Is the System of Equations Inconsistent Despite a Solved Example?

[SomeRandomGuy]

0a - 2b + 3c = 1
3a + 6b - 3c = -2
6a + 6b + 3c = 5

I got a = 3, b = 3/2, c = -4/3 but the book says that this system is inconsistent. It asks us to use Gauss-Jordan elimination (which I have used on the past 5 or so problems and got all the right answers). I know how to do these, I just don't see where my mistake is. I am probably looking over some very simplistic reason why it is inconsistent... Any help is appreciated.

 

[C0nfused]

The numbers you have found are not the solutions of this system , because they don't give correct results, if put in any of the equations of the system. And a solution of any system must give correct results for all the equations, so if you find a solution (a,b,c) this must be a solution for all the equations.
In your problem you can multiply the second equation with -2 and then add it with the third one(Gauss method). Your system ends up like this:
| 0a-2b+3c=1 | 0a-2b+3c=1
| -6a-12b+6c=4 <=>| -6a-12b+6c=4
| 6a+6b+3c=5 | -6b+9c=9

the last equation can be written like this: -2b+3c=3
So if you add the first one and the {[-2b+3c=3] multiplied with -1} you get 0=-2
This last one is actually this equation : 0a+0b+0c=-2. So the initial system has the same solutions as this system:
0a+0b+0c=-2
-6a-12b+6c=4
-6b+9c=9

obviously this system is inconsistent, so the initial one is also inconsistent as they are equivalent

 

[Integral]

For us to see your mistake you would have to show us your arithemtic. How many times have you repeated the calculations?

 

[dextercioby]

I don't know why you struggled so much,the determinant of the coefficients is zero,therefore,no unique solution.

Daniel.

 

[noppakhuns]

The detrminant of the coefficient matrix is equal to zero. Since the matrix is singular, the system is inconsistent.

 

[Hurkyl]

The detrminant of the coefficient matrix is equal to zero. Since the matrix is singular, the system is inconsistent.

Not always.

 

[SomeRandomGuy]

lol thanks guys but I don't know what a determinant is yet (that's chapter 2). I have tried it twice, and have gotten the same result both times...

EDIT: Thanks for your help guys, but on my 4th attempt, I found it to be inconsistent. I don't know what I did wrong the other way because I did it a different way this time. Anyway, thanks for your help.

 

[lokisapocalypse]

It's amazing how simple a problem can be the 4th time around.

If I had a dollar for every simple error like that I've made...