Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$\{\begin{array}{c}x+2y+3z=-5\\ 2x+y+z=1\\ x+y-z=8\end{array}$

Verified step by step guidance

1. Write the system of equations in matrix form as an augmented matrix \([A|\mathbf{b}]\), where \(A\) is the coefficient matrix and \(\mathbf{b}\) is the constants column vector.
2. Use Gaussian elimination to transform the augmented matrix into an upper triangular form by applying row operations: swapping rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another.
3. Once the matrix is in upper triangular form, use back-substitution to solve for the variables starting from the last row and moving upwards.
4. Alternatively, use Gauss-Jordan elimination to reduce the augmented matrix to reduced row echelon form (RREF), where the coefficient matrix becomes the identity matrix.
5. From the RREF matrix, directly read off the solutions for the variables, as each variable corresponds to a leading 1 in the identity matrix portion.

Key Concepts

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with 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 is a method to solve systems by transforming the coefficient matrix into an upper triangular form using row operations. Once in this form, back-substitution is used to find the variable values starting from the last equation upward. This stepwise approach simplifies solving linear systems.

Gauss-Jordan Elimination

Gauss-Jordan elimination extends Gaussian elimination by reducing the matrix further into reduced row-echelon form, where the coefficient matrix becomes the identity matrix. This method directly provides the solution without back-substitution, making it a systematic approach to solving linear systems using matrices.