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Worked example: equivalent systems of equations

Sal analyzes a couple of systems of equations and determines whether they have the same solution as a third given system.

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  • leafers ultimate style avatar for user Valentin Gödel Escher Bach
    I’m not getting the explainations, why are we sure that we get the same solution when we add the two equations and replace the second line with that ?
    Why are we sure the new line is going to cross the first line at the exact same point ?
    (2 votes)
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    • leafers ultimate style avatar for user Alex
      We know the new line crosses the first at the same point because we know that point satisfies the new equation. The solution point is the point that satisfies both equations, that is, the x and y that make the two sides equal in both equations. So if you add things that are equal to things that are equal, the result must be equal, so the same solution must satisfy the new equation.

      If you want to think of it algebraically, consider the two equations:
      ax + by = c and dx + ey = f
      When you add the left hand sides, you get ax + by + dx + ey, (you could combine like terms here, but it's easier to think of them separate). Assuming the system is solvable, we know there is some x and y that solve the system, so let's see what that same x and y do in the new expression. Since that x and y satisfy both of the original equations, the first part will evaluate to c, and the second part will evaluate to f, so the whole expression will evaluate to c + f. This means the same x and y will satisfy the new equation ax + by + dx + ey = c + f therefore line represented by that equation must pass through that point.
      (36 votes)
  • aqualine ultimate style avatar for user Divija R
    In any linear equation, why does multiplying or dividing give the same line on the graph but adding or subtracting does not give the same line?
    For example,
    m+n=a+b
    If we multiply by 2,
    2(m + n )= 2(a +b)
    If we plot on graph, both of these equations give the same line.
    But if
    m+n=a+b
    Then, m+n -a =a+b-a
    But these two equations have different lines
    (6 votes)
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    • duskpin ultimate style avatar for user Victor
      For your first example, you aren't taking away or adding anything to the equation, therefore, they remain equal. However, for your second example, a+b have an a, and when you subtract, you are removing the a, therefore changing the equation and line.
      I hope this helped!
      (6 votes)
  • male robot hal style avatar for user Marioland
    At , Sal said that the solution of the system is not going to change as long you as multiply both sides of the equation by a scaler. Would dividing both sides of the equation give the same result?
    (5 votes)
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  • male robot donald style avatar for user Vinay Sharma
    It doesn't make any sense to me . Sal subtracts one equation from the other and still, it gives the same solution when I graphed it in my notebook. Why does that make sense?
    (4 votes)
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    • female robot grace style avatar for user loumast17
      Adding one equation to th other or subtracting both work.

      The way i think of it is pretty simple, I just hope it makes sense to you. Let's stat witht he two euations we start with in the video.

      x + 2y = -1
      -4x + 5y = 1

      The point is to find x and y that make both true. Well if both are true we can actually get more creative. keep in mind, we are saying both are true.

      Let's start with x + 2y = -1

      Now, we could add 1 to both sides. so it would look like x + 2y + 1 = -1 + 1. So it's just like normal algebra here. Well, the second equation says 1 = -4x + 5y, and since we know it has to be true we can instead write one (or both) of those 1s as -4x + 5y. This would make x + 2y + 1 = -1 + 1 become x + 2y + (-4x + 5y) = -1 + 1 and then if you simplify everything you get -3x + 7y = 0. And this is the same as addng the two equations together.

      Let me know if this doesn't make sense. Also, keep in mind you can combine this with multiplying an equation by a number. So x + 2y = -1 is also saying 2x + 4y = -2, which is 2 times x + 2y = -1.

      Again, let me know if this doesn't make sense.
      (5 votes)
  • aqualine ultimate style avatar for user N
    If equivalent equations are equations that have the same solution , does that mean that the equations in a system of equations are equivalent equations?

    Please reply.
    Thanks in advance :)
    (4 votes)
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    • male robot hal style avatar for user Sun
      The answer for that question is no! because they can also be not equivalent when the left sides match but the right sides do NOT match! See the next video for more info! The point of having equivalent equations is to make sure one of the students equation should have the left AND right sides match to the teachers equation! and then you can multiply the other equation because we are permitted! Good question N!
      (3 votes)
  • female robot grace style avatar for user Anna
    Why would adding or subtracting 1 equation from another to get a different equation lead to the same solution? Would it be through simplification or something?

    And would you get the same solution if you multiply or divide 1 equation by another to get your second equation in the system?
    (2 votes)
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    • male robot hal style avatar for user james
      In the teacher's system if we add x and -4x, we will get -3x in the Vivek's second equation. No matter what "solution value" you give to the x in teacher's sysem, if you add two equations together (x - 4x) you will ALWAYS get -3 times that value you gave to x. The same is true for y values.

      For instance: The solution to the teacher's equation system in this video is: (x= -7/13, y= -3/13)
      When you add two equations and give the value -7/13 to x, the x part will be:
      x-4x = -7/13+28/13 =21/13 which exactly equals -3x

      If the x solution of the teacher's system was 1 million, nothing would change:
      x-4x = 1 million - 4 million = -3 milion which exactly equals -3x

      Also DON'T FORGET that when we add x and y's on the left side we also add the right sides of two equations: -1+1 = 0 in order to maintain equality.

      And yes you can also multiply or divide as long as you do the same operation to the right side
      (5 votes)
  • male robot hal style avatar for user Sun
    At , what is a legitimate operation?
    (2 votes)
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  • male robot hal style avatar for user Sun
    Vivek's second equation isn't actually been multiplied by the teacher's second equation! so I wonder if Vivek's system of equation is NOT equivalent to the teacher's system of equation? why would it be equivalent?
    (3 votes)
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  • male robot hal style avatar for user Sun
    At the end of the video, Sal said that both of their systems of equations are equivalent to the teacher system. Wouldn't Camila's system of equations is not equivalent to the teacher's system?
    (2 votes)
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    • stelly blue style avatar for user Kim Seidel
      Camila's is equivalent.
      The 1st equation matches the 2nd equation in the original. And, Camila's 2nd equation is the original 1st equation multiplied by -8. We are allowed to multiply equations by any value as long as we do the entire equation. It is still equivalent to the original.

      Hope this helps.
      (2 votes)
  • purple pi purple style avatar for user Darko
    For anyone that is interested in how Vivek´s system of linear equation looks like compared to Teacher´s and wants to see for themselves that Vivek system is still true and that Sal is correct, the new equation is still part of the teachers system of linear equation: here is a link to a very useful website to construct graphs quickly. https://www.desmos.com/calculator/4i6iw8731a
    (2 votes)
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Video transcript

- [Voiceover] "Vivek and Camila's teacher gave them a system "of linear equations to solve. "They each took a few steps that lead "to the systems shown in the table below." So we have the teacher's original system, what Vivek got after doing some operations, what Camila got after doing some operations. Which of them obtained a system that is equivalent to the teacher's system? So the first question we should ask ourselves is what does it mean to even have an equivalent system? For the sake of this question, or for our purposes, an equivalent system is a system that has the same solution. So if there's some X-Y pair that satisfies the teacher's system that is the solution to the teacher's system. Well Vivek's system, we're gonna call it equivalent if it has the same solution. Similarly, if Camila's system has the same solution, then we're gonna call it equivalent to the teacher's system. So let's make some comparisons here. So first let's look at Vivek. So his first equation is actually unchanged from the teacher's equation, is unchanged from the teacher's equation, so any solution that meets both of these equations is for sure gonna meet this top equation because it's literally the same as the top equation of the teacher, so that works out. Also look at the second one. The second one is definitely a different equation over here. We can check that it's not just being multiplied by some number on both sides. To go from one to zero if you were multiplying, you would have to multiply one times zero and then in order to maintain the equality, you would have to do that on both sides. But zero times this left-hand side would have been zero, you would have gotten zero equals zero, so he didn't just scale both sides by some number, looks like he did another operation. He probably looks like he's adding or subtracting something to both sides, so let's see how he could have gotten this right over here. So he took -4x plus 5y is equal to one. And it looks like from that he was able to get - 3x plus 7y is equal to zero. So let's see what he had to do to do that. Let's see, he would have had to, to go from -4x to -3x, he would have had to add an X, so I could just write an X right over there. To go from 5y to 7y, he would have had to add 2y. So on the left-hand side, he would have to add X plus 2y. Notice we have an X plus 2y right over there. And on the right-hand side, he would have had to add or subtract a one, or add a negative one. Notice we see a negative one right over there. So what he essentially did is he added the left-hand sides of these two equations to get this new left-hand side right over here, and he added the right-hand sides to get this new right-hand side. And that is a legitimate operation. This new equation that you got, this new linear equation, it's going to represent a different line than this one right over here, but the resulting system is going to have the same solution. Why do we feel confident that the resulting system is going to have the same solution? Well for an X-Y pair that satisfies both of these equations, that's what a solution would be, for that X-Y pair, X plus 2y is equal to negative one. So for that solution, we're adding the same thing to both sides. We're saying: "Look, I'm gonna add X plus 2y "to the left-hand side. "Well if I don't wanna change the solution, "I have to add the same thing to the right-hand side." Well they're telling us for the solution to this equation, X plus 2y is equal to negative one, so negative one is the same thing as X plus 2y for that solution, so we're not gonna change the resulting solution of the system, so it's a completely legitimate operation what Vivek did is adding the left-hand sides and adding the right-hand sides to get this new second equation. That's not going to change the solution of the system. In fact, that's a technique we often use to eventually find the solution of a system. So now let's look at Camila, or Camila. So her first equation is actually the exact same equation as the teacher's second equation. Now let's see, her second equation, how does it relate possibly to the first equation? So just looking at it offhand, it looks like it might just be, it looks like she just multiplied both sides times a number. And it looks like that number, she clearly multiplied the right-hand side times negative eight. So times negative eight. Negative one times negative eight is positive eight. And it looks like she also multiplied the left-hand side by negative eight. Negative eight times X is -8X. Negative eight times 2y is -16y. So she just multiplied both sides by the same value which actually doesn't change the equation. This actually is going to be-- It changes it the way it looks, but it actually represents the same line. So this is definitely still an equivalent system. These are still the same constraints. You're going to have the same solution. Whenever you're dealing with systems, you're not going to change the solution of the system as long as you either multiply both sides of an equation by a scaler, or you are adding and subtracting the equations. When I say add or subtract the equations, you're adding the left-hand side to the left-hand side, adding the right-hand side to the right-hand side like we had here, or subtracting the one from the other on the left-hand side and if we subtract the bottom from the top on the left-hand and we subtract the bottom from the top on the right-hand, it's not going to change our solution. So both of them obtained a system that is equivalent, meaning that it has the same solution as the teacher's system.