1. 7.3 Solving Systems of Equations in Three Variables
Or when planes crash together

2. So far we have solved for the intersection of lines
Do you remember what you get when planes intersect?

3. So far we have solve for the intersection of lines
Did you remember what you get when planes intersect?
You form lines

4. What happens when you intersect 3 planes?

5. What happens when you intersect 3 planes?
You sometimes get points with three variables.

6. What happens when you intersect 3 planes?
You sometimes get points with three variables. Of course they can intersect in different ways.
Here we get a
line again.

7. What happens when you intersect 3 planes?
You sometimes get points with three variables. Of course they can intersect in different ways.
Of course we
can get nothing.
This would be
No solution.

8. You could just have three planes that do not intersect at all
Parallel planes.

9. Solve the system of equations by Gaussian Elimination
What is Gaussian Elimination?
In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations.Gauss – Jordan elimination, an extension of this algorithm, reduces the matrix further to diagonal form, which is also known as reduced row echelon form.

10. Solve the system of equations by Gaussian Elimination
I am going to rewrite the system

11. Solve the system of equations by Gaussian Elimination
Going to multiply row 1 by -2 and add to row 2
Going to multiply row 1 by -5 and add to row 3

12. Solve the system of equations by Gaussian Elimination
Going to multiply row 1 by -2 and add to row 2
Going to multiply row 1 by -5 and add to row 3

13. Solve the system of equations by Gaussian Elimination
Going to multiply row 2 by (17/-7) and add to row 3

14. Solve the system of equations by Gaussian Elimination
Going to multiply row 2 by (17/-7) and add to row 3

15. Solve the system of equations by Gaussian Elimination
Going to multiply row 3 by (7/29)

16. Solve the system of equations by Gaussian Elimination
Going to multiply row 3 by -5 and add to row 2

17. Solve the system of equations by Gaussian Elimination
Going to multiply row 2 by (-1/7)

18. Solve the system of equations by Gaussian Elimination
Going to multiply row 3 by -2 and add to row 1

19. Solve the system of equations by Gaussian Elimination
Going to multiply row 2 by -4 and add to row 1

20. Solve the system of equations by Gaussian Elimination
Going to multiply row 2 by -4 and add to row 1

21. Solve the system 5x + 3y + 2z = 2 2x + y – z = 5 x + 4y + 2z = 16
The point of intersect for the system is
( - 2, 6, - 3)
These points make all the equations true.

22. Now one with infinite solutions
2x + y – 3z = 5 x + 2y – 4z = 7
6x + 3y – 9z = 15
Middle equation by – 6 added to the third equation.
-6x - 12y + 24z = - 42
When added together -9y + 15y = - 27

23. Solve the new system - 3y + 5z = - 9 -9y + 15z = - 27
Multiply the top equation by – 3 then add to the bottom equation
9y – 15z = 27
0 = 0 Infinite many solutions

24. One the has no solutions
3x – y – 2z = 4
6x + 4y + 8z = 11
9x + 6y + 12z = - 3
Multiply the first equation by – 2 and add to the middle equation.
-6x + 2y + 4z = - 8
6y + 12z = 3

25. One the has no solutions
3x – y – 2z = 4
6x + 4y + 8z = 11
9x + 6y + 12z = - 3
Multiply the first equation by – 3 and add to the last equation.
-9x + 3y + 6z = - 12
9y + 18z = - 15

26. Solve the new system 6y + 12z = 3 multiply by 3 18y + 36z = 9
Add together
0 = 39 Wrong!, No solution.

27. Homework
Page 507-
# 4, 16, 28, 38,
46, 54, 66

28. Homework
Page 507
# 10, 22, 32,
42, 50, 60