Matrix Eliminations to find intersections of planes

[lunabee]

Urgent: Find the intersection pts of these 3 planes.

I tried once to show my work of the first 4/5 steps of each question but apparently it was to hard to follow because of the way I set it up. I'm having trouble getting the 1's in all the rows and the zeros in the 3rd column of the 2nd row n 2nd column of the third row.

#1
2x-6y+4z=11
1x-3y+4z=-7
8x+18y-2z=-1

#2
1x+2y+1z=12
2x-1y+1z=5
3x+1y-2z=1

 

[pka]

You do have some very urgent problems!
You have given us a mess.
There are no planes in what you posted!
Please try again.

 

[soroban]

Hello, lunabee!

The second one isn't too bad . . .

\(\displaystyle 2)\;\:\begin{array}{ccc}x\,+\,2y\,+\;z\:=\:12\\ 2x\,-\,y\,+\,z\:=\:5\\ 3x\,+\,y\,-\,2z\:=\:1\end{array}\)

We have: \(\displaystyle \:\begin{pmatrix}1 & 2 & 1 & | & 12 \\ 2 & -1 & 1 & | & 5 \\ 3 & 1 & -2 & | & 1\end{pmatrix}\)

\(\displaystyle \begin{array}{ccccc} : \\: \\ R2-2\cdot R1\\ : \\ R3 - 3\cdot R1\end{array}\:\begin{pmatrix}1 & 2 & 1 & | & 12 \\ 0 & -5 & -1 & | & -19\\ 0 & -5 & -5 & | & -35\end{pmatrix}\)

\(\displaystyle \begin{array}{ccccc} :\\: \\ (-1)R2 \\ : \\ R3\div(-5)\end{array}\;\begin{pmatrix}1 & 2 & 1 & | & 12 \\ 0 & 5 & 1 & | & 19 \\ 0 & 1 & 1 & | & 7\end{pmatrix}\)

\(\displaystyle \begin{array}{ccc}: \\: \\R2\leftrightarrow R3\end{array}\:\begin{pmatrix}0 & 2 & 1 & | & 12 \\ 0 & 1 & 1 & | & 7 \\ 0 & 5 & 1 & | & 19\end{pmatrix}\;\;\) . . . yes, we can switch rows, too!

\(\displaystyle \begin{array}{cccc}R1-2\cdot R2\\: \\: \\ R3-5\cdot R2\end{array}\:\begin{pmatrix}1 & 0 & -1 & | & -2 \\ 0 & 1 & 1 & | & 7 \\ 0 & 0 & -4 & | & -16\end{pmatrix}\)

\(\displaystyle \begin{array}{cccc}:\\ :\\ :\\ R3\div(-4)\end{array}\:\begin{pmatrix}1 & 0 & -1 & | & -2 \\ 0 & 1 & 1 & | & 7\\ 0 & 0 & 1 & | & 4\end{pmatrix}\)

\(\displaystyle \begin{array}{cccc}R1+R3 \\R2-R3\\:\\:\end{array}\:\begin{pmatrix}1 & 0 & 0 & | & 2 \\ 0 & 1 & 0 & | & 3\\ 0 & 0 & 1 & | & 4\end{pmatrix}\)


Therefore: \(\displaystyle \,x\,=\,2,\;y\,=\,3,\;z\,=\,4\)

 

[Vampire99]

soroban, what text type do you use to write that answer? It's very neat lol.
thx

 

[stapel]

soroban, what text type do you use to write that answer? It's very neat

To learn how to use LaTeX, please follow the links in the "Forum Help" pull-down menu at the very top of the page.

Thank you.

Eliz.