Write the augmented matrix for each system of linear equations.
$\(\begin{cases}\)x - y + z = 8 \(\y\) - 12z = -15 \(\z\) = 1\(\end{cases}\)$

Verified step by step guidance

1. Identify the coefficients of each variable in the system of equations. For the first equation \(x - y + z = 8\), the coefficients are 1 for \(x\), -1 for \(y\), and 1 for \(z\).
2. For the second equation \(y - 12z = -15\), note that \(x\) is missing, so its coefficient is 0. The coefficients are 0 for \(x\), 1 for \(y\), and -12 for \(z\).
3. For the third equation \(z = 1\), both \(x\) and \(y\) are missing, so their coefficients are 0. The coefficients are 0 for \(x\), 0 for \(y\), and 1 for \(z\).
4. Write the augmented matrix by placing the coefficients of \(x\), \(y\), and \(z\) in the first three columns, and the constants on the right side of the equations in the last column.
5. The augmented matrix will have three rows (one for each equation) and four columns (three for variables and one for constants).

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 for forming matrices.

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. This format simplifies solving systems using matrix operations.

Matrix Notation and Construction

Matrix notation organizes coefficients and constants into rows and columns, making it easier to apply algebraic methods like row operations. Constructing the augmented matrix requires correctly placing coefficients of variables and constants in order, including zeros for missing variables.