# Solve: 2*x_1 + x_3 + 3*x_4 = 0;2*x_2 - x_1 - x_3 - 2*x_4 = 2; x_1 - x_2 + x_4 = 2 Show that x_2 = 0, x_3 = x_1 - 6 and x_4 = 2 - x_1 is a solution.

Asked on by lienetom

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The equations:

2*x_1 + x_3 + 3*x_4 = 0

2*x_2 - x_1 - x_3 - 2*x_4 = 2

x_1 - x_2 + x_4 = 2

have to be solved for x_1, x_2, x_3 and x_4

As there are 3 equations are 4 variables it is not possible to find unique solutions.

The solutions given are:  x_2 = 0, x_3 = x_1- 6 and x_4 = 2 - x_1

It can be shown that these are true by substituting in the equations.

2*x_1 + x_3 + 3*x_4 = 0

=> 2*x_1 + x_1 - 6 + 3*(2 - x_1)

=> 2*x_1 + x_1 - 6 + 6 - 3*x_1 = 0

2*x_2 - x_1 - x_3 - 2*x_4 = 2

=> 0 - x_1 - x_1 + 6 - 2(2 - x_1)

=> 0 - x_1 - x_1 + 6 - 4 + 2*x_1

=> 2

x_1 - x_2 + x_4

=> x_1 - 0 + 2 - x_1 = 2

This shows that the given solutions satisfy the given equations. It should be noted that there are many other ways in which the variables can be represented.

## 2 x_1 + x_3 + 3 x_4 = 0;2 x_2 - x_1 - x_3 - 2 x_4 = 2; x_1 - x_2 + x_4 = 2

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