Worked example: equivalent systems of equations

- [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.