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 coefficient part of the augmented matrix (excluding the last column, which represents the constants). For example, if the matrix has 4 columns, then there are 3 variables.
2. Label the variables as \(x_1\), \(x_2\), \(x_3\), etc., according to the number of variables identified.
3. Write each row of the augmented matrix as an equation. The entries in the row correspond to the coefficients of the variables, and the last entry is the constant term on the right side of the equation.
4. For each row, form an equation by multiplying each variable by its corresponding coefficient and setting the sum equal to the constant term. For example, if the first row is \([a_{11}, a_{12}, a_{13} | b_1]\), write the equation as \(a_{11}x_1 + a_{12}x_2 + a_{13}x_3 = b_1\).
5. Repeat this process for all rows to write 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 multiple linear equations involving the same set of variables. Writing the system means expressing each row of the augmented matrix as an equation, where coefficients multiply variables and equal the constant term from the augmented column.

Matrix Dimensions and Variables

The size of the augmented matrix indicates the number of equations and variables. For example, a 4x3 augmented matrix typically means 4 equations and 2 variables plus 1 column for constants. Recognizing this helps correctly assign variables and constants when writing the system.