1. 3.4 Solving equations Simultaneous equations

2. Snakes on planes or How to describe the geometric relationship between planes

3. Once upon a time there was a plane called snakes…. Let’s call it 2x + 4y + 3z = 10 A point that would sit on this plane is (x, y, z) = (2, 0, 2) x y z

4. The plane then cloned himself… 2x + 4y + 3z = 10

5. And tried to hide the fact that he wasn’t as different as he thought…. 2x + 4y + 3z = 10 4x + 8y + 6z = 20 10x + 20y + 15z = 50

6. But then decided on being inconsistent instead…… 2x + 4y + 3z = 10 4x + 8y + 6z = 30 10x + 20y + 15z = 70

7. And being inconsistent changed his mind again…… 2x + 4y + 3z = 10 4x + 8y + 6z = 30 x + 2y + 3z = 15

8. Story recap Write a system of three linear equations with three variables to represent the following geometric situations: Three planes parallel Three planes become one! Two planes parallel with the third plane intersecting both planes

9. Snakesy wanted to prove how unique he was so found two other planes with nothing in common with him…… 2x + 4y + 3z = 10 x – 2y + 5z = 8 4x - 3y - 2z = -10

10. But then decided it would be nice to be dependent on some friends….. 2x + 4y + 3z = 10 3x - 2y + 3z = 20 5x + 2y + 6z = 30

11. But also didn’t want anyone to know that he needed a little help from his friends…… 4x + 8y + 6z = 20 3x - 2y + 3z = 20 5x + 2y + 6z = 30

12. So decided to elevate himself above his friends and be once again inconsistent! 2x + 4y + 3z = 10 3x - 2y + 3z = 20 5x + 2y + 6z = 30 2x + 4y + 3z = 40

13. Story recap Write a system of three linear equations with three variables to represent the following geometric situations: Three planes that meet in a line Three planes that intersect at one point only Two planes meet in a line that is parallel to the third plane