Write the augmented matrix for each system of linear equations.
$\(\begin{cases}\)2w + 5x - 3y + z = 2 \\3x + y = 4 \(\w\) - x + 5y = 9 \\5w - 5x - 2y = 1\(\end{cases}\)$

Verified step by step guidance

1. Identify the variables in the system: here, the variables are \(w\), \(x\), \(y\), and \(z\).
2. Write the coefficients of each variable in each equation in order, making sure to include zeros for any variable that does not appear in that equation.
3. For the first equation \$2w + 5x - 3y + z = 2\(, the coefficients are \(2\), \(5\), \)-3$, and \(1\) respectively, with the constant term \(2\).
4. For the second equation \$3x + y = 4\(, note that \)w\( and \)z$ are missing, so their coefficients are \(0\). The coefficients are \(0\), \(3\), \(1\), \(0\) with constant \(4\).
5. Repeat this for the remaining equations, then write the augmented matrix by placing the coefficients in rows and the constants in the last column, separated by a vertical line.

Key Concepts

System 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 interpret and organize these equations is essential for solving or representing them in matrix form.

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 Representation of Variables and Constants

In forming an augmented matrix, variables are arranged in a consistent order across all equations, and their coefficients are placed in columns accordingly. The constants from the right side of the equations form the last column. Proper alignment ensures accurate representation and facilitates solution methods like row reduction.