Question

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

Accepted Answer

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:

$$\left[\begin{array}{cc|c} 1 & 1 & 5 \ 1 & -1 & -1 \end{array}ight]$$ 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:

$$ R_2 \leftarrow R_2 - R_1 $$ After the row operation, the matrix will look like:

$$\left[\begin{array}{cc|c} 1 & 1 & 5 \ 0 & -2 & -6 \end{array}ight]$$

Next, make the leading coefficient in the second row a 1 by dividing the entire second row by $$-2$$:

$$ R_2 \leftarrow \frac{1}{-2} R_2 $$ Now, use the second row to eliminate the $$ y -$$term in the first row by performing the operation:

$$ R_1 \leftarrow R_1 - (1) 	imes R_2 $$

This will give the matrix in reduced row echelon form, where you can read off the values of $$ x $$ and $$ y . $$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 $$ (x, y) .$$

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 video answer for a similar problem:

Key Concepts

Gauss-Jordan Elimination Method

Systems of Linear Equations

Parametric Solutions for Infinite Solutions

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 video answer for a similar problem:

Key Concepts

Gauss-Jordan Elimination Method

Systems of Linear Equations

Parametric Solutions for Infinite Solutions

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