Write the system of linear equations represented by the augmented matrix. Use x, y, and z, or, if necessary, w, x, y, and z, for the variables.
$\(\begin{bmatrix}\)5 & 0 & 3 & \(\vert\) & -11 \\0 & 1 & -4 & \(\vert\) & 12 \\7 & 2 & 0 & \(\vert\) & 3\(\end{bmatrix}\)$

Verified step by step guidance

1. Identify the variables corresponding to the columns of the matrix. Since there are three columns before the augmented part, use variables \(x\), \(y\), and \(z\).
2. Write the first row as an equation by multiplying each coefficient by its variable and setting it equal to the augmented value: \(5x + 0y + 3z = -11\).
3. Write the second row as an equation: \(0x + 1y - 4z = 12\).
4. Write the third row as an equation: \(7x + 2y + 0z = 3\).
5. Combine all three equations to form the system of linear equations represented by the augmented matrix.

Key Concepts

Augmented Matrix Representation

An augmented matrix combines the coefficients of variables and constants from a system of linear equations into a single matrix. Each row corresponds to an equation, and each column (except the last) corresponds to a variable. The last column represents the constants on the right side of the equations.

Writing Systems of Linear Equations from Matrices

To write a system from an augmented matrix, interpret each row as an equation by multiplying each coefficient by its variable and setting the sum equal to the constant in the last column. Variables are assigned in order to the columns, such as x, y, and z for three variables.

Variables and Their Order

The order of variables corresponds to the order of columns in the matrix. For a 3-variable system, the first column is x, the second y, and the third z. Correctly identifying this order is essential to accurately translate the matrix into equations.