Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$\(\begin{cases}\)w + x + y + z = 4 \\2w + x - 2y - z = 0 \(\w\) - 2x - y - 2z = -2 \\3w + 2x + y + 3z = 4\(\end{cases}\)$

Verified step by step guidance

1. Write the system of equations as an augmented matrix. The coefficients of the variables \( w, x, y, z \) and the constants on the right side form the matrix: \[\left[\begin{array}{cccc|c} 1 & 1 & 1 & 1 & 4 \\ 2 & 1 & -2 & -1 & 0 \\ 1 & -2 & -1 & -2 & -2 \\ 3 & 2 & 1 & 3 & 4 \end{array}\right]\]
2. Use Gaussian elimination to transform the matrix into an upper triangular form (row echelon form). Start by using the first row to eliminate the \( w \)-terms in rows 2, 3, and 4 by performing row operations such as: - Replace row 2 with (row 2) - 2*(row 1) - Replace row 3 with (row 3) - (row 1) - Replace row 4 with (row 4) - 3*(row 1)
3. Next, use the second row to eliminate the \( x \)-terms in rows 3 and 4. This involves finding a suitable multiplier to subtract a multiple of row 2 from rows 3 and 4 to create zeros below the pivot in the second column.
4. Then, use the third row to eliminate the \( y \)-term in row 4 by subtracting a multiple of row 3 from row 4 to get a zero in the third column of the fourth row.
5. Once the matrix is in upper triangular form, use back-substitution to solve for the variables starting from the last row and moving upwards. This means solving for \( z \) first, then \( y \), then \( x \), and finally \( w \).

Key Concepts

Systems of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. Understanding how to represent and interpret these systems is fundamental before applying matrix methods.

Matrix Representation of Systems

Systems of linear equations can be expressed in matrix form as AX = B, where A is the coefficient matrix, X is the column matrix of variables, and B is the constants matrix. This representation simplifies the use of matrix operations to solve the system efficiently.

Gaussian Elimination and Gauss-Jordan Elimination

Gaussian elimination transforms the augmented matrix into an upper triangular form to solve via back-substitution, while Gauss-Jordan elimination reduces it further to reduced row-echelon form for direct solution. Both methods use row operations to systematically solve linear systems.