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:

   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:

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!