Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$\(\begin{cases}\)x + 2y = z - 1 \(\x\) = 4 + y - z \(\x\) + y - 3z = -2\(\end{cases}\)$

Verified step by step guidance

1. Step 1: Write the system of equations in standard form, aligning variables on the left and constants on the right: \(x + 2y - z = -1\) \(x - y + z = 4\) \(x + y - 3z = -2\)
2. Step 2: Set up the augmented matrix representing the system: \[\left[\begin{array}{ccc|c} 1 & 2 & -1 & -1 \\ 1 & -1 & 1 & 4 \\ 1 & 1 & -3 & -2 \end{array}\right]\]
3. Step 3: Use Gaussian elimination to create zeros below the leading 1 in the first column. For example, subtract the first row from the second and third rows: Row2 = Row2 - Row1 Row3 = Row3 - Row1
4. Step 4: Continue the elimination process to get the matrix into upper triangular form, then use back-substitution to solve for variables starting from the last row upwards.
5. Step 5: Alternatively, perform Gauss-Jordan elimination by continuing row operations to get the matrix into reduced row echelon form, where the left side is the identity matrix, and the right side gives the solution 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.

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 variable matrix, 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 by 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.