Write the system of equations associated with each augmented matrix . Do not solve.

Verified step by step guidance

1. Identify the number of variables in the system by counting the number of columns in the augmented matrix, excluding the last column which represents the constants.
2. Label each variable, typically as \(x_1, x_2, x_3, \ldots\), corresponding to each column of the coefficient matrix.
3. Write an equation for each row of the matrix by multiplying each variable by its corresponding coefficient in that row and setting the sum equal to the constant term in the last column.
4. Express each equation in the form \(a_{i1}x_1 + a_{i2}x_2 + \cdots + a_{in}x_n = b_i\), where \(a_{ij}\) are the coefficients from the matrix and \(b_i\) is the constant from the augmented column.
5. Repeat this process for all rows to form the complete system of linear equations represented by the augmented matrix.

Key Concepts

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. Understanding this helps translate between matrix form and equation form.

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. Writing the system means expressing each row of the augmented matrix as an equation, with coefficients multiplying variables equal to the constants.

Matrix to Equation Conversion

Converting an augmented matrix to a system of equations involves interpreting each row as an equation by assigning variables to columns and equating the linear combination of variables to the constant term. This skill is essential for understanding the relationship between matrices and systems.