Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$\(\begin{cases}\)2x - y - z = 4 \(\x\) + y - 5z = -4 \(\x\) - 2y = 4\(\end{cases}\)$

Verified step by step guidance

1. Write the system of equations as an augmented matrix. For the system: \[\begin{cases} 2x - y - z = 4 \\ x + y - 5z = -4 \\ x - 2y + 0z = 4 \end{cases}\] The augmented matrix is: \[\left[ \begin{array}{ccc|c} 2 & -1 & -1 & 4 \\ 1 & 1 & -5 & -4 \\ 1 & -2 & 0 & 4 \end{array} \right]\]
2. Use Gaussian elimination to transform the matrix into an upper triangular form. Start by making the element in the first row, first column a leading 1 if possible (or use row operations to create zeros below it). For example, you can swap rows or multiply/divide rows to simplify calculations.
3. Perform row operations to eliminate the entries below the leading 1 in the first column. This means making the elements in the second and third rows, first column equal to zero by subtracting appropriate multiples of the first row from these rows.
4. Next, move to the second row and second column to create a leading 1 (pivot). Then use row operations to eliminate the entry below this pivot in the third row, second column, making it zero.
5. Once the matrix is in upper triangular form, use back-substitution to solve for the variables starting from the last row. Substitute the found values back into the previous equations to find all variable values.

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 using row operations, followed by back-substitution to find variable values. Gauss-Jordan elimination further reduces the matrix to reduced row-echelon form, allowing direct reading of solutions without back-substitution.