- "Vivek and also Camila'steacher gave them a mechanism "of direct equations to solve. "They each took a few steps that lead "to the systems displayed in the table below." therefore we have the teacher's initial system, what Vivek got afterdoing some operations, what Camila gained afterdoing part operations. I m sorry of them acquired asystem the is tantamount to the teacher's system? for this reason the very first questionwe need to ask oneself is what walk it average to evenhave an equivalent system? because that the benefits of thisquestion, or because that our purposes, an identical system is a systemthat has actually the exact same solution. Therefore if there's part X-Y pair the satisfies the teacher's device that is the equipment to the teacher's system. Fine Vivek's system, we'regonna speak to it indistinguishable if it has actually the same solution. Similarly, if Camila'ssystem has actually the very same solution, climate we're gonna contact itequivalent to the teacher's system. For this reason let's make some comparisons here. So first let's look at Vivek. So his an initial equationis actually unchanged from the teacher's equation, is unchanged from the teacher's equation, so any kind of solution the meetsboth of these equations is for certain gonna satisfy this optimal equation due to the fact that it's accurate thesame as the height equation of the teacher, so that works out. Additionally look at the 2nd one. The 2nd one is absolutely adifferent equation over here. Us can check that it'snot simply being multiply by some number ~ above both sides. To walk from one to zeroif you were multiplying, friend would need to multiply one times zero and then in order tomaintain the equality, you would have to do that on both sides. However zero times this left-handside would have been zero, you would certainly have gained zero equates to zero, so he didn't just scaleboth political parties by part number, looks prefer he did another operation. He more than likely looks likehe's including or individually something to both sides, so let's see just how he can have acquired this ideal over here. Therefore he took -4x to add 5y is equal to one. And it looks favor fromthat he was able to obtain - 3x add to 7y is equal to zero. For this reason let's see what he had to do to do that. Let's see, he would have had actually to, to walk from -4x come -3x, he would have had to add an X, therefore I could just writean X right over there. To go from 5y come 7y, he would certainly have had actually to include 2y. For this reason on the left-hand side,he would have actually to add X add to 2y. Notification we have an X plus2y appropriate over there. And on the right-hand side,he would certainly have had to add or subtract a one, or include a negative one. Notification we see a negativeone best over there. Therefore what he basically did ishe included the left-hand political parties of these 2 equations toget this brand-new left-hand side ideal over here, and also headded the right-hand political parties to obtain this brand-new right-hand side. And also that is a legit operation. This new equation the you got, this brand-new linear equation,it's going to stand for a various line thanthis one appropriate over here, but the resulting mechanism isgoing to have the very same solution. Why carry out we feeling confidentthat the resulting device is going to have actually the same solution? Well for an X-Y pair thatsatisfies both of these equations, that's what a solution would be, for the X-Y pair, X plus 2y is same to negative one. Therefore for the solution, we're including the exact same thing to both sides. We're saying: "Look,I'm gonna include X plus 2y "to the left-hand side. "Well if i don't wannachange the solution, "I have to add the exact same thingto the right-hand side." well they're telling us forthe systems to this equation, X to add 2y is equal to an adverse one, so negative one is thesame point as X add to 2y for the solution, sowe're not gonna adjust the resulting solution of the system, therefore it's a fully legitimateoperation what Vivek go is including the left-hand political parties and including the right-hand sides to acquire this brand-new second equation. That's not going come changethe equipment of the system. In fact, that's a method we regularly use to eventually discover thesolution the a system. So currently let's look at Camila, or Camila. Therefore her an initial equation isactually the specific same equation as the teacher's 2nd equation. Now let's see, her 2nd equation, exactly how does the relate possiblyto the an initial equation? So simply looking at it offhand, it looks like it could just be, the looks favor she simply multipliedboth sides times a number. And it looks favor thatnumber, she clearly multiplied the right-hand side times an adverse eight. So times negative eight. An adverse one time negativeeight is optimistic eight. And also it looks favor she likewise multiplied the left-hand side by an adverse eight. Negative eight time X is -8X. An unfavorable eight time 2y is -16y. Therefore she just multiplied bothsides by the very same value which in reality doesn'tchange the equation. This in reality is going to be-- It transforms it the means it looks, but it in reality represents the same line. For this reason this is definitelystill an equivalent system. These space still the exact same constraints. You're walk to have actually the exact same solution. Anytime you're managing systems, you're not going to changethe solution of the device as lengthy as you either multiplyboth sides of one equation through a scaler, or you are addingand subtracting the equations.

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As soon as I say add or subtract the equations, you're adding the left-handside to the left-hand side, including the right-hand sideto the right-hand side choose we had actually here, or subtractingthe one from the other on the left-hand next andif us subtract the bottom indigenous the optimal on the left-hand and we subtract the bottomfrom the top on the right-hand, it's no going to readjust our solution. Therefore both of them acquired asystem that is equivalent, definition that it has actually the exact same solution together the teacher's system.