Gaussian Elimination Examples

Solve this system of equations using Gaussian Elimination.

-2x + 3y + 4z = 12
-x + 5y – 2z = 9
3x – y + 6z = 1

Multiply Row 2 by 3, add Row 2 and Row 3.

-2x + 3y + 4z = 12
-x + 5y – 2z = 9
y = 2

Multiply Row 2 by -2, add Row 1 and Row 2.

-2x + 3y + 4z = 12
-7y + 8z = -6
y = 2

Now backwards solve by substituting y = 2 into the second equation to solve for z. Then, solve for x. The solution is (-1, 2, 1).

Solve this system of equations using Gaussian Elimination.

2x – 3y + 7z = 10
8x – 12y + 28z = 40
6x – 9y + 21z = 30

Multiply Row 1 by -4, add Row 1 and Row 2.

If you end up with something like this, there are infinitely many solutions. A little detective work on the original equations might shed some light on why. Did you notice that the Row 2 is 4 times Row 1 and Row 3 is 3 times Row 1? This means that they are all on the same exact plane. They intersect in an infinite number of spots.

Solve this system of equations using Gaussian Elimination.

2x – y + 3 = 4
-2x + y – 3 = 6
x – 2y + 2z = 3

This looks simple to start. Add Row 1 and Row 2 and watch the impossible.

Ummm. That's not true. 0 ≠ 10. If you end up with something like this, there is no solution. Those three planes never intersect.