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.
x + y = 5
x - y = -1

Verified step by step guidance

1. Write the system of equations as an augmented matrix. For the system \( \begin{cases} x + y = 5 \\ x - y = -1 \end{cases} \), the augmented matrix is: \[\left[\begin{array}{cc|c} 1 & 1 & 5 \\ 1 & -1 & -1 \end{array}\right]\]
2. Use row operations to create a leading 1 in the first row, first column (which is already 1), and then eliminate the \( x \)-term in the second row by subtracting the first row from the second row. This means: \[ R_2 \leftarrow R_2 - R_1 \]
3. After the row operation, the matrix will look like: \[\left[\begin{array}{cc|c} 1 & 1 & 5 \\ 0 & -2 & -6 \end{array}\right]\] Next, make the leading coefficient in the second row a 1 by dividing the entire second row by \(-2\): \[ R_2 \leftarrow \frac{1}{-2} R_2 \]
4. Now, use the second row to eliminate the \( y \)-term in the first row by performing the operation: \[ R_1 \leftarrow R_1 - (1) \times R_2 \] This will give the matrix in reduced row echelon form, where you can read off the values of \( x \) and \( y \).
5. Translate the final matrix back into equations to find the solution for \( x \) and \( y \). Since this is a system of two equations with two variables, the solution will be a unique ordered pair \( (x, y) \).

Key Concepts

Gauss-Jordan Elimination Method

Gauss-Jordan elimination is a systematic procedure to solve systems of linear equations by transforming the augmented matrix into reduced row-echelon form. This method uses row operations to simplify the matrix, making it easier to find the values of variables directly. It extends Gaussian elimination by continuing until each leading coefficient is 1 and all other entries in the column are zero.

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. Solutions are the values of variables that satisfy all equations simultaneously. Systems can have a unique solution, infinitely many solutions, or no solution, depending on the relationships between the equations.

Parametric Solutions for Infinite Solutions

When a system has infinitely many solutions, some variables are expressed in terms of arbitrary parameters to describe the solution set. For two-variable systems, one variable (like y) is set as arbitrary; for three-variable systems, another variable (like z) is chosen. This approach clearly represents all possible solutions.