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

7.3 Solving 7.3 Solving Systems of Systems of

Equations in Three Equations in Three VariablesVariablesOr when planes crash Or when planes crash

togethertogether

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?


You sometimes get points with three variables.


You sometimes get points with three variables. Of course they can intersect in different ways.

Here we get a

line again.


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


I am going to rewrite the system

1624

52

2235

zyx

zyx

zyx


Going to multiply row 1 by -2 and add to row 2


2235

52

1624

zyx

zyx

zyx


Going to multiply row 2 by (17/-7) and add to row 3

78817

2757

1624

zy

zy

zyx


Going to multiply row 2 by (17/-7) and add to row 3

)7/87()7/29(

2757

1624

z

zy

zyx


Going to multiply row 3 by (7/29)

)7/87()7/29(

2757

1624

z

zy

zyx



3

2757

1624

z

zy

zyx


Going to multiply row 2 by (-1/7)

3

427

1624

z

y

zyx



3

6

224

z

y

yx



3

6

2

z

y

x

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 = 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

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 = 46x + 4y + 8z = 119x + 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 = 46x + 4y + 8z = 119x + 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.

HomeworkHomework

Page 507-Page 507-

## 4, 16, 28, 38,4, 16, 28, 38,

46, 54, 6646, 54, 66

HomeworkHomework

Page 507 Page 507

## 10, 22, 32, 10, 22, 32,

42, 50, 60 42, 50, 60

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

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