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

- anonymous

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

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- schrodinger

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- 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?

- anonymous

@jim_thompson5910

- anonymous

@ParthKohli @UnkleRhaukus

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## More answers

- UnkleRhaukus

x_3=0

- anonymous

Right. But whats wrong with my other answers?

- anonymous

Here is what the online page looks like:

##### 1 Attachment

- anonymous

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

- anonymous

What did were your elimination steps?

- anonymous

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

- anonymous

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

- 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

- anonymous

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

- anonymous

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

- anonymous

Did you solve it like this?

- anonymous

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

- anonymous

I just added in the equation \(x_3=t\)

- anonymous

Sorry, that doesn't make sense.

- anonymous

I added in the equation: \[
0x_1+ 0x_2 +1x_3 = t
\]

- anonymous

Ohh I see.

- anonymous

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

- anonymous

So I do a back substitution?

- anonymous

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

- anonymous

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

- anonymous

then you can back substitute

- anonymous

1 second.

- anonymous

x₁=2.5+3.5t ?

- anonymous

x₂=6-2t?

- anonymous

yeah I'm getting that too

- anonymous

did you try it?

- anonymous

One attempt left :3 . Fingers crossed.

- anonymous

you can double check it you know

- anonymous

How?

- anonymous

plug in each equation into the equations

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

let me look at wolfram

- anonymous

You can do this on wolfram? O_O .

- anonymous

http://www.wolframalpha.com/input/?i=2x%2B5y%2B3z%3D35%2C+12x%2B31y%2B20z%3D216%2C+z+%3D+t

- anonymous

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

- anonymous

THANKS!!! It's right :) .

- anonymous

No problem. Matrix is your friend.

- anonymous

I love Linear algebra <3 .

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