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8th grade (Eureka Math/EngageNY)

Unit 4: Lesson 4

Topic D: Systems of linear equations and their solutions

Number of solutions to a system of equations graphically

Sal determines how many solutions the following system of equations has by considering its graph: 10x-2y=4 and 10x-2y=16. Created by Sal Khan.

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  • leaf red style avatar for user Jack McClelland
    I'm sorry, I still don't get how Sal solved the problem around .
    They are "fundamentally different ratios"...what does that mean?
    (14 votes)
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    • mr pants teal style avatar for user Wrath Of Academy
      He's comparing the "5 to 1" and "4 to 1" ratios of y to x, and saying that they have different slopes. Therefore, the two lines must intersect somewhere at one point.

      If you've watched enough videos on here, you'll notice that Sal frequently (over)uses the word "fundamentally," to just mean "certainly" or "definitely." He didn't mean anything special by the use of the word "fundamentally" here.
      (13 votes)
  • aqualine ultimate style avatar for user Joel Alexander
    Possibly not the right place to ask this, but - at , what's Arbegla? Some manner of American cultural reference, I'm assuming.
    (7 votes)
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  • male robot hal style avatar for user mama976
    So two linear equations will ALWAYS intersect at one point if thier slope is different? I don't really understand what Sal said at . Please help. I will upvote you comment if you help me!
    (3 votes)
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    • piceratops ultimate style avatar for user Denis Orlov
      Correct! If two linear equations have different slopes, they will ALWAYS intersect. Even if their slopes are different by a very small margin and lines themselves are far apart - they will still intersect. These lines extend into infinity and so when the slopes are different, lines will eventually meet, maybe somewhere very high on the graph or somewhere very low but they WILL intersect.

      Linear Equations with the same slope are parallel lines and will NEVER intersect, no matter how far they reach into infinity. They will always run parallel to one another.

      Linear Equations with the same slope AND same y-intercept (x=0) is one line running on top of another line, into infinity. Any (x,y) point on one line will also satisfy equation for the other line - because both lines are identical - to infinity. Thus infinite number of solutions.


      TLDR: Unless lines are parallel (same slope) they will ALWAYS have one solution (intersect).
      (5 votes)
  • piceratops ultimate style avatar for user Kara
    How would one solve an equation with no y in it for example:
    x=3
    y=2x=7
    or
    x=3.14
    x= -3/2
    (1 vote)
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    • aqualine ultimate style avatar for user Robby Olivam
      If there is no y, or any second variable, then it would just appear as a straight line that crosses the x axis at whatever constant is in the equation. Therefore, two lines that only have an x and no y would either never intersect or they would completely overlap.
      (8 votes)
  • blobby green style avatar for user dreyera
    what and why is math so mathy
    (2 votes)
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  • starky tree style avatar for user Athens, Adah
    the heck are these intros
    (4 votes)
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  • duskpin ultimate style avatar for user Just Bob
    At , what if it is a curved line
    (3 votes)
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  • blobby blue style avatar for user Joanna Ni
    At , if the line are exactly the same, what is the point of drawing/graphing two of them?
    (3 votes)
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  • male robot hal style avatar for user xajaud6
    so what happens when two equations have the same y-intercept but different slopes?
    (2 votes)
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  • aqualine ultimate style avatar for user QUIDES
    how does having different ratios mean having different slopes. how can we say that.
    (2 votes)
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    • blobby green style avatar for user Trey Shaffer
      Remember "rise over run"? That is a ratio that represents the slope. So, if the ratio (or "rise over run") is different, then the equations have different slopes. Mathematicians LOVE to use a million words for the same thing (Slope, "rise over run", ratio, gradient, etc.). They are all practically the same depending on the context!
      (2 votes)

Video transcript

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.