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. See Examples 1-4. 6x - 3y - 4 = 0 3x + 6y - 7= 0

Verified step by step guidance

1. Step 1: Write the system of equations in augmented matrix form: $$\left[\begin{array}{cccc}6& -3& |& 4\\ 3& 6& |& 7\end{array}\right]$$.
2. Step 2: Use row operations to get a leading 1 in the first row, first column. Divide the first row by 6: $$\left[\begin{array}{cccc}1& -\frac{1}{2}& |& \frac{2}{3}\\ 3& 6& |& 7\end{array}\right]$$.
3. Step 3: Eliminate the first column of the second row by replacing the second row with the result of the second row minus 3 times the first row: $$\left[\begin{array}{cccc}1& -\frac{1}{2}& |& \frac{2}{3}\\ 0& 7.5& |& 5\end{array}\right]$$.
4. Step 4: Get a leading 1 in the second row, second column by dividing the second row by 7.5: $$\left[\begin{array}{cccc}1& -\frac{1}{2}& |& \frac{2}{3}\\ 0& 1& |& \frac{2}{3}\end{array}\right]$$.
5. Step 5: Eliminate the second column of the first row by replacing the first row with the result of the first row plus $\frac{1}{2}$ times the second row: $$\left[\begin{array}{cccc}1& 0& |& 1\\ 0& 1& |& \frac{2}{3}\end{array}\right]$$.

Key Concepts

Gauss-Jordan Elimination

The Gauss-Jordan elimination method is a systematic procedure used to solve systems of linear equations. It involves transforming the system's augmented matrix into reduced row echelon form, allowing for easy identification of solutions. This method can handle both unique solutions and cases with infinitely many solutions by manipulating the rows of the matrix through elementary row operations.

Infinitely Many Solutions

A system of equations has infinitely many solutions when at least one equation can be derived from the others, leading to dependent equations. In such cases, the solution can be expressed in terms of one or more free variables, such as 'y' or 'z', which can take any value. This indicates that there are multiple combinations of variable values that satisfy all equations in the system.

Augmented Matrix

An augmented matrix is a matrix that represents a system of linear equations, including the coefficients of the variables and the constants from the equations. It is formed by appending the constant terms as an additional column to the coefficient matrix. This format simplifies the application of row operations during methods like Gauss-Jordan elimination, facilitating the process of finding solutions to the system.