systems of equations
Download
Skip this Video
Download Presentation
Systems of equations

Loading in 2 Seconds...

play fullscreen
1 / 29

Systems of equations - PowerPoint PPT Presentation


  • 236 Views
  • Uploaded on

Systems of equations . With Gaussian elimination. System of equations. Find all pairs of x and y values that make the equations true. System of equations. Swap the order of the rows R1 <-> R2. System of equations. Multiply a row by a number -4 * R1  R1. System of equations.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Systems of equations ' - pepin


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
systems of equations

Systems of equations

With Gaussian elimination

system of equations
System of equations

Find all pairs of x and y values that make the equations true.

system of equations1
System of equations

Swap the order of the rows

R1 <-> R2

system of equations2
System of equations

Multiply a row by a number

-4*R1  R1

system of equations3
System of equations

Add a row to another row

R1 + R2  R2

system of equations4
System of equations

Multiply a row by a number

-¼*R1  R1

the add multiply shortcut
The add-multiply shortcut

Multiply a row by a number and add it to another row

-4*R1 + R2  R2

row operations
Row operations
  • Swap rows R1<-> R2
  • Multiply a row by a number k*R1  R1
  • Add rows together R1 + R2  R2
  • Multiply-add shortcut k*R1 + R2  R2
gaussian elimination
Gaussian Elimination
  • A method that you can use to solve ANY system of equations (no matter how big), using only two rules.
  • Multiply a row by a number k*R1  R1
  • Multiply-add shortcut k*R1 + R2  R2
how to solve a system of any number of linear equations
How to solve a system of (any number of) linear equations

Method: Gaussian Elimination

  • Today’s fun irrelevant fact: Gauß is my great-great-great-great-great-great-great-grand-advisor
  • Gauß Gerling Plucker Klein  Bocher Ford  Engen Steffe Thompson  Castillo-Garsow
the method
The method
  • Write equations in standard form
  • Use multiply to get 1x in the top equation
  • Use multiply-add to get 0x in all other equations.
  • Use multiply to get 1y in the second equation
  • Use multiply-add to get 0y in all other equations.
  • Repeat for all variables.
gaussian elimination1
Gaussian Elimination
  • Get your system in standard form

(All the variables on one side, all the constants on the other)

4x + 8y - 4z = 8

2x + 3y + 4z = 4

5x + 8y + 1z = 7

gaussian elimination2
Gaussian Elimination
  • Use multiply to get 1x in the top equation

4x + 8y - 4z = 8 (1/4) * R1 --> R1

2x + 3y + 4z = 4

5x + 8y + 1z = 7

1x + 2y - 1z = 2

2x + 3y + 4z = 4

5x + 8y + 1z = 7

gaussian elimination3
Gaussian Elimination
  • Use multiply-add to get 0xs everywhere else

1x + 2y - 1z = 2

2x + 3y + 4z = 4 -2 * R1 + R2 --> R2

5x + 8y + 1z = 7 -5 * R1 + R3 --> R3

1x + 2y - 1z = 2

0x - 1y + 6z = 0

0x - 2y + 6z = -3

gaussian elimination4
Gaussian Elimination
  • Use multiply to get 1y in the second equation

1x + 2y - 1z = 2

0x - 1y + 6z = 0 -1 * R2 --> R2

0x - 2y + 6z = -3

1x + 2y - 1z = 2

0x + 1y - 6z = 0

0x - 2y + 6z = -3

gaussian elimination5
Gaussian Elimination
  • Use multiply-add to get 0ys in all other equations
  • You can do all of these now, but I’m going to put one off for later.

1x + 2y - 1z = 2

0x + 1y - 6z = 0

0x - 2y + 6z = -3 2 * R2 + R3 --> R3

1x + 2y - 1z = 2

0x + 1y - 6z = 0

0x + 0y - 6z = -3

gaussian elimination6
Gaussian Elimination
  • Use multiply to get 1z in the third equation

1x + 2y - 1z = 2

0x + 1y - 6z = 0

0x + 0y - 6z = -3 (-1/6) * R3 --> R3

1x + 2y - 1z = 2

0x + 1y - 6z = 0

0x + 0y + 1z = 0.5

gaussian elimination7
Gaussian Elimination
  • Get 0z in all other equations

1x + 2y - 1z = 2 1 * R3 + R1 --> R1

0x + 1y - 6z = 0 6 * R3 + R2 --> R2

0x + 0y + 1z = 0.5

1x + 2y + 0z = 2.5

0x + 1y + 0z = 3

0x + 0y + 1z = 0.5

gaussian elimination8
Gaussian Elimination
  • Finish my incomplete step
  • Get 0y in all other equations

1x + 2y + 0z = 2.5 -2 * R2 + R1 --> R1

0x + 1y + 0z = 3

0x + 0y + 1z = 0.5

1x + 0y + 0z = -3.5

0x + 1y + 0z = 3

0x + 0y + 1z = 0.5

slide20

Solve the system of equations

-3x − 9y = -6-3x − 13y = -8

  • x = -2, y = 0
  • x = 0, y = 8/13
  • x = 1/2, y = 1/2
  • x = -1/2, y = -1/2
  • None of the above
slide21

-3x − 9y = -6 (-1/3)*R1 ->R1

-3x − 13y = -8

1x+ 3y = 2

-3x − 13y = -8 3R1 + R2 -> R2

1x + 3y = 2

0x − 4y = -2 (-1/4)R2 -> R2

1x + 3y = 2 (-3)R2 + R1 -> R1

0x+ 1y = ½

1x + 0y = 1/2

0x + 1y = 1/2

C

slide22

-3x − 9y = -6 (-1/3)*R1 ->R1

-3x − 13y = -8

1x+ 3y = 2

-3x − 13y = -8 3R1 + R2 -> R2

1x + 3y = 2

0x − 4y = -2 (-1/4)R2 -> R2

1x + 3y = 2 (-3)R2 + R1 -> R1

0x+ 1y = ½

1x + 0y = 1/2

0x + 1y = 1/2

what is the system of equations corresponding to the augmented matrix below
What is the system of equations corresponding to the augmented matrix below?
  • 2x+3y = 4, x + 2y = 3
  • 3x+2y = 4, 2x + y = 3
  • 2x+y = 4, 3x + 2y = 3
  • x+y = 4, x + 2y = 3
  • None of the above
what is the system of equations corresponding to the augmented matrix below1
What is the system of equations corresponding to the augmented matrix below?
  • 2x+3y = 4, x + 2y = 3
solving a system of equations on your calculator and showing work
Solving a system of equations on your calculator (and showing work)

In my calculator, I set the matrix [A]

  • Solve

4x + 8y - 4z = 8

2x + 3y + 4z = 4

5x + 8y + 1z = 7

Then I used the command rref([A])

The calculator output was

So the answer is

x=-3.5

y=3

z=0.5

special situations
Special situations
  • If, at the end you wind up with something impossible, then there are NO SOLUTIONS
      • Example:

The last row:

0x + 0y = 1 is impossible,

So there are NO SOLUTIONS.

special situations1
Special situations
  • If, at the end you wind up with something that is always true, then there are INFINITELY MANY SOLUTIONS
      • Example:

The last row:

0x + 0y = 0 is always true,

So there are INFINITELY MANY SOLUTIONS.

solve the following system
Solve the following system.
  • x = 0, y = 3, z = 2
  • x = 5, y = 3, z = 2
  • x = 1, y = 3, z = 2
  • x = -2, y = 3, z = 2
  • None of the above
slide29

x=-2

y=3

z=2 D

ad