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An augmented matrix is a compact way to represent a system of linear equations. It combines the coefficients of the variables and the constants from the equations into a single matrix. This simplifies the process of solving the system, especially when using techniques like Gaussian elimination. Each row in the augmented matrix corresponds to an equation, and each column represents either a variable's coefficient or the constant term.

For instance, the augmented matrix \[\left[\begin{array}{cccc}1 & 3 & 4 & 7 \3 & 9 & 7 & 6 \end{array}\right]\]represents a system of two equations in three variables \(x_1, x_2,\) and \(x_3\). The last column in this matrix is the constants column that follows the equal sign in the equations. Understanding how to form and interpret an augmented matrix is fundamental to solving systems of equations efficiently.

Transforming an augmented matrix into Row Echelon Form (REF) is a critical step in solving a system of equations. REF is an organized structure that makes it easier to solve equations through back-substitution. The key features of REF include having leading ones in each row, and zeros below these leading ones.

Achieving this form involves using row operations that are similar to performing arithmetic operations on equations. For example, given the matrix \[\left[\begin{array}{cccc}1 & 3 & 4 & 7 \3 & 9 & 7 & 6 \end{array}\right]\]a common step would be to eliminate the value in the first position of the second row to create a zero. This results in:\[\left[\begin{array}{cccc}1 & 3 & 4 & 7 \0 & 0 & -5 & -15 \end{array}\right]\]
This systematic elimination allows us to focus on each variable starting from the top, making the problem much more manageable.

A system of equations consists of multiple equations that are solved together, usually because they share common variables. The goal is to find values for those variables that satisfy all equations simultaneously. Systems can be linear or non-linear, with linear systems being the focus of linear algebra.

For example, using the system represented by the matrix \[\left[\begin{array}{cccc}1 & 3 & 4 & 7 \3 & 9 & 7 & 6 \end{array}\right]\]the corresponding equations are:

• \(x_1 + 3x_2 + 4x_3 = 7\)
• \(3x_1 + 9x_2 + 7x_3 = 6\)

Solving these equations typically involves strategies like substitution or elimination, and often involves representing them using matrices to streamline the process with matrix operations. Each solution to the system corresponds to a point in space where the equations intersect.

In the context of solving systems of linear equations, a free variable is one that is not bound by a unique solution but can take any real number value. This typically occurs in underdetermined systems, where there are more variables than equations. Free variables lead to a range of possible solutions, often represented as a parameterized expression.

For example, in the system arising from the augmented matrix we worked with, there are three variables but fewer independent equations. This results in \(x_2\) being a free variable because it does not have a specific solution from the system. Hence, the general solution can be expressed in terms of \(x_2\):

• \(x_1 = -5 - 3x_2\)
• \(x_2 = t\)
• \(x_3 = 3\)

where \(t\) is any real number. Free variables enable the formation of a general solution, giving insights into the behavior and solution space of the system.