91Ó°ÊÓ

A system of equations is a collection of two or more equations with the same set of unknowns. Solving a system means finding values for the unknowns that satisfy all the equations simultaneously. In this exercise, the system is:
\( -w - 3y + z + 2x = -8 \)
\( x + y - z - w = -4 \)
\( w + y + z + x = 22 \)
\( x - y - z - w = -14 \)

These equations contain variables \(x, y, z,\) and \(w\). We can solve this system using various methods, such as substitution, elimination, or matrices. Matrices offer a structured and sometimes more straightforward approach, especially for larger systems.

To solve a system of equations using matrices, we first convert the system into matrix form. Matrix form arranges the coefficients of the variables into a matrix, facilitating easier manipulations. Here, we rewrite the given system to clearly order the variables:
\( 2x - w - 3y + z = -8 \)
\( x + y - z - w = -4 \)
\( x + y + z + w = 22 \)
\( x - y - z - w = -14 \)
Next, we extract two matrices: a coefficient matrix \(A\) and a constant matrix \(B\):
\[ A = \begin{pmatrix} 2 & -3 & 1 & -1 \ 1 & 1 & -1 & -1 \ 1 & 1 & 1 & 1 \ 1 & -1 & -1 & -1 \end{pmatrix} \]
\[ B = \begin{pmatrix} -8 \ -4 \ 22 \ -14 \end{pmatrix}\]
With these matrices, the original system becomes \(A \cdot X = B\), where \(X\) is the vector of variables:
\[ X = \begin{pmatrix} x \ y \ z \ w \end{pmatrix} \]

To solve the matrix equation \(A \cdot X = B\), we need the inverse of matrix \(A\), denoted as \(A^{-1}\). The inverse of a matrix is a matrix that, when multiplied by the original matrix, yields the identity matrix. Finding \(A^{-1}\) can be done using various methods like Gaussian elimination or adjoint and determinant formulas.
Once we have \(A^{-1}\), we can solve for \(X\) by multiplying both sides of the equation by \(A^{-1}\):
\[ A^{-1} \left(A \cdot X\right) = A^{-1} \cdot B \]
Since \(A^{-1} \cdot A\) is the identity matrix, we get:
\[ X = A^{-1} \cdot B\]
This multiplication will give us the exact values for \(x, y, z,\) and \(w\), completing the solution of the system of equations. Through this process, we leverage matrix operations to simplify and solve complex systems of equations efficiently.