Do the three lines 2x1 + 4x2 + 4x3 = 4, x2 - 2x3 = -2, and 2x1 + 3x2 = 0 have at least one common point of intersection? It's linear algebra, and has to do with determining the consistency of systems. Like a matrices problem, the 1 2 and 3 represent rows, not exponents.

- anonymous

- Stacey Warren - Expert brainly.com

Hey! We 've verified this expert answer for you, click below to unlock the details :)

- chestercat

I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!

- rvc

do you know matrices?

- tkhunny

Have you considered a determinant?

- anonymous

Hi rvc, I know a little about matrices since we just started going over them, but this one has thrown me for a loop.

Looking for something else?

Not the answer you are looking for? Search for more explanations.

## More answers

- rvc

equations are consistent if the determinant equals to zero

- rvc

hello :)

- anonymous

Okay, so how do I make that happen? I was sick and missed the lecture, so I'm a bit lost. I sorry for bothering you. -_-;

- rvc

do you remember how to find determinant?

- anonymous

No, not really. I'm sorry.

- anonymous

I missed it. How do I find the determinant?

- rvc

|dw:1441072730208:dw|

- anonymous

Hm... So how does that translate into my word problem?

- anonymous

if we break it down into a matrices
2 4 4 4
0 1 2 -2
2 3 0 2

- anonymous

But I'm not quite sure I understand how the other numbers within the matrices are broken up to be multiplied(?) by either a11 a12 or a13

- rvc

atm i need to go.
i have my exam in the next hour.
all the best and sorry to leave you in between this problem.
i m sure @mathmate will help you. :)

- anonymous

@tkhunny

- dan815

hey have you figured out how to take a determinant yet

- anonymous

Nope, I have not. Do you have any time to help me figure out how to do it?

- dan815

yeah

- anonymous

Wow! That'd be great. I'm trying to work through the problems in the book for the lecture I missed but I'm kind of lost.

- dan815

okay basically to start off, you have an equation which is a linear function of 3 variables

- dan815

for example
1x+2y+3z=5 lets say this equation instead of x1,x2,x3

- anonymous

Alright

- dan815

have you seen equations like this before its the qeuation of a plane in 3d space

- anonymous

Yes, a long time ago. It's been a while since I've had to work through anything like this though.

- dan815

okay i see well dont worry

- dan815

now how about a gradient have you heard about gradients or normal vectors to planes

- anonymous

Nope. I have not dealt with that. We're just starting out the school year with matrices.

- dan815

okay well then for now i cant fully explain to you why this matrix method is working, but lets just go through the determinant

- dan815

a non zero determinant means that all 3 of your systems are independant of each other meaning they are not parallel or one of the system is not a combination of the 2 remaining systems

- anonymous

That's okay! I appreciate your willingness to explain!

- anonymous

That makes sense.

- dan815

okay we dont really care about the constant term for now, again this has something do with the normal vectors how they wont depend on the constant

- dan815

so your matrix is the constants of all the coefficceints in order

- anonymous

Alright.
So we break it up into a matrix...
2 4 4 4
0 1 2 -2
2 3 0 2

- dan815

|dw:1441074412998:dw|

- dan815

we dont care about the constants in this case

- anonymous

Alright, so just the first three columns then.

- dan815

yeah the numbers infront of variables only

- anonymous

2 4 4
0 1 2
2 3 0

- dan815

now to take a determinant of a matrix

- dan815

first u learn to take the determinant of 2 by 2 matrix

- anonymous

How do you take the determinant.

- dan815

|dw:1441074586338:dw|

- dan815

this is the determinnat of 2 by 2 matrix

- dan815

difference* of the product of the diagonals in that order

- dan815

we will use this to take the determinant of 3by3 matrix

- anonymous

Okay. How would you combine a 3 by 3 though?
A B C
D E F
G H I... _> AE-BD, BF-CE, DH-EG, EI-FH?

- dan815

|dw:1441074679006:dw|

- dan815

do get the mini determinants from the big determinant what you do is cover this way

- dan815

|dw:1441074913573:dw|

- dan815

|dw:1441074951428:dw|

- dan815

https://www.youtube.com/watch?v=ROFcVgehEYA

- dan815

just watch that its a lot better

- dan815

since the determinant is non zero you can conclude there will be only 1 intersection as they are all independant

- dan815

because if u look at 3 planes

- anonymous

Wow... Thank you so much! That's amazing!
It makes a lot of sense, way more than I thought it would. This is a great addition to my notes!

- dan815

|dw:1441075204298:dw|

- dan815

now if u throw in another plane, that means u are really checking the intersection of a line and a plane

- dan815

so if that plane is independant that means it will be a line passing through a plane so just a point

- dan815

if the plane was not independant that means the line also belongs in that plane, which means the plane stated is actually redundant which is why ud get a 0 determinant, that wont really make sense yet but ya something to keep in mind for later

- anonymous

Okay!

- dan815

just know the formal definition of independance

- dan815

wait

- dan815

http://prntscr.com/8b85p2

- anonymous

Okay, what else would you like to explain. I'm not leaving until you're finished ^_^. I just "fanned" you and wrote what I hope will be a glowing testimonial of your efforts.

- dan815

basically this a bunch of vectors are said to be linearly independant, if there is no way to do add them or some factor of them up together to end up with 0

- anonymous

I will make sure to add the formal definition of independence in my notes as well, so that I remember it for the next lecture.

- dan815

for example
v1,v2 linearly independant if
av1+bv2=0 only happens when both a and b are 0 which is the trivial case, or basically taking nothing to get nothing

- anonymous

That makes sense.

- dan815

yep basically the definition in there is the same thing not stated for all sizes of vectors

- dan815

you can have v1,v2,v3...vn
and they all independant if
a1*v1+a2*v2+..._an*vn=0 the only solution is that a1,a2,a3...an are all 0

- dan815

if that is the only solution, meaning taking nothing to get nothing, everything else doesnt product 0 that means they are lienarly independant

- dan815

it is is easy to see somthing like this when u talk about orthogonal vectors which are vectors than are as indepednant as possible of each other

- dan815

for example if u take a vector in the x direction and another vector in the y direction, theres no way to add them to get 0,

- dan815

once u have the existance of a vector in x, how can you even add some vector in the y direction to make it go back to 0

- anonymous

Neat! Thank you!

- dan815

you're welcome

- dan815

good luck with this stuff

- anonymous

Thank you! If nothing else, this experience has taught me to never get sick ever again. That will be the day lecture will be over something I don't understand.

Looking for something else?

Not the answer you are looking for? Search for more explanations.