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The system of equations in your homework problem can be written as:

\[
\begin{aligned}
x_1 + x_2 &= 2 \\
x_2 + x_3 &= 4 \\
x_3 + x_4 &= -1 \\
x_1 + x_4 &= -3
\end{aligned}
\]

We can solve this system step by step.

### Step 1: Solve for \(x_1\) from the first equation
From the first equation:

\[
x_1 = 2 - x_2
\]

### Step 2: Solve for \(x_3\) from the second equation
From the second equation:

\[
x_3 = 4 - x_2
\]

### Step 3: Solve for \(x_4\) from the third equation
Substitute the expression for \(x_3\) into the third equation:

\[
(4 - x_2) + x_4 = -1
\]
\[
x_4 = -1 - 4 + x_2 = -5 + x_2
\]

### Step 4: Solve for \(x_2\) using the fourth equation
Substitute the expressions for \(x_1\) and \(x_4\) into the fourth equation:

\[
(2 - x_2) + (-5 + x_2) = -3
\]
Simplify:

\[
2 - x_2 - 5 + x_2 = -3
\]
\[
-3 = -3
\]

This equation is always true, meaning that \(x_2\) can take any value. Therefore, \(x_2\) is a free variable.

### Step 5: General solution
Since \(x_2\) is a free variable, we can express the solution in terms of \(x_2\). Let's set \(x_2 = s\), where \(s\) is any real number. Then, the general solution is:

\[
\begin{aligned}
x_1 &= 2 - s \\
x_2 &= s \\
x_3 &= 4 - s \\
x_4 &= -5 + s
\end{aligned}
\]

Thus, the solution to the system is:

\[
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix}
=
\begin{bmatrix}
2 - s \\
s \\
4 - s \\
-5 + s
\end{bmatrix}
\]

This is a parametric solution with \(s\) as the free parameter.

Would you like further details or have any questions?

Here are 5 related questions you can explore:
1. How can this system be represented in matrix form?
2. What are the implications of having a free variable in a system of equations?
3. How would the solution change if an additional constraint were applied to the system?
4. How does the system rank help determine whether a system is underdetermined or has a unique solution?
5. What are the possible geometrical interpretations of this solution?

**Tip:** When solving systems of linear equations, always check for free variables to see if the system has infinite solutions.

Solve the system of equations given in Homework-3: Problem 2. The system consists of the following four equations: x1 + x2 = 2, x2 + x3 = 4, x3 + x4 = -1, and x1 + x4 = -3.

Learn how to solve a system of four linear equations with variables x1, x2, x3, and x4. The problem involves finding the parametric solution with a free variable using algebraic substitution. Suitable for high school and early college students studying linear algebra and systems of equations.

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