## swiftskier96 Group Title 43. What is the solution of the system of equations? 2x + 2y + 3z = -6 3x + 5y + 4z = 3 2x + 3y + 4z = -10 one year ago one year ago

1. swiftskier96 Group Title

@phi I forgot how to do this. Can you help?

2. swiftskier96 Group Title

@ghazi

3. phi Group Title

elimination is the technique

4. phi Group Title

are you allowed a calculator?

5. swiftskier96 Group Title

Yeah i have one.

6. phi Group Title

then the best way is just "brute force" write down just the numbers keeping the same order 2 2 3 -6 3 5 4 3 2 3 4 -10

7. phi Group Title

the idea is we want the first column to have zeros below the first 2 (call 2 in 1st row, 1st col the pivot) (that means we will zero out the 3 and the 2 beneath it) to zero out the 3 in the 2nd row we do this: calculate -3/pivot (in other words, -3/2= -1.5) and save this number call the number the "factor" so I can talk about it. now for each number in the 2nd row: multiply the factor times the number in the 1st row, and add it to the number in the 2nd row. write down the answer Here is what I mean for the 2nd row the factor is -3/2. multiply -3/2 * 2 (the 1st number in row 1) and add it to 3 (1st # in row 2) we get 0 now multiply -3/2 * 2 (2nd # in 1st row) and add it to 5 (2nd # in 2nd row). we get 2 multiply -3/2 * 3 (3rd # in 1st row) and add it to 4 (3rd # in 2nd row). we get -0.5 finally, multiply -3/2 *-6 and add it to 3 we get 9+3= 12 these numbers are our new 2nd row. So far we have 2 2 3 -6 0 2 -0.5 12 2 3 4 -10

8. phi Group Title

now we do the same thing for row 3. still use 2 in the first row as the pivot, calculate the factor for the bottom row: it is -2/pivot or -1 can you figure out what the bottom row becomes?

9. swiftskier96 Group Title

So i take the first number on the bottom row (2) and multiply it by the factor (-2/-1)?

10. phi Group Title

This is complicated. The factor is the # we want to zero out (2 in the bottom) row divided by minus the pivot. the pivot is the first # in the 1st row. It is 2. -pivot is -2. the factor is 2 (bottom row)/-2 or -1 the factor is -1 Now multiply the factor times each number in the pivot row (1st row) and add to the number in the bottom row. Does that make sense?

11. swiftskier96 Group Title

Yeah that makes more sense. Ill figure it out here. One sec...

12. swiftskier96 Group Title

In the 3rd row we'd have -2 -3 -4 10.

13. phi Group Title

you multiplied the bottom row by the factor. what you want to do is multiply the pivot row by the factor then add that result to the bottom row you know you are on the right track if the first # in the answer is 0 so multiply the 1st row by -1 add these new numbers to the 3rd row. what do you get?

14. swiftskier96 Group Title

Ohhh ok. I got 0 1 1 -4

15. phi Group Title

yes, so now the numbers look like this 2 2 3 -6 0 2 -0.5 12 0 1 1 -4 notice that the 1st column has zeros except to the 2. we now move to the 2nd row to its first non-zero and call that the pivot row 2 is the pivot row. its first not zero is the 2. 2 is the new pivot. we want to zero out the 1 in the 3rd row. Find the factor: #/-pivot can you tell me the factor?

16. swiftskier96 Group Title

So are you saying that i should take the 1 in the 3rd row and divide it by the pivot (2)?

17. phi Group Title

almost, the factor is the 1 in the 3rd row (it is the number just below the pivot) but divide by minus the pivot.

18. swiftskier96 Group Title

Sorry, im confused. What does divide by minus the pivot mean? Are you saying that the pivot should be negative?

19. phi Group Title

I am saying when we want the factor (for the row we are working on) we take the number in the row (1 in this case) and divide by minus the pivot. (pivot is 2, minus the pivot is -2) the factor is going to be 1 / -2

20. phi Group Title

Yes, the factor is -0.5 now multiply the factor times the pivot row add those numbers to the 3rd row. this will be the new 3rd row. what do you get ?

21. swiftskier96 Group Title

So id do -0.5 * 0, -0.5 * 2, -0.5 * -0.5, and -0.5 * 12. That would be 0 -1 0.25 -6. Then i add those numbers to the 3rd row which would make the 3rd row 0 0 1.25 -10

22. phi Group Title

perfect

23. swiftskier96 Group Title

Yesss! Lol

24. phi Group Title

The numbers look like this 2 2 3 -6 0 2 -0.5 12 0 0 1.25 -10 now remember we left out the variables. Let's look at just the bottom row and put back the variables: 0x + 0y + 1.25z = -10 0 times x and 0 times y are both 0 so what we really have is 1.25z= -10 can you solve for z ?

25. swiftskier96 Group Title

Yesh i can! 1.25z/1.25 = -10/1.25 z = -8

26. phi Group Title

what we are doing is called "back substitution". We work backwards (or upwards) the 2nd row is 0 2 -0.5 12 with variables (and leaving out the 0) 2y -0.5*z = 12 The first thing to do is replace z with its value -8 2y -0.5* -8 =12 can you solve for y ?

27. swiftskier96 Group Title

-0.5 * -8 = 4 So would it be 2y - 4 = 12 or 2y + 4 = 12?

28. phi Group Title

if you get confused about signs, don't use subtraction. I think 3-2 as 3 + -2 (three plus a minus 2) so the equation 2y -0.5* -8 =12 I think of as 2y + (-0.5* -8) =12

29. phi Group Title

I really should have written as 2y + -0.5z = 12 in the first place

30. swiftskier96 Group Title

Ok. So it would be 2y + 4 = 12 2y + 4 - 4 = 12 - 4 2y = 8 2y/2 = 8/2 y = 4

31. phi Group Title

yes. Do you want to tackle the first row ?

32. swiftskier96 Group Title

Sure, ill try. The first row is 2 2 3 -6. So it would be 2x + (2 + 3) = -6 right?

33. swiftskier96 Group Title

Or is it 2x + (2*3) = -6?

34. phi Group Title

don't skip steps. Put back in the variables y and z

35. phi Group Title

remember each row has the variables, we just left them out because we don't calculate with them until this last "back substitution"

36. swiftskier96 Group Title

Oh right. So 2 2 3 -6 becomes 2x + 2(4) + 3(-8) = -6?

37. phi Group Title

yes, but I would write it as 2x + 2y + 3z = -6 first, then write it as 2x + 2(4) + 3(-8) = -6 (until I did a few more problems).

38. phi Group Title

Just to help remember what is going on

39. swiftskier96 Group Title

Ok. So that would look like 2x + 8 + (-24) = -6 2x + (-16) = -6 2x + (-16) - (-16) = -6 - (-16) 2x = 10 2x/2 = 10/2 x = 5

40. phi Group Title

Yes, so that is the answer: x=5, y=4, z=-8 A bit painful, but 3 equations and 3 unknowns are painful when you do them by hand. If you have the energy,see if you can redo the first column on your own. I gave the steps, but it may have been as clear as mud the first time through.

41. swiftskier96 Group Title

Ok. Will do. Thanks for all the help! That was a really hard question to go through!

42. phi Group Title

The good news is this technique works for any size, 2 x 2 upto big x big (where big is as many numbers as you want to think about). These days computers solve 1,000,000 by 1,000,000 or even bigger....

43. swiftskier96 Group Title

Man... That would take a really long time to solve... O_O