Question

Substitution and elimination trouble. How do I set myself up for this?

"For what value of 'k' a system below not have a unique solution?"

-1x+7y+2z=6
2x+1y+kz=-11
3x-6y-2z=-25

Answer choices:
-2

0

-8

5

Answer 1

Either my eyesight fails me or I don't see any k in the equations in question.

Answer 2

Sorry!! typo. The second equation it is kz, not is

Answer 3

...can't stop thinking about matrices D:

...but never mind that.

We can try solving this without minding the k (we can just keep it as part of the answers if need be). This will be systematic, but time-consuming.

Or we can try inspecting the equations, this doesn't take long at all, but there is no clearly-defined method of going at it, we can just inspect it and look for certain signs that point to the answer.

Which way do you want to go? :>

Answer 4

Inspecting equations sounds like my kind of method haha. Not to sound lazy, but i've been doing homework for two hours and my brain is slowly shriveling. So, yes, option two please.

Answer 5

All right, first things first. For a system of equations to not have a unique solution, it has to be what's called INCONSISTENT

To be such, one of the equations could be redundant (it doesn't tell us anything the two other equations don't already; A system like this is called DEPENDENT, and has infinite solutions)

or completely off course (it goes off in a different direction, so to speak, and doesn't result in any solution that fits all three equations, this is called INDEPENDENT)

Answer 6

Okay im following

Answer 7

That was fast. Okay, we need to FORCE this system to be inconsistent somehow. All we need to do is use the left side of the two other equations (the ones without k):
\[\Large -x +7y +2z\\\Large 3x-6y-2z\]

And use them to arrive at the one WITH the k.
\[\Large 2x + y + \color{red} kz\]

Answer 8

The simplest possibility is if the k-expression is a MULTIPLE of one of the two other expressions.

Is that possible?

Could we just take one of the two other expressions, multiply them by some number and arrive at the k-expression

Answer 9

It looks like the two above could make the k expression. Like if 3x and -x were combined it would make 2x, and -6y+7y would make y

Answer 10

oh wait you mean just one expression?

Answer 11

Very good.
It certainly DOES look like you can just add the two expressions

Answer 12

Oh okay good haha

Answer 13

And you'd get the k-expression.
But for that to be true, what does k HAVE to be?

Answer 14

0?

Answer 15

Bingo.

We're done here :P

Answer 16

Thank you so much!!!

Answer 17

Try to master doing this the long way, though, not all like-problems are so simple, sometimes you can get the k-expression using some wacky sum of weird multiples of the other two.

Just to be safe ^_^

Answer 18

Will do! Very grateful for your help:)