Question

What is the solution to the system of equations shown below?
2x-y+3z=8
x-6y-z=0
-6x+3y-9z=24

Answer 1

Do you want the solution or the method on how to do it?

Answer 2

Let me just explain it real fast to you. Since these equations are made of the same variables you can add them. So when you subtract 2x(eq2) from eq(1) you get:
11y +5z= 8. Continuing this will get you the answers you need.

Answer 3

sorry I still need help

Answer 4

have you studied matrices and/or determinants?

Answer 5

no

Answer 6

ok.
multiply the first eq. with "3" and add it to the third eq.
what do you see?

Answer 7

wait why are we multiplying a 3?

Answer 8

you will realize a hard fact about this system :)

Answer 9

ok. so when multiplying a 3 to the first eq. I get
6x-3y+9z=24
-6x+3y-9z=24
wait so they cancel out? where do we go from there?

Answer 10

exactly.
this is what is called as an "under-determined" system. there are infinite solutions possible.

Answer 11

you have "3" unknown variables and essentially, only "2" unique relations.

Answer 12

where does the second equation come into play?

Answer 13

for a system of "n" variables, you will need ATLEAST "n" unique equations.

Answer 14

?

Answer 15

well, we'd have used it if there were "3" unique equations like before (the Kirchhoff's laws problem)

Answer 16

but since we have established that there is no "unique" solution, we cannot solve it.

Answer 17

why wouldn't it be no solution?

Answer 18

but you can go about this way...

\[\text{let}\qquad z=a\\
2x-y=8-3a\qquad\text{and}\qquad
x-6y=a\\
\;\\
-12x+6y=-48+18a\\
\oplus\\
x-6y=a\\
---------------\\
-11x=-48+9a\implies\boxed{x={48\over11}-{18\over11}a}
\]

Answer 19

you can try the rest of your life, if you've got that time, to try to come up with a unique solution but you wont . so, lets get our basics set up before we loose our minds :)

Answer 20

so its infinitely many, opposed to no solution because? sorry

Answer 21

because it is an "under-determined" system.
(just out of curiosity, double check the equations that you wrote here with the problem, are the exactly same?)

Answer 22

yes. They are the same

Answer 23

ok. then there you go... like I explained, there are "3" unknowns and only two unique equations. so, the system is under-determined and no unique solution exists.