7.3 Solving 7.3 Solving Systems of Systems of Equations in Three Equations in Three Variables Variables Or when planes crash Or when planes crash together together
7.3 Solving Systems of Equations in Three Variables
Dec 30, 2015
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. - PowerPoint PPT Presentation
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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?
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
I am going to rewrite the system
1624
52
2235
zyx
zyx
zyx
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
2235
52
1624
zyx
zyx
zyx
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
2235
52
1624
zyx
zyx
zyx
Solve the system of equations by Gaussian Elimination
Going to multiply row 2 by (17/-7) and add to row 3
78817
2757
1624
zy
zy
zyx
Solve the system of equations by Gaussian Elimination
Going to multiply row 2 by (17/-7) and add to row 3
)7/87()7/29(
2757
1624
z
zy
zyx
Solve the system of equations by Gaussian Elimination
Going to multiply row 3 by (7/29)
)7/87()7/29(
2757
1624
z
zy
zyx
Solve the system of equations by Gaussian Elimination
Going to multiply row 3 by -5 and add to row 2
3
2757
1624
z
zy
zyx
Solve the system of equations by Gaussian Elimination
Going to multiply row 2 by (-1/7)
3
427
1624
z
y
zyx
Solve the system of equations by Gaussian Elimination
Going to multiply row 3 by -2 and add to row 1
3
6
224
z
y
yx
Solve the system of equations by Gaussian Elimination
Going to multiply row 2 by -4 and add to row 1
3
6
224
z
y
yx
Solve the system of equations by Gaussian Elimination
Going to multiply row 2 by -4 and add to row 1
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
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