Is the system of

linear equations below dependent or independent? And they give us two

equations right here. And before I tackle

this specific problem, let’s just do a little

bit a review of what dependent or independent means. And actually, I’ll compare that

to consistent and inconsistent. So just to start

off with, if we’re dealing with systems of linear

equations in two dimensions, there’s only three possibilities

that the lines or the equations can have relative to each other. So let me draw the

three possibilities. So let me draw three

coordinate axes. So that’s my first

x-axis and y-axis. Let me draw another one. That is x and that is y. Let me draw one

more, because there’s only three possibilities

in two dimensions. x and y if we’re dealing

with linear equations. So you can have the situation

where the lines just intersect in one point. Let me do this. So you could have

one line like that and maybe the other line

does something like that and they intersect at one point. You could have the

situation where the two lines are parallel. So you could have a situation–

actually let me draw it over here– where you have one

line that goes like that and the other line has the

same slope but it’s shifted. It has a different y-intercept,

so maybe it looks like this. And you have no points

of intersection. And then you could

have the situation where they’re actually the same

line, so that both lines have the same slope and

the same y-intercept. So really they

are the same line. They intersect on an

infinite number of points. Every point on

either of those lines is also a point

on the other line. So just to give you a little

bit of the terminology here, and we learned this in the

last video, this type of system where they don’t intersect,

where you have no solutions, this is an inconsistent system. And by definition, or

I guess just taking the opposite of

inconsistent, both of these would be considered consistent. But then within consistent,

there’s obviously a difference. Here we only have one solution. These are two different lines

that intersect in one place. And here they’re essentially

the same exact line. And so we differentiate

between these two scenarios by calling this one

over here independent and this one over

here dependent. So independent– both lines

are doing their own thing. They’re not dependent

on each other. They’re not the same line. They will intersect

at one place. Dependent– they’re

the exact same line. Any point that satisfies one

line will satisfy the other. Any points that

satisfies one equation will satisfy the other. So with that said, let’s see if

this system of linear equations right here is dependent

or independent. So they’re kind of

having us assume that it’s going

to be consistent, that we’re going to

intersect in one place or going to intersect in an

infinite number of places. And the easiest way to

do this– we already have this second equation here. It’s already in

slope-intercept form. We know the slope is negative

2, the y-intercept is 8. Let’s put this first equation

up here in slope-intercept form and see if it has a different

slope or a different intercept. Or maybe it’s the same line. So we have 4x plus

2y is equal to 16. We can subtract 4x

from both sides. What we want to do is isolate

the y on the left hand side. So let’s subtract

4x from both sides. The left hand side– we

are just left with a 2y. And then the right hand side,

we have a negative 4x plus 16. I just wrote the negative

4 in front of the 16, just so that we have it in the

traditional slope-intercept form. And now we can divide both

sides of this equation by 2, so that we can isolate

the y on the left hand side. Divide both sides by 2. We are left with y is equal

to negative 4 divided by 2 is negative 2x plus

16 over 2 plus 8. So all I did is algebraically

manipulate this top equation up here. And when I did that, when

I solved essentially for y, I got this right

over here, which is the exact same thing

as the second equation. We have the exact same slope,

negative 2, negative 2, and we have the exact

same y-intercept, 8 and 8. If I were to graph these

equations– that’s my x-axis, and that is my y-axis– both

of them have a y-intercept at 8 and then have a

slope of negative 2. So they look

something– I’m just drawing an approximation of it–

but they would look something like that. So maybe this is the graph

of this equation right here, this first equation. And then the second equation

will be the exact same graph. It has the exact

same y-intercept and the exact same slope. So clearly these two

lines are dependent. They have an infinite

number of points that are common to both of them,

because they’re the same line.

Sal, I don't understand. If there are infinite solutions, then how can it be CONSISTENT?

@rinwhr The number of solutions for a dependent system of linear equations is infinite, but all equations have the exact same infinite set of solutions. That's pretty consistent 😉

But really we just learn the definitions of the technical terms explained above so we can talk about systems of equations without having to go through the rigmarole of explaining what exactly we mean every time. They aren't necessarily supposed to make sense to you on a philosophical level, they're just handy jargon.

thanks

Thanks. You guys really help

Consistent means there is at least one solution. And one infinite number of solution is way greater than one lol. (jk, but i think you got it!)

Si vous voulez la traduction française, cliquer sur le lien en dessous de la vidéo pour la voir directement sur leur site, puis appuyez sur Options.

I noticed that youtube khan videos don't purpose the actual already translated languages available for that video (currently there is chinese, arabic and French), go on their website if you are looking for these (link in the description) and hit option.

I don't get my teacher thxs

guys I think the line should across the x-axis on the negative side because the equations refer y=-2+8 … what do you think

parallel lines are not included in any of dependent and independent lines??? please tell me

Um I think this is algabra 2 but still really helpful