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Systems of equations With Gaussian elimination

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

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System of equations Swap the order of the rows R1 R2

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

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

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

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

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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

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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

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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

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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.

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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

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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

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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

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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

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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

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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

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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

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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

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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

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-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

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-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

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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

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

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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

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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.

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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.

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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

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x=-2 y=3 z=2 D

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EXAMPLE 4 Solve linear systems with many or no solutions Solve the linear system. a.x – 2y = 4 3x – 6y = 8 b.4x – 10y = 8 – 14x + 35y = – 28 SOLUTION a.

EXAMPLE 4 Solve linear systems with many or no solutions Solve the linear system. a.x – 2y = 4 3x – 6y = 8 b.4x – 10y = 8 – 14x + 35y = – 28 SOLUTION a.

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