7.3 Solving Systems of Equations in Three Variables

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.

http//en.wikipedia.org/wiki/Gaussian_elimination
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 5x 2y 4z 7
• 6x 3y 9z 15
• Middle equation by 6 added to the third
  equation.
• 6x 3y 9z 15
• -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
• -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
• 6x 4y 8z 11
• 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
• 9x 6y 12z - 3
• 9y 18z - 15

26
Solve the new system

• 6y 12z 3 multiply by 3
• 18y 36z 9
• 9y 18z - 15 multiply by 2
• -18y 36z 30
• Add together
• 18y 36z 9
• -18y 36z 30
• 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