Undergrad This linear system has a solution? (Gauss' elimination)

[requied]

TL;DR

I have the following linear equation
-3x1 +x2-2x3 =-7
5x1 + 3x2-4x3 = 2
x1 + 2x2-3x3 = -1

And I wonder it has a solution, because I found the third line as:
0 0 0 = -49/6

I don't know the terms so I'm sorry if the informations at summary above is unclear. But I add a detailed photo of my calculations below. I use Gauss' Elimination laws.

 

[mathwonk]

I didn't check your work, but I urge you to do it again. I taught this for years and always made mistakes on every calculation unless I did every step in complete detail. no shortcuts at all.

 

[requied]

I didn't check your work, but I urge you to do it again. I taught this for years and always made mistakes on every calculation unless I did every step in complete detail. no shortcuts at all.

I think that the calculations shown in photo is enough to someone who already know a little solving linear systems.

but I urge you to do it again.

Here you refer to solve again the question?

 

[mathwonk]

"I think that the calculations shown in photo is enough to someone who already know a little solving linear systems."

I think you are missing the point, i don't have a problem, you do.

 

[requied]

"I think that the calculations shown in photo is enough to someone who already know a little solving linear systems."

I think you are missing the point, i don't have a problem, you do.

Sir, what kind of problem are you speaking of? I don't get it what I actually should do ?

 

[mathwonk]

Forgive me, I am only suggesting if you are in doubt as to whether a gaussian elimination is correct, you should do it again, and see if you get the same answer.

 

[requied]

Forgive me, I am only suggesting if you are in doubt as to whether a gaussian elimination is correct, you should do it again, and see if you get the same answer.

I get it but I had wondered this case which I add below. Isn't it time wasting making an effort to solve whole question again

 

[requied]

So this period of time I researched and found a linear system calculator which would solve step by step and says "Equation 3 cannot be solved, therefore, the system has no solution (i.e. the system is inconsistent).
"http://www.math.odu.edu/~bogacki/cg...at=R3;whid=V;whid=1;submit21=Submit&hideops=0

So, I think I get what I want.

 

[mathwonk]

"Isn't it time wasting making an effort"? indeed whose time do you speak of?

 

[requied]

"Isn't it time wasting making an effort"? indeed whose time do you speak of?

Man it's 4 am here :D I'm doing a homework for 5 hours. So my time...

 

[requied]

Man it's 4 am here :D I'm doing a homework for 5 hours. So my time...

I don't want to be misunderstood, I am wasting my time, not you.

 

[mathwonk]

pardon me for being flippant. i share your experience at trying to get gaussian calculations correct as i said. but the sad truth is there is no other way but to do them over and over until all errors are eliminated. and if you are up that late doing hw, perhaps you could learn to do it earlier next time?

 

[requied]

if you are up that late doing hw, perhaps you could learn to do it earlier next time?

You are right at this point, thanks to you for all motivations. All the best :)

 

[mathman]

Calculate the determinant (I get 0). There is no solution! That's the problem!

 

[requied]

There is no solution!

So the actual answer is this, right?

 

[WWGD]

Why don't you compute the determinant of the original matrix to see if it is 0 too? Not a guarantee if it is, but shows a problem if it is not.

 

[requied]

The determinant of the original matrix is 0. I don't know what happens if a determinant of a matrix is 0.

 

[WWGD]

View attachment 263424
The determinant of the original matrix is 0. I don't know what happens if a determinant of a matrix is 0.

One of the consequences is that there is a vector b for which Ax=b has no solution. And the matrix is not invertible.

 

[mathman]

The determinant is 0 means that the equations are not linearly independent, so you can't get a solution. Getting (0,0,0) for the left side after Gauss elimination is equivalent to a 0 determinant.

 

[FactChecker]

A zero determinant means that there is no unique solution. There could be many solutions or there may be none.
In this case, there is no solution. Your calculation leading to an obvious false equation is proof of that (although I haven't checked your work). An easier calculation process would be to add 3*row3 to row1 and subtract 5*row3 from row2. That will give you new row1 and row2 which are clearly contradictory: $7x_2-11x_3=-10$ and $-7x_2+11x_3=7$.