Worked example: equivalent systems of equations

the vacant camelus teacher gave them a system of linear equations to solve they H took a few steps that led to the systems shown in the table below so we have the teachers original system what vivec got after doing some operations what Kamel got after doing some operations which of them obtained a system that is equivalent to the teacher 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 there's some XY pair that satisfies the teachers system that is the solution to the teachers system well the VEX system we're going to call it equivalent if it has the same solution say similarly if Camilla's system has the same solution then we're going to call it equivalent to the teachers system so let's make some comparisons here so first let's look at vivec so the first his first equation is actually unchanged from the teachers equation is unchanged from the teachers equation so any any solution that meets this meets both of these equations is for sure going to meet this top equation because it's literally the same as the top equation of the teacher so that works out now let's look at the second one the second one is definitely this is a different this the second equation is definitely a different equation over here we can check that it's not just being multiplied by some number on both sides if we to go from 1 to 0 if you were multiplying you would have to multiply 1 times 0 and then if you want to in order to maintain the equality we'd have to do that on both sides but 0 times this left-hand side would have been 0 you would have gotten 0 equals 0 so he didn't just scale it scale both sides by some number looks like he did another operation he probably 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 negative 4x plus 5y is equal to 1 and it looks like from that he was able to get negative 3x plus 7y is equal to 0 so let's see what he might have what he had to do to do that let's see he would have had to to go from negative 4x to negative 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 - why so on the left-hand side he would have had to add X plus 2y notice we have an X plus 2i right over there and on the right-hand side he would have had to add or subtract a 1 or add a negative 1 notice we see a negative 1 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 the this new equation that you got this new 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 XY pair that satisfies that satisfies both of these equations that's what a solution would be for that XY pair x plus 2y is equal to negative 1 so for that solution we're adding the same thing to both sides we're saying look I'm going to add X plus 2y to the left-hand side well I need to add if I'm if I don't want to if I don't want to 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 1 so negative 1 is the same thing as X plus 2y for that solution so we're not going to change the resulting solution of the system so it's a completely legitimate operation what vivec 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 orcam Allah so her first equation is actually the exact same equation as the teachers as the teachers second equation now let's see the 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 actually just multiplied both sides times a number and it looks like that number she clearly multiplied the right-hand side times negative 8 so times negative 8 negative 1 times negative 8 is positive 8 looks like she also multiplied the left-hand side by negative 8 negative 8 times X is negative 8x negative 8 times 2i is negative 16 y 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 you're going to have these are still the same constraint 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 scalar or you are adding and subtracting the equations when I say add or subtract 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 thus it you know 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 both of them obtain the system that is equivalent meaning that it has the same solution as the teachers system