- By
**pepin** - Follow User

- 236 Views
- Uploaded on

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

With Gaussian elimination

System of equations

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

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

- 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

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

- 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 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 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 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 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 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 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 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 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

-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

-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

-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?

- 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 below?

- 2x+3y = 4, x + 2y = 3

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

- 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 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.

- 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

Download Presentation

Connecting to Server..