Write each system as a matrix and solve it by Gauss-Jordan...

Write each system as a matrix and solve it by Gauss-Jordan elimination. If a system has infinitely many solutions, show a general solution.
$$\left\{\begin{array}{l}
x+2 y=-3 \\
x+4 y=-2 \\
2 x+z=-8
\end{array}\right.$$

Key Concepts

System of Linear Equations

A collection of one or more linear equations involving the same set of variables is known as a system of linear equations. These systems can have a unique solution, infinitely many solutions, or no solution at all, and understanding the structure of these systems is fundamental to many areas of mathematics and applied sciences.

Augmented Matrix

An augmented matrix is a compact way to represent a system of linear equations by combining the coefficients of the variables and the constants into a rectangular array. This representation facilitates the use of systematic procedures, such as elimination methods, to solve the system.

Gauss-Jordan Elimination

The Gauss-Jordan elimination method is a technique used to solve systems of linear equations by transforming the augmented matrix into its reduced row echelon form (RREF). This involves using elementary row operations to simplify the system until the solution can be read directly from the matrix, making the method both systematic and efficient.

General Solution

When a system of linear equations has infinitely many solutions, its solutions can be expressed in terms of one or more free parameters. This general solution captures all possible solutions by representing the variables with parameters, indicating the dependency among the equations and reflecting the system's underlying structure.

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