- 143

- 13

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.

2X

_{1}-4X

_{4}=-10

3X

_{2}+3X

_{3}=0

X

_{3}+4X

_{4}=-1

-3X

_{1}+2X

_{2}+3X

_{3}+X

_{4}=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 X

_{4}=1 and this means it is consistent?!

I don't see how this means it is consistent.

Thanks!