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.
2x - y = 6
4x - 2y = 0

Verified step by step guidance

1. Write the system of equations as an augmented matrix. For the system \( 2x - y = 6 \) and \( 4x - 2y = 0 \), the augmented matrix is: \[ \left[\begin{array}{cc|c} 2 & -1 & 6 \\ 4 & -2 & 0 \end{array}\right] \]
2. Use row operations to get a leading 1 in the first row, first column. You can divide the first row by 2: \[ R_1 \rightarrow \frac{1}{2} R_1 \] which gives: \[ \left[\begin{array}{cc|c} 1 & -\frac{1}{2} & 3 \\ 4 & -2 & 0 \end{array}\right] \]
3. Eliminate the first variable (x) from the second row by replacing the second row with \( R_2 - 4R_1 \): \[ R_2 \rightarrow R_2 - 4R_1 \] This will create a zero in the first column of the second row.
4. Simplify the second row after the operation to see if it leads to a row of zeros or a contradiction. This will help determine if the system has a unique solution, no solution, or infinitely many solutions.
5. If the system has infinitely many solutions, express one variable (in this case \( y \)) as arbitrary and write the other variable(s) in terms of this arbitrary variable. If there is a unique solution, back-substitute to find the values of \( x \) and \( y \).

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 until each variable corresponds to a leading 1, making the solution straightforward to read.

Types of Solutions in Linear Systems

A system of linear equations can have a unique solution, infinitely many solutions, or no solution. Infinitely many solutions occur when equations are dependent, leading to free variables that can take arbitrary values, which must be expressed explicitly in the solution.

Expressing Solutions with Arbitrary Variables

When a system has infinitely many solutions, one or more variables are free and can be assigned arbitrary parameters (like y or z). Writing the solution set involves expressing dependent variables in terms of these arbitrary variables to describe all possible solutions.