Determining if the system is consistent

[cookiemnstr510510]

Hello all,
I have the "correct" answer from Chegg. However, I am not satisfied that I really understand.
Heres the problem:

Determine if the system is consistent. Do not completely solve the system.

2X₁-4X₄=-10
3X₂+3X₃=0
X₃+4X₄=-1
-3X₁+2X₂+3X₃+X₄=5

Here is my attempt:
I first write the system in the augmented matrix form, then reduce it to the "gaussian elimination" form (I think that's what my teacher called it).

$\begin{vmatrix} 2 & 0 & 0 & -4 & -10 \\ 0 & 3 & 3 & 0 & 0\\ 0 & 0 & 1 & 4 & -1\\ -3 & 2 & 3 & 1 & 5 \end{vmatrix}$
Now I multiply Row 1 by 1/2 ((R1)(1/2))
$\begin{vmatrix} 1 & 0 & 0 & -2 & -5 \\ 0 & 3 & 3 & 0 & 0\\ 0 & 0 & 1 & 4 & -1\\ -3 & 2 & 3 & 1 & 5 \end{vmatrix}$
Now I Add 3Row 1's to Row 4 (R4+3R1)
$\begin{vmatrix} 1 & 0 & 0 & -2 & -5 \\ 0 & 3 & 3 & 0 & 0\\ 0 & 0 & 1 & 4 & -1\\ 0 & 2 & 3 & -5 & -10 \end{vmatrix}$
Now I multiply Row 2 by 1/3 ((R2)(1/3))
$\begin{vmatrix} 1 & 0 & 0 & -2 & -5 \\ 0 & 1 & 1 & 0 & 0\\ 0 & 0 & 1 & 4 & -1\\ 0 & 2 & 3 & -5 & -10 \end{vmatrix}$
Now I subtract 2Row1 from Row 4 (R4-2R1)
$\begin{vmatrix} 1 & 0 & 0 & -2 & -5 \\ 0 & 1 & 1 & 0 & 0\\ 0 & 0 & 1 & 4 & -1\\ 0 & 0 & 1 & -5 & -10 \end{vmatrix}$
Now I subtract Row 3 from Row 4 (R4-R3)
$\begin{vmatrix} 1 & 0 & 0 & -2 & -5 \\ 0 & 1 & 1 & 0 & 0\\ 0 & 0 & 1 & 4 & -1\\ 0 & 0 & 0 & -9 & -9 \end{vmatrix}$
Now I multiply Row 4 by (-1/9) ((R4)(-1/9))
$\begin{vmatrix} 1 & 0 & 0 & -2 & -5 \\ 0 & 1 & 1 & 0 & 0\\ 0 & 0 & 1 & 4 & -1\\ 0 & 0 & 0 & 1 & 1 \end{vmatrix}$

So at this point we see that in Row 4 that X₄=1 and this means it is consistent?!
I don't see how this means it is consistent.
Thanks!

 

[Abhishek11235]

Hint: What is relation of rank of matrix with the dimension of matrix? What should be kernel of this matrix?

 

[cookiemnstr510510]

Hint: What is relation of rank of matrix with the dimension of matrix? What should be kernel of this matrix?

Hmm, I don't know what any of that means :(

 

[Erland]

Determine if the system is consistent.

Du you know what "consistent" means in this context?

 

[cookiemnstr510510]

Du you know what "consistent" means in this context?

I know that for something to be inconsistent your final answer couldn't contain anything like:
0X₁+0X₂...+0X_n=a, where a does not equal zero.

From this definition of an inconsistent system, I can see that as long as you have a coefficient in front of X equal to a number, that makes it consistent.

Since the last entry is in the form of a consistent equation, this makes the entire system consistent?

 

[Erland]

Since the last entry is in the form of a consistent equation, this makes the entire system consistent?

Yes. That an equation system is consistent means that it has at least one solution. And you realize that $X_4=1$ in your system, and that you can find values of the other variables by back substitution, so the system is consistent.

 

[cookiemnstr510510]

Yes. That an equation system is consistent means that it has at least one solution. And you realize that $X_4=1$ in your system, and that you can find values of the other variables by back substitution, so the system is consistent.

awesome! THank you!