So that we don’t get stumped

by the Arbeglas in our life and especially when we don’t

have talking birds to help us, we should be able to

identify when things get a little bit weird with

our systems of equations. When we have scenarios that have

an infinite number of solutions or that have no

solutions at all. And just as a little bit of a

review of what could happen, these are the– think

about the three scenarios. You have the first

scenario which is kind of where we started off,

where you have two systems that just intersect in one place. And then you have

essentially one solution. So if you were to

graphically represent it you have one solution right

over there, one solution. And this means that the two

constraints are consistent and the two constraints are

independent of each other. They’re not the exact same line,

consistent and independent. Then you have the other scenario

where they’re consistent, they intersect,but they’re

essentially the same line. They intersect everywhere. So this is one of the

constraints for one of the equations,

and the other one if you look at it,

if you graph it, it is actually the

exact same one. So here you have an infinite

number of solutions. It’s consistent, you

do have solutions here, but they’re dependent equations. It’s a dependent system. And then the last

scenario, and this is when you’re dealing

in two dimensions, the last scenario is where

your two constraints just don’t intersect with each other. One might look like

this, and then the other might look like this. They have the exact

same slope but they have different intercepts. So this there is no solution,

they never intersect. And we call this an

inconsistent system. And if you wanted to think

about what would happen just think about what’s

going on here. Here you have different slopes. And if you think about

it, two different lines with different

slopes are definitely going to intersect

in exactly one place. Here they have the same

slope and same y-intercept, so you have an infinite

number of solutions. Here you have the same slope

but different y-intercepts, and you get no solutions. So the times when

you’re solving systems where things are going

to get a little bit weird are when you have

the same slope. And if you think about it,

what defines the slope, and I encourage you to test this

out with different equations, is when you have– if you

have your x’s and y’s, or you have your a’s and b’s or

you have your variables on the same side of

an equation, where they have the same ratio

with respect to each other. So with that, kind of

keeping that in mind, let’s see if we can think

about what types of solutions we might find. So let’s take this down. So they say determine

how many solutions exist for the

system of equations. So you have 10x minus

2y is equal to 4, and 10x minus 2y is equal to 16. So just based on what we

just talked about the x’s and the y’s are on the

same side of the equation and the ratio is

10 to negative 2. Same ratio. So something strange is

going to happen here. But when we have the

same kind of combination of x’s and y’s in the first one

we get 4, and on the second one we get 16. So that seems a

little bit bizarre. Another way to

think about it, we have the same number of

x’s, the same number of y’s but we got a different number

on the right hand side. So if you were to simplify

this, and we could even look at the hints

to see what it says, you’ll see that

you’re going to end up with the same slope but

different y-intercepts. So we convert both the slope

intercept form right over here and you see one, the blue one

is y is equal to 5x minus 2, and the green one is y

is equal to 5x minus 8. Same slope, same ratio

between the x’s and the Y’s, but you have different

values right over here. You have different y-intercepts. So here you have no solutions. That is this scenario right over

here if you were to graph it. So no solutions,

check our answer. Let’s go to the next question. So let’s look at this

one right over here. So we have negative 5 times

x and negative 1 times y. We have 4 times x and 1 times y. So it looks like the

ratio if then we’re looking at the x’s and y’s

always on the left hand side right over here, it looks like

the ratios of x’s and y’s are different. You have essentially

5 x’s for every one y, or you could say negative 5

x’s for every negative 1 y, and here you have 4

x’s for every 1 y. So this is fundamentally

a different ratio. So right off the bat

you could say well these are going to intersect

in exactly one place. If you were to put this

into slope intercept form, you will see that they

have different slopes. So you could say

this has one solution and you can check your answer. And you could look at the

solution just to verify. And I encourage you to do this. So you see the blue one if you

put in the slope intercept form negative 5x plus 10 and

you take the green one into slope intercept

form negative 4x minus 8. So different slopes,

they’re definitely going to intersect

in exactly one place. You’re going to

have one solution. Let’s try another one. So here we have 2x plus

y is equal to negative 3. And this is pretty

clear, you have 2x plus y is equal

to negative 3. These are the exact

same equations. So it’s consistent information,

there’s definitely solutions. But there’s an infinite number

of solutions right over here. This is a dependent system. So there are infinite

number of solutions here and we can check our answer. Let’s do one more because that

was a little bit too easy. OK so this is interesting

right over here, we have it in different forms. 2x plus y is equal

to negative 4, y is equal to

negative 2x minus 4. So let’s take this

first blue equation and put it into

slope intercept form. If we did that you would

get, if you just subtract 2x from both sides you get y is

equal to negative 2x minus 4, which is the exact same thing as

this equation right over here. So once again they’re

the exact same equation. You have an infinite

number of solutions. Check our answer, and you can

look at the solution right here. You convert the blue

one into slope intercept and you get the exact

same thing as what you saw in the green one.

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Thanks for these videos, they really help out. Truly amazing ðŸ™‚ Keep up the awesome tutorials!

awesome video!

Hey I WANT A TALKING BIRD (PREFERABLY A SAGE TALKING BIRD).

Feel deprived :((((

If they have different slopes and no solutions that would mean they are not parallel. Can this happen? What if both slopes are somehow diverging like a fractal?

No, even though it is a minimal difference, eventually they will cross. The values may be in the positive millions or larger but they will intercept. For example, with these y = 2x + 1000 and y = 2.001 – 10000, at some point they will intersect.

If the slopes are divergent, they aren't lines. If two otherwise identical curves have different slopes but don't intersect, they don't exist in the same plane. These examples are dealing with linear equations in two dimensions, where neither of those situations is possible, but a parabola for example will have two different points of intersection with most continuous curves, and of course in three or more dimensions there are noncoplanar nonparallels that don't intersect.

Right from wikipedia, "Two lines in a plane that do not intersect or touch at a point are called parallel lines."

y = 1/x

y = -x

The top one isn't a line, but this is linear algebra right? Ok, forget about that one. I cheated =)

i disliked the video by accident

My head has exploded.

i see no dislikes. did you rectify the problem?

I see 29 likes, 0 dislikes.

Please invite thenewboston in your team.

In these last few videos your cursor hasn't been showing on the screen, so when you point to something on screen we cannot see what you are pointing to which can be confusing.

I just noticed you do all the videos for the channel, how do you know all of this information?!

does anyone know what kind of hardware khanacademy uses to write and draw digitally? I've often been in the situation where I had to explain something to someone via internet and a tool like that would be very handy..

I need your help.really bad im about to take my asvab in 3 days and I cant find any help on arithmatic reasoning any were can you help

i love the concept. now apply it to daily life so we can make sense of it. dennis hayslip

Now I know what to look for if I can't solve a system of equations.