Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$\(\begin{cases}\)3w - 4x + y + z = 9 \(\w\) + x - y - z = 0 \\2w + x + 4y - 2z = 3 \\-w + 2x + y - 3z = 3\(\end{cases}\)$

Verified step by step guidance

1. Write the system of equations as an augmented matrix. The system is: \[\begin{cases} 3w - 4x + y + z = 9 \\ w + x - y - z = 0 \\ 2w + x + y + z = 1 \\ -w + 2x + y - 3z = 3 \end{cases}\] The augmented matrix is: \[\left[ \begin{array}{cccc|c} 3 & -4 & 1 & 1 & 9 \\ 1 & 1 & -1 & -1 & 0 \\ 2 & 1 & 1 & 1 & 1 \\ -1 & 2 & 1 & -3 & 3 \end{array} \right]\]
2. Use Gaussian elimination to transform the matrix into an upper triangular form. Start by using the first row to eliminate the entries below the leading 1 in the first column. For example, use row 1 to eliminate the w-terms in rows 2, 3, and 4 by performing row operations such as: - Replace row 2 with (row 2) - (1/3) * (row 1) - Replace row 3 with (row 3) - (2/3) * (row 1) - Replace row 4 with (row 4) + (1/3) * (row 1)
3. Next, move to the second row and use it to eliminate the x-terms in rows 3 and 4. This involves making the element in the second column of row 2 a leading 1 (if it is not already), then using it to eliminate the corresponding entries below it by appropriate row operations.
4. Continue this process for the third row to eliminate the y-term in row 4, making the matrix upper triangular (all zeros below the main diagonal).
5. Once the matrix is in upper triangular form, use back-substitution to solve for the variables starting from the last row up to the first. This means solving for z from the last equation, then substituting back to find 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.