## anonymous 3 years ago Solve the following system by Gauss–Jordan elimination. 2x₁+5x₂+3x₃=35 12x₁+31x₂+20x₃=216

1. anonymous

I got x₁=5/2 x₂=6 x₃=t (free variable) The answer says I am wrong. Care to tell me what I did wrong?

2. anonymous

@jim_thompson5910

3. anonymous

@ParthKohli @UnkleRhaukus

4. UnkleRhaukus

x_3=0

5. anonymous

Right. But whats wrong with my other answers?

6. anonymous

Here is what the online page looks like:

7. anonymous

Shouldn't the other variables be in terms of the free variable?

8. anonymous

What did were your elimination steps?

9. anonymous

It's difficult to post the elimination steps here. I checked with my calculator though. It's correct.

10. anonymous

But yeah, that's what I think @wio . I don't know how to express it though.

11. UnkleRhaukus

when we check your solution , we see t must be 0 x₁=5/2 x₂=6 x₃=t _________ 2x₁+5x₂+3x₃=35 5+30+3t = 35 35 +3t = 35 _________ 12x₁+31x₂+20x₃ = 216 60 + 186 + 20t = 216 216 + 20t = 216

12. anonymous

but t is a free variable is it not? SO it can be any value it wants.

13. anonymous

$\begin{bmatrix} 2&5&3 \\ 12&31&20 \\ 0&0&1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 35 \\ 216 \\ t \end{bmatrix}$

14. anonymous

Did you solve it like this?

15. anonymous

No... How did you get 0,0 and 1?

16. anonymous

I just added in the equation $$x_3=t$$

17. anonymous

Sorry, that doesn't make sense.

18. anonymous

I added in the equation: $0x_1+ 0x_2 +1x_3 = t$

19. anonymous

Ohh I see.

20. anonymous

All of the elimination you did before should work for the most part.

21. anonymous

So I do a back substitution?

22. anonymous

You first want to get the second row to have a zero in it's first colum

23. anonymous

because x3=t so solve the others n terms of t.

24. anonymous

then you can back substitute

25. anonymous

1 second.

26. anonymous

x₁=2.5+3.5t ?

27. anonymous

x₂=6-2t?

28. anonymous

yeah I'm getting that too

29. anonymous

did you try it?

30. anonymous

One attempt left :3 . Fingers crossed.

31. anonymous

you can double check it you know

32. anonymous

How?

33. anonymous

plug in each equation into the equations

34. anonymous

Assign a random number to t and see if it works?

35. anonymous

like, plug in t for x_3 and 6-2t for x_2 into the equations

36. anonymous

hmm actually I'm not quite sure how to double check it

37. anonymous

let me look at wolfram

38. anonymous

You can do this on wolfram? O_O .

39. anonymous
40. anonymous

wolfram agrees... it's just a matter of the stupid answer verify thing not being a wingspan

41. anonymous

THANKS!!! It's right :) .

42. anonymous

No problem. Matrix is your friend.

43. anonymous

I love Linear algebra <3 .