In Exercises 1–8, write the augmented matrix for each system of linear equations.

Verified step by step guidance

1. Identify the coefficients of each variable and the constants in the system of linear equations.
2. Write the coefficients of the variables in a matrix form, aligning them according to their respective variables.
3. Include the constants from the right-hand side of the equations as an additional column in the matrix.
4. Construct the augmented matrix by combining the coefficient matrix and the constants column.
5. Double-check the augmented matrix to ensure all coefficients and constants are correctly placed.

Key Concepts

Augmented Matrix

An augmented matrix is a matrix that represents a system of linear equations, where each row corresponds to an equation and each column corresponds to the coefficients of the variables, along with an additional column for the constants on the right side of the equations. This format allows for efficient manipulation and solution of the system using matrix operations.

Linear Equations

Linear equations are mathematical statements that express a relationship between variables in which each term is either a constant or the product of a constant and a single variable. In the context of a system of equations, these equations can be solved simultaneously to find the values of the variables that satisfy all equations in the system.

Row Operations

Row operations are techniques used to manipulate the rows of a matrix to simplify it or to solve a system of equations. The three primary row operations are swapping two rows, multiplying a row by a non-zero scalar, and adding or subtracting the multiple of one row to another. These operations are fundamental in transforming an augmented matrix into row-echelon form or reduced row-echelon form.