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Systems of equations

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

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  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 = 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

  20. 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

  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? • 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

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

  25. 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

  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. • 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

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

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