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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.

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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


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


-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


-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 showing work)

  • 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 showing work)

  • 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. showing work)

  • 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


x showing work)=-2

y=3

z=2 D


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