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The given system of equations is:

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

We need to solve this system for \(x_1, x_2, x_3, x_4\).

### Step 1: Solve for one variable in terms of others
Let's begin by solving the first equation for \(x_1\):

\[
x_1 = -1 - x_2
\]

### Step 2: Substitute into the fourth equation
We can substitute \(x_1 = -1 - x_2\) into the fourth equation:

\[
(-1 - x_2) + x_4 = 1
\]
\[
x_4 = 1 + 1 + x_2 = 2 + x_2
\]

Now we have expressions for \(x_1\) and \(x_4\) in terms of \(x_2\).

### Step 3: Use the second and third equations
Next, substitute these into the second and third equations:

1. The second equation is \(x_2 + x_3 = 2\). Solve for \(x_3\):
\[
x_3 = 2 - x_2
\]

2. The third equation is \(x_3 + x_4 = 4\). Substitute \(x_3 = 2 - x_2\) and \(x_4 = 2 + x_2\) into this:
\[
(2 - x_2) + (2 + x_2) = 4
\]
This simplifies to:
\[
4 = 4
\]

This equation is always true, so it doesn't provide new information.

### Step 4: Conclusion
From the analysis, we can express all variables in terms of \(x_2\):

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

Thus, the solution to the system is:

\[
\mathbf{x} = \begin{pmatrix} -1 - s \\ s \\ 2 - s \\ 2 + s \end{pmatrix}
\]
where \(s\) is any real number (a free parameter).

Would you like more details on the steps or further clarifications?

### Related Questions:
1. How do you check if a system has infinitely many solutions?
2. What are the conditions for a system to have a unique solution?
3. How can you interpret the free parameter \(s\) geometrically?
4. What are different methods to solve systems of linear equations?
5. How does this system relate to matrix representation of linear equations?

### Tip:
When dealing with systems of equations, it's useful to reduce the system to one or two variables by substitution or elimination to simplify the solution process.

Solve the system of equations: x1 + x2 = -1, x2 + x3 = 2, x3 + x4 = 4, x1 + x4 = 1.

Learn how to solve a system of linear equations with four variables and express the solution in terms of a free parameter. This problem involves substitution and simplification techniques. Suitable for students in Grades 9-12.

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