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Course: 8th grade (Eureka Math/EngageNY) > Unit 4

Lesson 4: Topic D: Systems of linear equations and their solutions

Solutions to systems of equations: consistent vs. inconsistent

A consistent system of equations has at least one solution, and an inconsistent system has no solution. Watch an example of analyzing a system to see if it's consistent or inconsistent. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • leaf green style avatar for user Hello
    Does anyone know why they call it "consistent" or "inconsistent", not some other word?
    (49 votes)
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    • female robot grace style avatar for user J Helston
      "Inconsistent" is because it is not possible for both equations to hold simultaneously. They contradict each other in the sense that if one holds, the other must fail. Thus their graphs never intersect and there is no solution to their system.

      "Consistent" is then the opposite. There does exist solution(s) to the system.
      (89 votes)
  • mr pants teal style avatar for user Adithya Seshan
    when you graph a system of equation can you have 2 solutions?
    (11 votes)
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    • purple pi purple style avatar for user doctorfoxphd
      Yes, if the system includes other degrees (exponents) of the variables, but if you are talking about a system of linear equations, the lines can either cross, run parallel or coincide because linear equations represent lines.
      If you are graphing a system with a quadratic and a linear equation, these will cross at either two points, one point or zero points.

      If you have a quadratic like y = x² + 2x -3 and a linear equation like y = -x + 1 , this example intersects at two points, (-4,5) and (1, 0), so this system does have two solutions.

      If you have a quadratic like y = x² - 2x +1 and a linear equation like y = 2x - 3, this example intersects at one point, x = 2. y = 1 so the point (2,1) is the only solution to this system of equations.

      If you have a quadratic like y = x² - 2x + 1 and a linear equation like y = (1/5)x - 2
      these never cross and so there will be no solution for this system of equations.
      A straight line and a quadratic will not coincide, because a quadratic equation represents a parabola--a very un-linear curve! They cannot match at every point.
      (19 votes)
  • male robot donald style avatar for user Isaiah Greene
    I really don't get how a consistent system can be overlapped
    (6 votes)
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    • female robot grace style avatar for user C C
      Hi Isaiah,
      When you say "overlapped" do you mean the lines crossing? That's essentially the definition of a consistent system - that there is a solution, which is that point where the lines cross.

      Or when you say "overlapped", maybe you mean that the two lines are the same and they overlap each other from start to finish? That is considered a consistent system too. When your two equations graph to the same line, the solution is all the points on the line, a solution set, rather than just one point. The video "Infinite solutions to systems" has an example of that situation.
      (14 votes)
  • blobby green style avatar for user JRBarhydt
    Unless it is the same line, couldn't you just look at the slopes?
    If the slope is the same, then it is inconsistent. (unless it's the same line)
    If the slope is different, then they have to intersect.
    It seems like you could answer this question almost immediately in your head.
    (5 votes)
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  • blobby green style avatar for user Martin Erlić
    What do I do if there are 3 variables? Is there a way to figure this out by intuition, other than, of course, generating these vectors in my mind!?
    (3 votes)
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  • mr pants teal style avatar for user Harsh Madaik
    So a consistent line either has one solution or infinite solutions, right?
    (3 votes)
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  • blobby green style avatar for user mirahlinajohannes11
    x-y=2 is it inconsistent?
    (2 votes)
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  • piceratops ultimate style avatar for user Leonardo Padro
    Technically speaking these equations are being mapped on a 2-dimensinal scale right? Is there such a thing as mapping a 3-dimensional equation, or even a 4-dimensional one? Also I would like to know how those would be used and why.
    (2 votes)
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    • mr pink green style avatar for user David Severin
      Yes these are based in planar geometry. So imagine a square room with 4 walls, a floor, and a ceiling. You see all sorts of lines that could be parallel (if on the same plane which could include planes you do not see like the wall and ceiling on one side of the room and the wall and floor on the opposite side of the room). Similarly, there are a lot of intersecting lines at the corner of the room. Any two lines would still be on a single plane.
      The difference in 3d would be that each corner is the intersection of 3 lines which are not on the same plane and there are some lines that are skew to each other, not on the same plane, but also not parallel like the wall and ceiling of one wall and an adjacent wall and floor. So if you have two rooms that are aligned, you could end up with the same like with one long wall divided by an internal wall. I do not know about 4d. They are used in architecture, game design, and anything that needs 3 dimensions.
      (5 votes)
  • spunky sam blue style avatar for user Divyav22
    But what if we have a system of equations of a plane (ie. 3 variables)? Then how do we determine whether the planes are consistent or inconsistent?
    (3 votes)
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  • female robot ada style avatar for user Christy Han
    But also, can't you solve for it algebraically too? There are the ways "elimination" and "substitution" to solve for a system of equations.
    (2 votes)
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    • hopper jumping style avatar for user Michael Laman
      Yes. Good question.

      Sal explains that a solution is not needed (~ ) since we're only asked whether the system is consistent or inconsistent. I am glad you are thinking about also solving it and how. Yes, both elimination and substitution would work. I would favor substituting with the value of x from the first equation but your mileage may vary :-).

      Keep thinking beyond like that!
      (3 votes)

Video transcript

Is the system of linear equations below consistent or inconsistent? And they give us x plus 2y is equal to 13 and 3x minus y is equal to negative 11. So to answer this question, we need to know what it means to be consistent or inconsistent. So a consistent system of equations. has at least one solution. And an inconsistent system of equations, as you can imagine, has no solutions. So if we think about it graphically, what would the graph of a consistent system look like? Let me just draw a really rough graph. So that's my x-axis, and that is my y-axis. So if I have just two different lines that intersect, that would be consistent. So that's one line, and then that's another line. They clearly have that one solution where they both intersect, so that would be a consistent system. Another consistent system would be if they're the same line, because then they would intersect at a ton of points, actually at an infinite number of points. So let's say one of the lines looks like that. And then the other line is actually the exact same line. So it's exactly right on top of it. So those two intersect at every point along those lines, so that also would be consistent. An inconsistent system would have no solutions. So let me again draw my axes. Let me once again draw my axes. It will have no solutions. And so the only way that you're going to have two lines in two dimensions have no solutions is if they don't intersect, or if they are parallel. So one line could look like this. And then the other line would have the same slope, but it would be shifted over. It would have a different y-intercept, so it would look like this. So that's what an inconsistent system would look like. You have parallel lines. This right here is inconsistent. So what we could do is just do a rough graph of both of these lines and see if they intersect. Another way to do it is, you could look at the slope. And if they have the same slope and different y-intercepts, then you'd also have an inconsistent system. But let's just graph them. So let me draw my x-axis and let me draw my y-axis. So this is x and then this is y. And then there's a couple of ways we could do it. The easiest way is really just find two points on each of these that satisfy each of these equations, and that's enough to define a line. So for this first one, let's just make a little table of x's and y's. When x is 0, you have 2y is equal to 13, or y is equal to 13/2, which is the same thing as 6 and 1/2. So when x is 0, y is 6 and 1/2. I'll just put it right over here. So this is 0 comma 13/2. And then let's just see what happens when y is 0. When y is 0, then 2 times y is 0. You have x equaling 13. x equals 13. So we have the point 13 comma 0. So this is 0, 6 and 1/2, so 13 comma 0 would be right about there. We're just trying to approximate-- 13 comma 0. And so this line right up here, this equation can be represented by this line. Let me try my best to draw it. It would look something like that. Now let's worry about this one. Let's worry about that one. So once again, let's make a little table, x's and y's. I'm really just looking for two points on this graph. So when x is equal to 0, 3 times 0 is just 0. So you get negative y is equal to negative 11, or you get y is equal to 11. So you have the point 0, 11, so that's maybe right over there. 0 comma 11 is on that line. And then when y is 0, you have 3x minus 0 is equal to negative 11, or 3x is equal to negative 11. Or if you divide both sides by 3, you get x is equal to negative 11/3. And this is the exact same thing as negative 3 and 2/3. So when y is 0, you have x being negative 3 and 2/3. So maybe this is about 6, so negative 3 and 2/3 would be right about here. So this is the point negative 11/3 comma 0. And so the second equation will look like something like this. Will look something like that. Now clearly-- and I might have not been completely precise when I did this hand-drawn graph-- clearly these two guys intersect. They intersect right over here. And to answer their question, you don't even have to find the point that they intersect at. We just have to see, very clearly, that these two lines intersect. So this is a consistent system of equations. It has one solution. You just have to have at least one in order to be consistent. So once again, consistent system of equations.