How do you solve systems of linear equations with three variables?

Answer 1

You have to eliminate one variable between two equations at a time. For example:

5x + 2y + z = 10

10x - 2y - z = 12

10x - 4y + z = 14

You can add the first two equations together to get 15x = 22,

then divide by 15 to get x = 22/15.

Now combine the second and third equations:

10x - 2y - z = 12

10x - 4y + z = 14

20x - 6y = 26

Now you can plug 22/15 in for x. When you solve for y, you should have y = 5/9.

When you substitute y and z into any of the top three equations, you should find that z = 14/9.

You can check by plugging in 22/15 for x, 5/9 for y, and 14/9 for z in any of the top three equations, and they should fit perfectly.

In other words the solution is (22/15, 5/9, 14/9).