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8th grade (Eureka Math/EngageNY)

Unit 4: Lesson 4

Topic D: Systems of linear equations and their solutions

Solutions to systems of equations: dependent vs. independent

A dependent system of equations has infinite solutions, and an independent system has a single solution. Watch an example of analyzing a system to see if it's dependent or independent. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • starky ultimate style avatar for user Tiago Dias
    I understand that dependent systems have an infinite amoun of solutions and independent ones only have one solution, but why are they called that way? What is the logic behind these different classifications?
    (19 votes)
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    • starky ultimate style avatar for user Stefan
      According to Wikipedia (https://en.wikipedia.org/wiki/System_of_linear_equations#Independence), "independent" means that none of equations in a system of equations can be derived from each other.

      Say you have these two equations:
      3x + 2y = 6
      6x + 4y = 12

      then these equations would be dependent, since we can derive the second one from the first, or vice versa (by either multiplying the first one by a factor of two, or dividing the second one by a factor of two). Although no explanation is given for the choice of words "dependent" and "independent", maybe you can somehow relate these terms to this description.
      (2 votes)
  • blobby green style avatar for user Thomas Jones
    I may be jumping ahead a bit here, but what if we deal with curves that intersect twice?
    (9 votes)
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    • leafers ultimate style avatar for user Sirgargamel24
      You are jumping ahead, but that's a good thing, not a bad thing.

      These curves are like parabolas (a kind of U-shape on a graph), a type of conic section. When a line goes through a parabola in such a way that it intersects with the parabola twice, there are two different answers that the system has.
      (13 votes)
  • mr pants teal style avatar for user Wiebke Janßen
    Is the inconsistent graph independent or dependent?
    (6 votes)
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  • female robot ada style avatar for user Samantha
    How do you when to use substitution or elimination?
    (6 votes)
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    • blobby green style avatar for user Justin B
      well, when you solve for "Y" with y=mx +b to determine whether or not your lines are running parallel to each other, and you determine that they ARE NOT and that there IS A SOLUTION or SOLUTIONS, then it is easy to SUBSTITUTE because you have already solved for "y" with y=mx+b.

      You can use ELIMINATION when you ALREADY KNOW that your lines have a solution. For example, if you see a picture.

      If you find y=mx+b and determine that the lines ARE running parallel to each other, then there is no need to try and solve the equations, THERE IS NO SOLUTION.

      So, either one works. Knowing both Substitution and Elimination helps you not to try and solve your equations only one way. Thereby helping you work less.

      I hope this was helpful!
      (4 votes)
  • aqualine ultimate style avatar for user Sachait Arun
    what happens in the graph if the two lines are not parallel but one is slightly tilted. Is that an inconsistent or do you just have to increase your graph to mark the intercept
    (3 votes)
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  • leaf green style avatar for user Jestine Agodilos
    how can you determine the solution directly without using any graph ??
    (its our lesson in math now)
    (3 votes)
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    • mr pants teal style avatar for user Tim
      It all becomes clear if they are in the same form. It doesn't matter which, but let's get them into y = mx + c form.

      Eq1) 4x + 2y = 16
      Eq2) y = -2x + 8

      Well, Eq2 is already in y = mx + c form. Let's convert Eq1:
      4x + 2y = 16
      2y = -4x + 16
      y = -2x + 8

      Aha! This looks suspicious! We have THE SAME EQUATION TWICE! So this system is dependant and consistent. But that's great for this example, can we make a general rule? Sure can!

      If the coefficients (the 'm's) are different, that means the slopes are different. That means that the two lines are not parallel, and so they must meet eventually. It might be at a huge value of x or y, but they will meet! So:

      Rule 1: If the slopes (the 'm's) are different, the system is independent (and therefore also consistent)

      If the slopes are the same, the lines must either be on top of each other, or parallel. If they are on top of each other, the equations will be the same, so they will also have the same intercept (the 'c'). That means:

      Rule 2: If the slopes (the 'm's) are the same, and the intercepts (the 'c's) are the same also, the system is dependent.

      If the equations are parallel but not the same they must be paralle, but not on top of each other. Therefore:

      Rule 3: If the slopes are the same, but the intercepts aren't (the 'c's), the system is inconsistent.

      So, step 1: convert to y = mx + c form, step 2: apply the above three rules.

      Hope that helps :)
      (4 votes)
  • male robot hal style avatar for user macy hudgins
    Why did Sal not substitute in the y equation?
    (3 votes)
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  • spunky sam green style avatar for user Zion J
    At around , if Consistent solution #1 is independent, Consistent solution #2 is dependent, then what is an INconistent solution?
    (3 votes)
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  • blobby green style avatar for user Stacy Payne
    If the graph of a system of linear equations shows three lines such that each pair of lines intersects at a point different from the intersection point for any other pair, how many solutions does the system have?
    (3 votes)
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    • leaf green style avatar for user Jacob Mikkelsen
      It means there are no solutions for the system. A solution would be a point where all three lines intersect. The points of intersection between two lines are valid solutions for that part of the system (you could substitute it into these equations and it would be valid), but not the overall system.
      (3 votes)
  • leaf green style avatar for user jasonsanctis
    would the parallel lines also be independent?
    As they do their own things..
    Is there no classification as independent or dependent for Inconsistent lines?
    (3 votes)
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    • mr pants teal style avatar for user IanS
      A system of parallel lines can be inconsistent or consistent dependent.

      If the lines in the system have the same slope but different intercepts then they are just inconsistent.

      Though if they have the same slope and intercepts (in other words, they are the same line) then they are consistent dependent.
      (0 votes)

Video transcript

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.