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How many solutions does a system of linear equations have if there are at least two?

Sal answers this question for you! Created by Sal Khan.

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  • winston baby style avatar for user Valeriano, Franco
    I found a problem in the skill “Understanding systems of equations word problems” that I believe has an incorrect solution according to khan academy.
    It asked the following: “You are solving a system of two linear equations in two variables, and you discover that there are no solutions to the system. Which of the following graphs could describe the system of equations?”
    It asked me to select all the answers that apply. I selected only the blue graph and submitted my answer. It was incorrect. I then added the green graph to my answer and submitted. It was correct despite the fact that the green graph had only two vertical lines whose equations were x= -7 and x= -4. This is inconsistent with the statement that I was solving a system of "two linear equations in *two* variables” because the lines on the green graph only used the x variable.
    Is the supposed solution actually erroneous or is there something about the problem I am failing to understand?
    (4 votes)
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    • leaf grey style avatar for user Dan  Stadden
      That's tricky. I see why you would feel cheated, because of the way the subject has been taught. The important thing to remember is that ANY two parallel lines will have no intersection point. And ANY line (vertical or otherwise) can be represented by a linear equation. And ANY linear equation IS an equation in two variables (whether both variables are explicitly written or not). For example, the equations you mention would be written as follows in standard form: x + 0y = -7, and x + 0y = -4
      Because the coefficients on the y terms are zero, that term is not written. But it's still there.
      Also what Arlic stated below about slope is not true, vertical lines have an undefined slope. Because there is no change in x for any change in y, the slope is: "change in y" divided by zero. Division by zero is undefined.
      (16 votes)
  • starky sapling style avatar for user Road Runner
    What is the purpose of the video?
    (0 votes)
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  • aqualine ultimate style avatar for user Charlotte
    Are there any more solutions beside "no solutions","infident solutions",and "one solution"? Just wondering!! (sorry for spelling errors)
    (4 votes)
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    • piceratops ultimate style avatar for user Rachel
      For linear equations, no, that's it. Because through any two points is a line. And then they'll have all the same points so infinite solutions. You can get more solutions with different types of functions but that's it for linear functions.
      (4 votes)
  • aqualine ultimate style avatar for user ashley.ronnell
    In using the elimination method, say it was clear that one factor could be easily eliminated. Could I also set both equations equal to zero, then set them equal to one another, and eliminate this factor that way?
    (2 votes)
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    • piceratops ultimate style avatar for user LSC
      Do you mean a set of equations like the following:
      3x -2y = 5
      5x-2y = 7
      For this, yes, you could rewrite as:
      3x - 2y - 5 = 0
      5x - 2y - 7 = 0
      and then set them equal:
      3x - 2y - 5 = 5x - 2y - 7
      Adding 2y to each side gives
      3x - 5 = 5x - 7
      Then we can solve for x by consolidating the x and constant terms:
      -2x = -2
      x = 1
      Then we can use one of the original equations to solve for y
      3 - 2y = 5
      -2y = 2
      y = -1
      I hope this is what you meant...
      (5 votes)
  • male robot hal style avatar for user Leo
    What if it weren't linear?
    (3 votes)
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  • blobby green style avatar for user jamesaeronlipan
    whickof the following is a system of lienear equations in two variables
    (2 votes)
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    • leaf orange style avatar for user Evan S
      there can be either no solution, exactly one solution, or an infinite number of solutions. If you are dealing with two lines then the lines will either never intersect, intersect at only one point, or be on top of each other. If the latter occurs, there are an infinite number of solutions. If they only intersect at one point, then the coordinates of that point is the solutions. If there is no solution, then the lines will never intersect.
      (1 vote)
  • leaf grey style avatar for user Elizabeth
    If I come up with zero as the denominator of an equation, would that system have an infinite number of solutions or no solution? (I truly wonder, because it seemed that whichever answer I put in those cases the other was deemed correct.)
    (1 vote)
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    • piceratops ultimate style avatar for user Just Keith
      Division by 0 is undefined. If you encounter division by 0, then whatever you were doing that resulted in division by 0 has no mathematical meaning. You generally cannot draw any conclusions if you have division by 0.

      With linear equations, there are only three possibilities:
      There are zero solutions.
      There is one solution.
      There are infinitely many solutions.
      Thus, anytime you know there is more than one solution, you instantly know there are infinitely many solutions.
      NOTE: This only applies to straight lines. If you have any other kind of function, the rules for how many solutions there can be are different.
      (3 votes)
  • mr pink red style avatar for user Muhammed Umair Jafri
    What is the difference between "Zero Solutions" and "Infinitely" many solutions? Like I know the answer is one of those two whenever I get 0=0 or 5=8. But how do you interpret the results. How can we say that something has infinitely many solutions? I don't get it.
    (1 vote)
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    • aqualine ultimate style avatar for user Jack Messina
      Think of it this way: when are those statements true? Is 5 ever equal to 8? Nope. No matter what, 5 does not equal 8, so in that situation, there are no solutions. What about 0 = 0? Well, by the reflexive property of equality, we know that this is always true. 0 always equals 0, so there are infinitely situations where this is true. Does that answer your question?
      (2 votes)
  • blobby blue style avatar for user ✨ Sofia Utama 💯
    What is a linear equation? What does the problem mean by 'You are solving a system of two linear equations in two variables?' What does two variables mean? x and y?
    (1 vote)
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    • stelly blue style avatar for user Kim Seidel
      A linear equation is any equation that can be written as: Ax+By=C, where A,B, and C are real numbers, and at least A or B can't equal 0. If they both happen to equal 0, you have no variables in your equation. Examples of linear equation:
      2x+3y=8
      x=6
      y=11

      Many linear equation have 2 variables, and the variables are usually X and Y which enables you to graph the linear equation in a coordinate plane. You can have linear equations that use other variables.
      (2 votes)
  • leafers tree style avatar for user later.caleb
    So when you are using the substitution method and you get "x" or "y" equals another number then that is one "point". When you get two numbers that aren't equal that is "no solution," and when you have a number that is equal to the number on the other side of the equation then that is "infinite solutions," correct?
    (1 vote)
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Video transcript

You are solving a system of two linear equations in two variables. You have found more than one solution that satisfies the system. Which of the following statements is true? So before even reading these statements, let's just think about what's going on. So let me draw my axes here. Let's draw my axes. So this is going to be my vertical axis. That could be one of the variables. And then this is my horizontal axis. That's one of the other variables. And maybe, for sake of convention, this could be x, and this could be y, but they're whatever our two variables are. So it's a system of two linear equations. So if we're graphing them, each of the linear equations in two variables can be represented by a line. Now, there's only three scenarios here. One scenario is where the lines don't intersect at all. So the only way that you're going to have two lines in two dimensions that don't intersect is if they have the same slope and they have different y-intercepts. So that's one scenario, but that's not the scenario that's being described here. They say, you have found more than one solution that satisfies the system. Here there are no solutions. So that's not the scenario that we're talking about. There's another scenario where they intersect in exactly one place. So they intersect in exactly one place. There's one point, one xy-coordinate right over there that satisfies both of these constraints, but this also is not the scenario they're talking about. They're telling us that you have found more than one solution that satisfies the system. So this isn't the scenario either. So the only other scenario that we can have-- we don't have parallel lines that don't intersect. We don't have lines that only intersect in one place. The only other scenario is that we're dealing with a situation where both linear equations are essentially the same constraint. They both are essentially representing the same xy-relationship. That's the only way that I can have two lines, and this only applies to linear relationship and lines. But the only way that two lines can intersect more than one place is if they intersect everywhere. So in this situation, we know that we must have an infinite number of solutions. So which of these choices say that? This one right here-- "there are infinitely many more solutions to the system"-- right over there.