Question

A system of equations is given as follows:
x+y+2z=1
kx+2y+4z=k
For what values of k does the system have an infinite number of solutions? Determine the solution to the system for this value of k.

Answer 1

Well, infinite solutions means you do elimination or substiution and you end up with 2 sides being perfectly equal. It means they'll be the same line essentially. So we want values of k that can either make everything cancel or make both sides equal after we do elimination. So it looks like, based on the y and z values, that our k line may just be a multiple of 2 of the first line. So if we let k = 2, we get:
x + y + 2z = 1
2x + 2y + 4z = 2
Now this looks perfect. By elimination method, multiply the top equation by -2 and eliminate everything :3

Answer 2

yes! I'm right(^_^)
thanks, just confirming if I understand this correctly..

Answer 3

Yeah, np :P Did we leave our vectors section? xD

Answer 4

nope, this is still vectors, planes and etc.

Answer 5

Ah, okay xD Yeah, Im still stuck there....because im going slow, haha. Almost on to the next section in the chapter -_-

Answer 6

good for you!☺, i'm waiting for you a while ago, to answer my questions:)

Answer 7

Well, whatever you got that needs answering :3

Answer 8

Determine the vector equation of the line that passes through A(-2, 3,6) and is parallel to the line of intersection of the planes \[\pi _{1}:2x-y+z=0\] and \[\pi _{2}: y+4z=0\]

Answer 9

Well, the line of intersection of theplanes requires a system of equations using the two plane equations themselves. From there we need tosolve for x, y, and z and put it in terms of t so we can have the line in parametricform.

Answer 10

brb, i need to do something..

Answer 11

kkz

Answer 12

, i'm back:) i'm doing another question and i'm stuck this is the question:
determine the solution of this system of equation:
2x-y+2z=2 and -x+2y+z=1

Answer 13

All it says is solution, it doesnt say line of intersection or anything?

Answer 14

nope..

Answer 15

Yeah, often times you find a line of intersection solution. Alright, lemme take a look then.

Answer 16

sure:)

Answer 17

Yeah, I can get ya a line of intersection in parametric form, but if thats what is needed, doubt it, lol.

Answer 18

that's fine.. let me see:)

Answer 19

looks like it is a very long solution XD

Answer 20

So this is kind of a way you would go about finding a line of intersection. The first thing we do is eliminate one of the 3 variables and then choose to solve for one of the remaining two variables. It is not necessary that we come up with a numerical solution, just need to solve for a variable. Once we solve for a variable, we substitute this into one of the two original plane equations and solve for one of the other variables. The second variable we solve for, say we get x = z/7, we let that z/7 value = t. Then we solve for all 3 variables in terms of t. You'll understand the process as we go, though.
2x - y + 2z = 2
-x + 2y + z = -1
So we can just eliminate one of the variables to start with. I'll just eliminate y by multiplying the top plane by 2
4x - 2y + 4z = 4
-x + 2y + z = -1
= 3x + 5z = 3
So now we can just solve for either variable. I'll just solve for x saying:
\[x=\frac{ 3-5z }{ 3 }\]
So this x value is substituted into one of the two plane equations. Since the bottom equation only has a negative x I can do that one:
\[-\frac{ 3-5z }{ 3 }+2y+z=-1\]
= - 3 + 5z + 6y + 3z = -3
8z + 6y = 0
\[z=\frac{ -3y }{ 4 }\]
Now we say that t = -3y/4
Therefore z = t
y = -4t/3
x= 1- 5t/3
Kinda messy ish, but thats kinda the method xDD

Answer 21

Oh WOW!=O
give me time to read this

Answer 22

Yeah, np xD

Answer 23

aah, I found my mistake, instead of multiplying first equation by 2, I multiply -2 to the second one:) thanks again..

Answer 24

i mean 2.. to eliminate x

Answer 25

Lol, easy to make mistakes for sure xD

Answer 26

i wish i could be like you:)

Answer 27

Lol, got many people on here way more advanced in math than I am, though, lol.

Answer 28

at least your very helpful to me,XD

Answer 29

I try xD Just got a lot to learn, lol.

Answer 30

lol, I peek to the next section, IT'S THREE PLANES!, it looks very hard O_0,
did you do three planes?

Answer 31

Nope. Ive been going too slow. Ive been doing, like, all the rpoblems in this one section because I need to get it down xD

Answer 32

same here:)

Answer 33

wait, i checked the answer, it's different..
x=5/4 s
y=s
z=1-3/4 s

Answer 34

They solved for different variables. Its the same thing, but they have y as the main instead of z like we had.

Answer 35

ok?, i'll check again

Answer 36

Can just solve for different variables and see what happens. Im not sure if theres a requirement or specific way to say solve for THIS ONE

Answer 37

i got \[x=\frac{ 5-5z }{ 3}\]?