Transcript 3.5 Solving Systems of Equations in Three Variables

7.3 Solving Systems of
Equations in Three
Variables
Or when planes crash together
So far we have solved for the
intersection of lines
Do you remember what you get when
planes intersect?
So far we have solve for the
intersection of lines
Did you remember what you get when
planes intersect?
You form lines
What happens when you intersect
3 planes?
What happens when you intersect
3 planes?
You sometimes get points with three
variables.
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.
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.
You could just have three planes
that do not intersect at all
Parallel planes.
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
Solve the system of equations by
Gaussian Elimination
5 x  3 y  2 z  2

 2x  y  z  5
 x  4 y  2 z  16

I am going to rewrite the system
Solve the system of equations by
Gaussian Elimination
 x  4 y  2 z  16

 2x  y  z  5
5 x  3 y  2 z  2

Going to multiply row 1 by -2 and add to row 2
Going to multiply row 1 by -5 and add to row 3
Solve the system of equations by
Gaussian Elimination
 x  4 y  2 z  16

 2x  y  z  5
5 x  3 y  2 z  2

Going to multiply row 1 by -2 and add to row 2
Going to multiply row 1 by -5 and add to row 3
Solve the system of equations by
Gaussian Elimination
 x  4 y  2 z  16

  7 y  5z  27
 17 y  8z  78

Going to multiply row 2 by (17/-7) and add to
row 3
Solve the system of equations by
Gaussian Elimination
 x  4 y  2 z  16

  7 y  5z  27
(29 / 7) z  (87 / 7)

Going to multiply row 2 by (17/-7) and add to
row 3
Solve the system of equations by
Gaussian Elimination
 x  4 y  2 z  16

  7 y  5z  27
(29 / 7) z  (87 / 7)

Going to multiply row 3 by (7/29)
Solve the system of equations by
Gaussian Elimination
 x  4 y  2 z  16

  7 y  5 z  27

z  3

Going to multiply row 3 by -5 and add to row 2
Solve the system of equations by
Gaussian Elimination
 x  4 y  2 z  16

  7 y  42

z  3

Going to multiply row 2 by (-1/7)
Solve the system of equations by
Gaussian Elimination
 x  4 y  22

 y6
 z  3

Going to multiply row 3 by -2 and add to row 1
Solve the system of equations by
Gaussian Elimination
 x  4 y  22

 y6
 z  3

Going to multiply row 2 by -4 and add to row 1
Solve the system of equations by
Gaussian Elimination
 x  2

 y6
 z  3

Going to multiply row 2 by -4 and add to row 1
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.
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 + 3y – 9z = 15
-6x - 12y + 24z = - 42
When added together -9y + 15y = - 27
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
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
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
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.
Homework
Page 507# 4, 16, 28, 38,
46, 54, 66
Homework
Page 507
# 10, 22, 32,
42, 50, 60