Write the augmented matrix for each system of linear equations.
$\(\begin{cases}\)2x + y + 2z = 2 \\3x - 5y - z = 4 \(\x\) - 2y - 3z = -6\(\end{cases}\)$

Verified step by step guidance

1. Identify the coefficients of each variable in the system of equations. For the first equation, the coefficients are 2 for x, 1 for y, and 2 for z.
2. For the second equation, the coefficients are 3 for x, -5 for y, and -1 for z (since it is '- z', the coefficient is -1).
3. For the third equation, the coefficients are 1 for x, -2 for y, and -3 for z.
4. Write these coefficients in a matrix form where each row corresponds to an equation and each column corresponds to a variable: x, y, and z respectively.
5. Add the constants from the right side of the equations as the last column to form the augmented matrix.

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more 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 interpret and manipulate these systems is fundamental in algebra.

Augmented Matrix

An augmented matrix represents a system of linear equations by combining the coefficient matrix and the constants into one matrix. Each row corresponds to an equation, and each column corresponds to a variable or the constants, facilitating systematic methods like row operations.

Matrix Representation of Equations

Matrix representation translates linear equations into a compact form using rows and columns. This allows the use of matrix operations to solve systems efficiently, such as Gaussian elimination or matrix inversion, making it easier to handle multiple equations and variables.