Lesson Objectives
• Learn how to solve a linear system in four variables using Gauss-Jordan Elimination

## How to Solve a Linear System in Four Variables Using Gauss-Jordan Elimination

Over the course of the last two lessons, we have learned how to solve a linear system in two and three variables using Gaussian Elimination and Gauss-Jordan Elimination. In this lesson, we will look at an example of how to solve a linear system in four variables using Gauss-Jordan Elimination.

### Gauss-Jordan Four-Variable System

• Obtain a 1 as the first element in the first column
• Use the first row to transform the remaining elements in the first column into zeros
• Obtain a 1 as the second element in the second column
• Use the second row to transform the remaining elements in the second column into zeros
• Obtain a 1 as the third element in the third column
• Use the third row to transform the remaining elements in the third column into zeros
• Obtain a 1 as the fourth element in the fourth column
• Use the fourth row to transform the remaining elements in the fourth column into zeros
Example #1: Solve each linear system. $$2x_1+x_2 -x_3 + 2x_4=-6$$ $$3x_1+4x_2 + x_4=1$$ $$x_1 + 5x_2 + 2x_3 + 6x_4=-3$$ $$5x_1 + 2x_2 - x_3 - x_4=3$$ Let's set up our augmented matrix: $$\left[ \begin{array}{cccc|c}2&1&-1&2&-6\\ 3&4&0&1&1\\ 1&5&2&6&-3 \\5&2&-1&-1&3 \end{array}\right]$$ Our main goal is to get 1's down the main diagonal, with zeros above and below. Our end result: $$\left[ \begin{array}{cccc|c}1&0&0&0&1\\ 0&1&0&0&0\\ 0&0&1&0&4 \\0&0&0&1&-2 \end{array}\right]$$ $$(1, 0, 4, -2)$$

#### Skills Check:

Example #1

Solve each system. $$3x_1 + 2x_2 + 7x_3 - x_4=44$$ $$2x_1 + 6x_2 - 4x_3 - 8x_4=34$$ $$x_1 + x_2 - 2x_3 + 3x_4=0$$ $$-4x_1 - 7x_2 - 9x_3 + 2x_4=-80$$

A
$$(3, 5, -1, 7)$$
B
$$(2, -2, 4, -1)$$
C
$$(3, 1, 2, 5)$$
D
$$(4, 5, 3, -1)$$
E
$$(8, 9, 2, 7)$$

Example #2

Solve each system. $$x_1 + 2x_2 + 2x_3 + 4x_4=11$$ $$3x_1 + 6x_2 + 5x_3 + 12x_4=30$$ $$x_1 + 3x_2 - 3x_3 + 2x_4=-5$$ $$6x_1 - x_2 - x_3 + x_4=-9$$

A
$$(5, 4, 6, 8)$$
B
$$(-2, 1, 0, 4)$$
C
$$(0, 7, 5, -3)$$
D
$$(-1, 1, 3, 1)$$
E
$$(2, 7, 5, 0)$$