Question

If the system of equation x-ky-z=0, kx-y-z=0, x+y-z=0 has a non-zero solution, the value of k are:
a)-1,2
b)1,2
c)0,1
d)-1,1

Answer 1

Start by Subtracting first and third equation...

Answer 2

By SUBTRACTING FIRST and THIRD equation you will get k..

And SUBTRACTING FIRST and SECOND you will get other value of k..

Answer 3

what r u getting

Answer 4

yes and then equate them..

Answer 5

when K=-1, the solution is (0,0,0) and when K=1, we don't have a solution. So, none of the options seem correct to me.

we have -1 in (a) and 1 in rest three options..

Answer 6

I think we have atleast k = -1

Answer 7

right ?

Answer 8

are you not getting the D option??

Answer 9

K=-1 gives a solution but the solution is (0,0,0)..
The qsn is asking for non-zero solution.

And @waterineyes , when k=1 we don't have a solution at all. so, how can d be correct?

Answer 10

k wait let me think about this

Answer 11

At K = -1 I think the two lines have Infinitely many solutions because they are overlapping parallel lines..

Answer 12

k = 1 that is not possible

Answer 13

My mistake... K=-1 doesn't give a solution either.

Answer 14

Yes..
You think right..

Answer 15

hmn !! so i will have to think more :)

Answer 16

I guess all options are incorrect!

Answer 17

I found a site it is saying that d is right !!!

Answer 18

No last option is right I think..

Answer 19

Yes D is right @mathslover
What is the problem in that..

Answer 20

umm.. how?

Answer 21

Yes i proved that also !!! in the above images

Answer 22

What is the problem you are facing??

Answer 23

k = -1
http://assets.openstudy.com/updates/attachments/4ff95de4e4b058f8b76394cb-mathslover-1341743036742-k1.jpg
k = 1 ,
http://assets.openstudy.com/updates/attachments/4ff95de4e4b058f8b76394cb-mathslover-1341743214585-y.jpg

Answer 24

so, when k=1 and k=-1 what are the solutions?

Answer 25

For k =1,

It is easy way to find @mathslover
\[x - kx -ky + y = 0\]
\[(x-y)(k-1) = 0\]

\[k = 1\]

Answer 26

yes i found that above !!!! just made it longer

Answer 27

did u get this now @ujjwal

Answer 28

yeah, but what are the solutions to the equations when k=1 and k=-1. Th eqns are supposed to have a solution, right?

Answer 29

this is your final solution that i got from internet

Answer 30

I hope that this will lead u towards right answer and also right solution !!! @waterineyes what do you think about that image that i got from google books !! do u think that it is right at least i do !

Answer 31

Yes it is..
On putting k = 1,
first two lines become parallel but overlapping so it can have infinite number of solutions that is my logic..

Answer 32

@ujjwal what is ur respond towards this answer now? any confusion now ?

Answer 33

Google book helps! so, we have infinite solutions in both cases, i.e. when k=1 and k=-1.. And that would be regarded as non zero solution.

Answer 34

yes !!! thanks to google !!!

Answer 35

Well, what does non- zero solution exactly mean? Does it mean there are more than zero solutions? or, does it mean (0,0,0) is not a solution?

Answer 36

http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/system/system.html

You can prefer this

Answer 37

what do you exactly mean by non zero solution? That link doesn't help much..

Answer 38

for example : x+1=0 , x=-1 this is non zero solution

Answer 39

and if x+1=1 , x = 0 then there is no --> non-zero solution got it now ?

Answer 40

got it now @ujjwal ?

Answer 41

Note: if we have 2 non zero solutions a and b then we will write them as (a,b)

Answer 42

x+1=0 and x=-1 have infinite solutions.. so does non-zero solution mean infinite solution?

Answer 43

Or, does it simply mean solution other than (0,0)?

Answer 44

see let me take another : x+y=0
if x = -1 , then y =1
x = 1 , then y = -1 and so on .. this has infinite solutions
x+1=0 has finite solution that is -1

Answer 45

Got it now ? @ujjwal

Answer 46

btw thanks for that medal !!! so does it mean that u got it ?

Answer 47

I know what infinite solution is.. But just say me one thing:

which is correct?
a) Non-zero solution means that the solution to the equations (in x and y), whether it be finite or infinite, it should not be (0,0)
b)Non-zero solution means there are infinite solutions to the equations (in terms of x and y)
c)neither a nor b

Answer 48

i think a is correct

Answer 49

But in above set of equations, in both cases when k=1 and k=-1, (0,0,0) satisfies all the equations...

Answer 50

I somehow get it.. finally!! thanks @mathslover and @waterineyes ..

Answer 51

no problem and gr8 to know that