https://file.aiquickdraw.com/mathbot/file/1726424908064ndk2ujhk.jpg

The system of linear equations given in the image is:

\[
\begin{aligned}
x_1 + x_2 &= 1, \\
x_2 + x_3 &= 4, \\
x_3 + x_4 &= -4, \\
x_1 + x_4 &= -7.
\end{aligned}
\]

To solve this system, we can use substitution or elimination methods. Let’s proceed step by step:

### Step 1: Express \( x_1 \) in terms of \( x_2 \) from the first equation.
From the first equation \( x_1 + x_2 = 1 \), we solve for \( x_1 \):
\[
x_1 = 1 - x_2.
\]

### Step 2: Substitute \( x_1 = 1 - x_2 \) into the fourth equation.
The fourth equation is \( x_1 + x_4 = -7 \). Substituting \( x_1 = 1 - x_2 \) into this equation:
\[
(1 - x_2) + x_4 = -7,
\]
which simplifies to:
\[
x_4 = -8 + x_2.
\]

### Step 3: Substitute \( x_4 = -8 + x_2 \) into the third equation.
The third equation is \( x_3 + x_4 = -4 \). Substituting \( x_4 = -8 + x_2 \):
\[
x_3 + (-8 + x_2) = -4,
\]
which simplifies to:
\[
x_3 = 4 - x_2.
\]

### Step 4: Substitute \( x_3 = 4 - x_2 \) into the second equation.
The second equation is \( x_2 + x_3 = 4 \). Substituting \( x_3 = 4 - x_2 \):
\[
x_2 + (4 - x_2) = 4,
\]
which simplifies to:
\[
4 = 4.
\]
This equation is always true, meaning that \( x_2 \) is a free variable.

### Final Solution:
Since \( x_2 \) is a free variable, let \( x_2 = s \), where \( s \) is any real number. Now express the other variables in terms of \( s \):

\[
x_1 = 1 - s,
\]
\[
x_3 = 4 - s,
\]
\[
x_4 = -8 + s.
\]

Thus, the solution to the system is:
\[
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix}
=
\begin{bmatrix}
1 \\
4 \\
-4 \\
-7
\end{bmatrix}
+ s \begin{bmatrix}
-1 \\
1 \\
-1 \\
1
\end{bmatrix},
\]
where \( s \) is a free parameter.

Would you like more details or have any questions?

### Related questions:
1. How does Gaussian elimination work for solving systems of linear equations?
2. Can systems of linear equations have no solution or infinitely many solutions?
3. What is the significance of free variables in solving systems of equations?
4. How can matrix notation help in solving systems of linear equations?
5. How does the row-reduction method compare to substitution for solving linear systems?

### Tip:
In systems of linear equations, if an equation simplifies to something always true (like \( 4 = 4 \)), it usually means that one or more variables are free, allowing for infinitely many solutions.

Solve the system of linear equations:
1. x1 + x2 = 1
2. x2 + x3 = 4
3. x3 + x4 = -4
4. x1 + x4 = -7

Learn how to solve a system of linear equations with four variables using substitution methods and matrix representation. This problem involves finding parametric solutions and applying linear algebra techniques.

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