Use the Gauss-Jordan method to solve each system of...

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with $z$ arbitrary. $$ \begin{array}{c} 2 x+y-z+3 w=0 \\ 3 x-2 y+z-4 w=-24 \\ x+y-z+w=2 \\ x-y+2 z-5 w=-16 \end{array} $$

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with $z$ arbitrary.
$$
\begin{array}{c}
2 x+y-z+3 w=0 \\
3 x-2 y+z-4 w=-24 \\
x+y-z+w=2 \\
x-y+2 z-5 w=-16
\end{array}
$$

Key Concepts

Gauss-Jordan Elimination

A systematic method for solving systems of linear equations by transforming the associated augmented matrix into reduced row echelon form using a series of elementary row operations. This method makes it straightforward to read off the solutions directly from the matrix once the appropriate form is reached.

Systems of Linear Equations

A collection of two or more linear equations involving the same set of variables. Solving such systems means finding values for the variables that satisfy all the equations simultaneously, which can result in a unique solution, infinitely many solutions, or no solution at all.

Augmented Matrix

A compact way to represent a system of linear equations where the coefficients of the variables and the constants from the right-hand side are combined into a single matrix. This format is particularly useful with elimination methods like Gauss-Jordan elimination.

Elementary Row Operations

The operations that can be performed on the rows of an augmented matrix without changing the solution set of the corresponding system of equations. These include swapping rows, multiplying a row by a nonzero scalar, and adding or subtracting a multiple of one row from another.

Parametric Representation of Solutions

When a system of equations has infinitely many solutions, it is often solved by expressing some variables in terms of free parameters. This approach clearly describes the entire solution set by allowing one or more variables to take arbitrary values.

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