Question

Hello, Can you please help me solve this probelm: Solve the system of equations. Please show...

Hello,

Can you please help me solve this probelm:

Solve the system of equations. Please show all work. ALSO PLEASE USE GAUSSIAN ELIMINATION METHOD TO SOLVE!!!

x+y+z=0, 2x+5y+2z=-9, -x+8y-3z=-25

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Answer #1

x+y+z=0

2x+5y+2z=-9

-x+8y-3z=-25

.

system Ax=b is

\begin{pmatrix}1&1&1\\ 2&5&2\\ -1&8&-3\end{pmatrix}\begin{pmatrix}x\\ y\\ z\end{pmatrix}=\begin{pmatrix}0\\ -9\\ -25\end{pmatrix}

augmented matrix is

1 1 1 0
2 5 2 -9
-1 8 -3 -25

by gauss elimination method convert into row echelon form

Add (-2 * row1) to row2

1 1 1 0
0 3 0 -9
-1 8 -3 -25


Add (1 * row1) to row3

1 1 1 0
0 3 0 -9
0 9 -2 -25


Divide row2 by 3

1 1 1 0
0 1 0 -3
0 9 -2 -25


Add (-9 * row2) to row3

1 1 1 0
0 1 0 -3
0 0 -2 2


Divide row3 by -2

1 1 1 0
0 1 0 -3
0 0 1 -1

reduced system is

\begin{pmatrix}1&1&1\\ 0&1&0\\ 0&0&1\end{pmatrix}\begin{pmatrix}x\\ y\\ z\end{pmatrix}=\begin{pmatrix}0\\ -3\\ -1\end{pmatrix}

.

from last row

{\color{Red} z=-1}

.

from second row

{\color{Red} y=-3}

.

from first row

x+y+z=0

x-3-1=0

x-4=0

{\color{Red} x=4}

.

.

solution is

{\color{Red} x=4}

{\color{Red} y=-3}

{\color{Red} z=-1}

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