Solve the system of equations using x = A−1b:
2x1 + x2 + 5x3 + x4 = 5,
x1 + x2 − 3x3 − 4x4 = −1,
3x1 + 6x2 − 2x3 + x4 = 8,
2x1 + 2x2 + 2x3 − 3x4 = 2

To solve the system of equations using \( x = A^{-1}b \), we follow these steps:

1. **Rewrite the system of equations in matrix form**:
   The system of equations is given by:

   \[
   \begin{aligned}
   2x_1 + x_2 + 5x_3 + x_4 &= 5 \\
   x_1 + x_2 - 3x_3 - 4x_4 &= -1 \\
   3x_1 + 6x_2 - 2x_3 + x_4 &= 8 \\
   2x_1 + 2x_2 + 2x_3 - 3x_4 &= 2
   \end{aligned}
   \]

   This can be expressed in matrix form as \( A \mathbf{x} = \mathbf{b} \), where:
   \[
   A = \begin{pmatrix}
   2 & 1 & 5 & 1 \\
   1 & 1 & -3 & -4 \\
   3 & 6 & -2 & 1 \\
   2 & 2 & 2 & -3
   \end{pmatrix}, \quad
   \mathbf{x} = \begin{pmatrix}
   x_1 \\
   x_2 \\
   x_3 \\
   x_4
   \end{pmatrix}, \quad
   \mathbf{b} = \begin{pmatrix}
   5 \\
   -1 \\
   8 \\
   2
   \end{pmatrix}
   \]

2. **Find the inverse of matrix \( A \), \( A^{-1} \)**:
   We calculate \( A^{-1} \), which is the inverse of the coefficient matrix.

3. **Solve for \( \mathbf{x} \)**:
   Once we have \( A^{-1} \), we use the formula \( \mathbf{x} = A^{-1} \mathbf{b} \) to find the solution.

Let me calculate the inverse and solve for \( \mathbf{x} \).The solution to the system of equations is:

\[
\mathbf{x} = \begin{pmatrix} 
x_1 \\ 
x_2 \\ 
x_3 \\ 
x_4 
\end{pmatrix} = \begin{pmatrix} 
2 \\ 
0.2 \\ 
0 \\ 
0.8 
\end{pmatrix}
\]

Thus, the values for each variable are:
- \( x_1 = 2 \)
- \( x_2 = 0.2 \)
- \( x_3 = 0 \)
- \( x_4 = 0.8 \)

Would you like further details on any of the steps or clarifications?

Here are 5 related questions to deepen understanding:
1. How do you calculate the inverse of a matrix?
2. What conditions must a matrix satisfy to have an inverse?
3. Can you solve a system using \( A^{-1}b \) when \( A \) is singular?
4. How do we interpret a solution when one of the variables is nearly zero?
5. What are alternative methods for solving systems of linear equations?

**Tip**: Always check if the matrix is invertible by calculating its determinant before trying to find the inverse!

Solve the system of equations using x = A−1b: 2x1 + x2 + 5x3 + x4 = 5, x1 + x2 − 3x3 − 4x4 = −1, 3x1 + 6x2 − 2x3 + x4 = 8, 2x1 + 2x2 + 2x3 − 3x4 = 2

Learn how to solve a system of linear equations using matrix inversion with a step-by-step breakdown. This solution involves key concepts from linear algebra, including the matrix inverse and determinant, and is suitable for advanced high school or undergraduate students.

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