Solving systems of equations     - Dave Coulson, 2012
Okay, suppose we are required to work out the weight of three atomic particles,Exogen, Wydrogen and Zenon (chemical symbol...
Okay, suppose we are required to work out the weight of three atomicparticles, Exogen, Wydrogen and Zenon (chemical symbol...
Experimentation provides these results:  So for example (looking at the first line), a rock weighing 91  grams has four ch...
The way to untangle these equations is tosystematically reduce the square to a triangle… and then later reduce the triangl...
The way to untangle these equations is tosystematically reduce the square to a triangle… and then later reduce the triangl...
Double the numbers in the second line so that the x terms are the same.
Then subtract the numbers in the second line from the numbers above them.Put the new numbers in the second line.
Now multiply the bottom line by four so that the x terms are the same as in thefirst line.
… and subtract the bottom line from the first line so that the x term disappears.Put the result in the third line
Can you see that I made the x terms disappear by matching them and thensubtracting them?
Can you see that I made the x terms disappear by matching them and thensubtracting them?I’m going to do the same with the ...
Triple the middle line so that the y term is the same as in the bottom line.
Subtract the bottom line from the middle lineand put the result in the bottom line.
A square of numbers has turned into a triangle of numbers.
Identify z by dividing the bottom line by 10.
You know what z is now, so you can put that value in the second line andidentify y.
You know what z is now, so you can put that value in the second line andidentify y.
You know what z is now, so you can put that value in the second line andidentify y.
You know what z is now, so you can put that value in the second line andidentify y.
And you can get x by substituting the values for y and z into the top line.
And you can get x by substituting the values for y and z into the top line.
And you can get x by substituting the values for y and z into the top line.
And you can get x by substituting the values for y and z into the top line.
A square of numbers became a triangle of numbers and then a diagonal line.This process is called Gaussian elimination, and...
This is a 3x3 system (three equations, three unknowns), and can be solved byhand in a few minutes. Larger systems take a l...
[END]
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Solving systems of linear e

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Solving systems of linear e

  1. 1. Solving systems of equations - Dave Coulson, 2012
  2. 2. Okay, suppose we are required to work out the weight of three atomic particles,Exogen, Wydrogen and Zenon (chemical symbols X, Y and Z).
  3. 3. Okay, suppose we are required to work out the weight of three atomicparticles, Exogen, Wydrogen and Zenon (chemical symbols X, Y and Z).X, Y and Z never occur freely innature, meaning you can never find ablock of pure X, Y or Z.They always occur in mixtures, whichmeans if we want to work out theweights of these things, we have tobe clever about it.
  4. 4. Experimentation provides these results: So for example (looking at the first line), a rock weighing 91 grams has four chunks of Exogen in it as well as nine chunks of Wydrogen and fourteen chunks of Zenon.
  5. 5. The way to untangle these equations is tosystematically reduce the square to a triangle… and then later reduce the triangle to a diagonal line.
  6. 6. The way to untangle these equations is tosystematically reduce the square to a triangle… and then later reduce the triangle to a diagonal line.(Which isn’t as dumb as it sounds)
  7. 7. Double the numbers in the second line so that the x terms are the same.
  8. 8. Then subtract the numbers in the second line from the numbers above them.Put the new numbers in the second line.
  9. 9. Now multiply the bottom line by four so that the x terms are the same as in thefirst line.
  10. 10. … and subtract the bottom line from the first line so that the x term disappears.Put the result in the third line
  11. 11. Can you see that I made the x terms disappear by matching them and thensubtracting them?
  12. 12. Can you see that I made the x terms disappear by matching them and thensubtracting them?I’m going to do the same with the y terms.
  13. 13. Triple the middle line so that the y term is the same as in the bottom line.
  14. 14. Subtract the bottom line from the middle lineand put the result in the bottom line.
  15. 15. A square of numbers has turned into a triangle of numbers.
  16. 16. Identify z by dividing the bottom line by 10.
  17. 17. You know what z is now, so you can put that value in the second line andidentify y.
  18. 18. You know what z is now, so you can put that value in the second line andidentify y.
  19. 19. You know what z is now, so you can put that value in the second line andidentify y.
  20. 20. You know what z is now, so you can put that value in the second line andidentify y.
  21. 21. And you can get x by substituting the values for y and z into the top line.
  22. 22. And you can get x by substituting the values for y and z into the top line.
  23. 23. And you can get x by substituting the values for y and z into the top line.
  24. 24. And you can get x by substituting the values for y and z into the top line.
  25. 25. A square of numbers became a triangle of numbers and then a diagonal line.This process is called Gaussian elimination, and can be used for systems ofequations no matter how big they are. You need to have as many equations asthere are unknowns in the equation.
  26. 26. This is a 3x3 system (three equations, three unknowns), and can be solved byhand in a few minutes. Larger systems take a lot more time, but the process isthe same.
  27. 27. [END]

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