I Determining if the system is consistent

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 thats 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!
 
Hint: What is relation of rank of matrix with the dimension of matrix? What should be kernel of this matrix?
 
Hint: What is relation of rank of matrix with the dimension of matrix? What should be kernel of this matrix?
Hmm, I dont know what any of that means :(
 
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...+0Xn=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

Science Advisor
735
135
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.
 
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!
 

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