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

Verified step by step guidance

1. Write the system of equations as an augmented matrix. For the system \( \begin{cases} x + y + z = 4 \\ x - y - z = 0 \\ x - y + z = 2 \end{cases} \), the augmented matrix is: \[ \left[\begin{array}{ccc|c} 1 & 1 & 1 & 4 \\ 1 & -1 & -1 & 0 \\ 1 & -1 & 1 & 2 \end{array}\right] \]
2. Use Gaussian elimination to create zeros below the first pivot (the element in the first row, first column). Subtract the first row from the second and third rows: - Row 2 = Row 2 - Row 1 - Row 3 = Row 3 - Row 1
3. After these row operations, the matrix will have zeros in the first column below the pivot. Next, focus on the second row and use it to create a zero below its pivot (second row, second column) by manipulating the third row accordingly.
4. Once the matrix is in upper triangular form (all zeros below the main diagonal), use back-substitution to solve for the variables starting from the last row upwards.
5. Alternatively, you can continue with Gauss-Jordan elimination by creating zeros above and below each pivot to get the matrix into reduced row echelon form, from which the solutions for \(x\), \(y\), and \(z\) can be read directly.

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.

Gaussian Elimination with Back-Substitution

Gaussian elimination transforms a system's augmented matrix into an upper triangular form using row operations. Once in this form, back-substitution solves for variables starting from the last equation upward. This method systematically reduces complexity to find the solution.

Gauss-Jordan Elimination

Gauss-Jordan elimination extends Gaussian elimination by further reducing the matrix to reduced row echelon form, where each leading variable is isolated. This method allows direct reading of solutions without back-substitution, providing a straightforward approach to solving systems.