Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$\(\begin{cases}\)2x + 2y + 7z = -1 \\2x + y + 2z = 2 \\4x + 6y + z = 15\(\end{cases}\)$

Verified step by step guidance

1. Write the system of equations as an augmented matrix. For the system: \[\begin{cases} 2x + 2y + 7z = -1 \\ 2x + y + 2z = 2 \\ 4x + 6y + z = 15 \end{cases}\] The augmented matrix is: \[\left[ \begin{array}{ccc|c} 2 & 2 & 7 & -1 \\ 2 & 1 & 2 & 2 \\ 4 & 6 & 1 & 15 \end{array} \right]\]
2. Use Gaussian elimination to transform the matrix into an upper triangular form. Start by using the first row to eliminate the \(x\)-terms in the second and third rows. For example, subtract the first row from the second row, and subtract twice the first row from the third row.
3. Continue the elimination process to get zeros below the pivot in the second column. Use the second row to eliminate the \(y\)-term in the third row by appropriate row operations.
4. Once the matrix is in upper triangular form, write the corresponding system of equations from the matrix. This system will have the form: \[\begin{cases} a_{11}x + a_{12}y + a_{13}z = b_1 \\ 0 + a_{22}y + a_{23}z = b_2 \\ 0 + 0 + a_{33}z = b_3 \end{cases}\]
5. Use back-substitution to solve for \(z\) from the third equation, then substitute \(z\) into the second equation to solve for \(y\), and finally substitute \(y\) and \(z\) into the first equation to solve for \(x\).

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.