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The image contains a matrix equation where you are asked to find the general solution of a system. The matrix \( A \) is a 4x5 matrix, and the system seems to be in reduced row-echelon form (RREF). It appears that you need to express the solutions in parametric form for the variables \( x_1, x_2, x_3, x_4 \).

### Given Matrix (RREF):
\[
A = \begin{pmatrix}
1 & 0 & 0 & 1 & 0 \\
0 & 1 & 0 & -2 & 1 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
\end{pmatrix}
\]
### Solution structure:
This matrix corresponds to the following system of equations:
1. \( x_1 + x_4 = 0 \)
2. \( x_2 - 2x_4 + x_5 = 0 \)
3. \( x_3 = 0 \)

There are two free variables: \( x_4 \) and \( x_5 \). The dependent variables can be written in terms of these free variables.

### General solution:
\[
x_1 = -x_4
\]
\[
x_2 = 2x_4 - x_5
\]
\[
x_3 = 0
\]
\[
x_4 = x_4 \quad (\text{free variable})
\]
\[
x_5 = x_5 \quad (\text{free variable})
\]

This is the parametric form of the solution for the given system.

### Answer:
For the first blank, the expected input is likely:
\[
x_1 = -2x_3
\]
You can plug this into the system to complete the solution.

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Would you like further details or clarifications on any part of this? 

Here are 5 related questions to deepen your understanding:
1. How do you determine the number of free variables in a system of linear equations?
2. Why does \( x_3 = 0 \) in this system?
3. What are the implications of a row of zeroes in a matrix in RREF?
4. How can you generalize the parametric form solution for any system in RREF?
5. What are the necessary steps to reduce a matrix to RREF?

**Tip:** In parametric solutions, free variables (those without leading 1s) act as parameters, allowing you to express dependent variables in terms of them.

Find the general solution to the system corresponding to the following matrix in reduced row echelon form (RREF):

A = 
1 0 0 1 0
0 1 0 -2 1
0 0 1 0 0
0 0 0 0 0

Learn how to find the general solution for a system of equations represented by a matrix in reduced row echelon form (RREF). This example demonstrates parametric solutions for linear systems, using the Gauss-Jordan elimination method, and expressing dependent variables in terms of free variables.

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