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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.

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Presentation on theme: "Systems of equations With Gaussian elimination. System of equations Find all pairs of x and y values that make the equations true."— Presentation transcript:

1 Systems of equations With Gaussian elimination

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

3 System of equations Swap the order of the rows R1 R2

4 System of equations Multiply a row by a number -4*R1  R1

5 System of equations Add a row to another row R1 + R2  R2

6 System of equations Multiply a row by a number -¼*R1  R1

7 The add-multiply shortcut Multiply a row by a number and add it to another row -4*R1 + R2  R2

8 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

9 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

10 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

11 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.

12 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

13 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

14 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

15 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

16 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

17 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

18 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

19 Gaussian Elimination Finish my incomplete step Get 0y in all other equations 1x + 2y + 0z = * R2 + R1 --> R1 0x + 1y + 0z = 3 0x + 0y + 1z = 0.5 1x + 0y + 0z = x + 1y + 0z = 3 0x + 0y + 1z = 0.5

20 Solve the system of equations -3x − 9y = -6 -3x − 13y = -8 a)x = -2, y = 0 b)x = 0, y = 8/13 c)x = 1/2, y = 1/2 d)x = -1/2, y = -1/2 e)None of the above

21 -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

22 -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

23 What is the system of equations corresponding to the augmented matrix below? a)2x+3y = 4, x + 2y = 3 b)3x+2y = 4, 2x + y = 3 c)2x+y = 4, 3x + 2y = 3 d)x+y = 4, x + 2y = 3 e)None of the above

24 What is the system of equations corresponding to the augmented matrix below? a)2x+3y = 4, x + 2y = 3

25 Solving a system of equations on your calculator (and showing work) Solve 4x + 8y - 4z = 8 2x + 3y + 4z = 4 5x + 8y + 1z = 7 In my calculator, I set the matrix [A] Then I used the command rref([A]) The calculator output was So the answer is x=-3.5 y=3 z=0.5

26 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.

27 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.

28 Solve the following system. a)x = 0, y = 3, z = 2 b)x = 5, y = 3, z = 2 c)x = 1, y = 3, z = 2 d)x = -2, y = 3, z = 2 e)None of the above

29 x=-2 y=3 z=2 D


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