_1\) and \(x_2\) in terms of \(t\):
1. From \(x_1 + 2x_3 = -1\):
   \[
   x_1 = -1 - 2t
   \]

2. From \(x_2 + 5x_3 = 4\):
   \[
   x_2 = 4 - 5t
   \]

### Solution
The solution in terms of the parameter \(t\) is:
\[
(x_1, x_2, x_3) = (-1 - 2t, 4 - 5t, t)
\]

Would you like more details or have any questions?

### Related Questions:
1. How can we determine if a system has infinitely many solutions?
2. What are the characteristics of a matrix in reduced echelon form?
3. How do free variables affect the solution set of a system?
4. What is the difference between row echelon form and reduced row echelon form?
5. How can this solution be interpreted geometrically?

#### Tip:
For a matrix to represent a consistent system, any row of zeros must correspond to a true statement like \(0 = 0\).

Write the system of equations represented by the matrix in reduced echelon form, then find the solution in terms of a parameter if there are infinitely many solutions.

Learn how to interpret a matrix in reduced echelon form to write a system of linear equations and find solutions in terms of a free variable. This guide covers key concepts in matrix operations and linear algebra, suitable for high school or college students studying systems of equations.

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