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let's explore a few more methods for solving systems of equations let's say I have the equation 3x plus 4y is equal to 2.5 and I have another equation 5x 5x minus 4y is equal to twenty five point five and we want to find an x and y value that satisfies both of these equations if we think of it graphically this would be the intersection of the lines that represent the solution sets to both of these equations so how can we proceed it when we saw in substitution we like to eliminate one of the variables we did it through substitution last time but is there anything we can add or subtract let's focus on this yellow on this top equation right here is there anything that we can add or subtract to both sides of this equation remember anytime you deal with the equation you have to add or subtract the same thing to both sides but is there anything that we could add or subtract to both sides of this equation that might eliminate one of the variables and then we would have one equation in one variable and we can solve for it and it's probably not obvious even though it's sitting right in front of your face well what if we just added what if we just added this equation to that equation what I mean by that is what if we were to add 5x minus 4y to the left-hand side and add 25.5 to the right-hand side so if I were to literally add this to the left-hand side and add that to the right-hand side and you're probably saying Sal hold on how can you just add two equations like that and remember when you're doing any equation when you have you know if I have any equation of the form well it really any equation ax plus B Y is equal to C if I want to do something to this equation I just have to add the same thing to both sides of the equation so I could for example I could add D to both sides of the equation because D is equal to D so I won't be changing the equation you would get ax plus B y plus D is equal to C plus D and we've seen that multiple multiple times you can add anything you do to one side of the Equator you have to do to the other side we're saying hey Sal wait you know on the left-hand side you're adding 5x minus 4y to the equation on the right hand side you're adding 25.5 to the equation aren't you adding two different things to both sides of the equation and my answer would be no we know that 5x minus 4y is 25.5 this quantity and this quantity are the same they are both 25.5 this second equation is telling me that explicitly so I could add this to the left-hand side I'm essentially adding 25.5 to it and I could add 25.5 to the right-hand side so let's do that if we were to add the left-hand side 3x plus 5x is 8x and then what is 4y minus 4y and this was the whole point I saw you know when I looked at these two equations so now I have a 4y I have a negative 4y if you just add these two together they are going to cancel out they're going to be plus 0y or that whole term is just going to go away and that's going to be equal to 2.5 plus 25.5 is 28 so you divide both sides so you get 8x is equal to 28 and you divide both sides by 8 and we get divide both sides you get X is equal to 28 over 8 or you divide the numerator and the denominator by 4 that's equal to 7 over 2 that's our x value now we want to solve for our Y value and we could we could substitute this back into either either of these two equations let's use the top one you could do it with the bottom one as well so we know that 3 times X 3 times 7 over 2 3 times 7 over 2 I'm just substituting the x value we figured out into this top equation 3 times 7 over 2 plus 4y is equal to 2.5 let me just write that as 5 halves we're a deal stay in the fraction world so this is going to be 21 over 2 plus 4y is equal to five halves subtract 21 over 2 from both sides so minus 21 over 2 minus 21 over to the left-hand side you're just left with a 4y because these two guys cancel out is equal to this is 5 minus 21 over 2 that's negative 16 over 2 so that's negative 16 over 2 which is the same thing well I'll write it out this is negative 16 over 2 or we could write that let's continue up here 4y I'm just continuing this train of thought up here 4y is equal to negative 8 divide both sides by 4 and you get Y is equal to negative 2 so the solution to this equation is X is equal to 7 halves y is equal to negative 2 this would be the coordinate of their intersection and you could try it out on both of these equations right here so let's see let's verify that it also satisfies this bottom equation five times seven halves is 35 over two minus 4 times negative 2 so minus negative eight that's equivalent to see this is 17.5 plus 8 right this is 17.5 this is +8 and that indeed does equal 25 point five so this satisfies both equations now let's see if we can use our newly found skills to tackle a word problem our newly found skills in elimination so he says Nadia and Peter visit the candy store Nadia buys three candy bars and four fruit roll-ups for 284 Peter also buys three candy bars but can only afford one additional fruit roll-up his purchase cost as a dollar 79 what is the cost of each candy bar and each fruit roll-up so let's define some variables let's just use x and y let's let X equal cost of candy bar I was going do a C and the F for fruit roll-up but I'll just stick with X&Y cost of candy bar and let y equal the cost of a fruit roll-up a fruit roll-up alright so what is this first this first statement tells us Nadia buys three candy bars so the cost of three candy bars is going to be three x three candy bars three times the cost of a candy bar and four fruit roll-ups plus four times y the cost of a fruit roll-up this is how much Nadia spends three candy bars four fruit roll-ups and it's going to cost two dollars and 84 cents two dollars and 84 cents that's what this first statement tells us it translates into that equation the second statement Peter also buys three candy bars peter also buys three candy bars we can only afford one additional fruit roll-up so plus one additional fruit roll-up his purchase cost is equal to one dollar and seventy nine what is the cost of each candy bar in each fruit roll-up and we're going to solve this using elimination you could solve this using any of the techniques we've seen so far substitution elimination even graphing although it's kind of hard to eyeball things with the graphing so how can we do this remember with elimination you're going to add let's let's focus on this top equation right here is there's something we can add to both sides of this equation that will help us eliminate one of the variables or let me put it this way there's something we could add or subtract to both sides of this equation that will help us eliminate one of the variables well like in the problem we did a little bit earlier in the video what if we were to subtract this equation or what if we subtract 3x plus y from 3x plus 4y on the left-hand side and subtract $1 79 from the right-hand side and remember I'm by doing that I would be subtracting the same thing from both sides of the equation this is a dollar 79 how do I know because it says this is equal to a dollar 79 so if we did that we were to be subtracting the same thing from both sides of the equation so subtract 3x plus y from the left-hand side of the equation so I'm going to subtract it from the left-hand side and let me just do this over the right if I subtract 3x plus y that is the same thing as negative 3x minus y if you just distribute the negative sign so let's subtract it so you get negative 3x minus y and maybe I should maybe I should get make make it very clear this is not a plus sign I'm multiplying you can imagine I'm multiplying the second equation by negative one is equal to negative dollar seventy-nine I'm just taking the second equation you can imagine I'm multiplying it by negative one and now I'm going to add this side the left hand side to the left hand side of this equation and the right hand side to the right hand side of that equation and what do we get when you add 3x plus 4y minus 3x minus y the 3x is canceled out 3x minus 3x is 0 X now when you even write it down you get 4x - I'm sorry 4y minus y that is 3y and that is going to be equal to $2 84 - a dollar 79 what is that that is like that's a dollar a dollar five dollar five so 3y is equal to dollar five divide both sides by 3 divide both sides by three Y is equal to what's a dollar five divided by three so three goes into a dollar five it goes into 1 0 times 0 times 3 is 0 bring 1 minus 0 is 1 bring down a 0 3 goes into 10 three times 3 times 3 is 9 subtract 10 minus 9 is 1 bring down the five 3 goes into 15 5 times 5 times 3 is 15 subtract we have no remainder so y is equal to 35 cents so the cost of a fruit roll-up is 35 cents now we can substitute back into either of these equations to figure out the cost of a candy bar so let's use well let's use this bottom equation right here which was originally if you remember before I multiplied it by negative by negative one it was three X plus y is equal to a dollar seventy-nine so that means that 3x plus the cost of a fruit roll-up 0.35 is equal to a dollar seventy-nine if we subtract 0.35 from both sides what do we get the left-hand side you just left with the 3x these cancel out is equal to let's see this is a dollar seventy nine minus 35 that's a dollar forty four and three goes into a dollar forty four I think it goes well three goes into one dollar forty four goes into one 0 times 1 times 3 is zero bring down the 1 or subtract bring down the 4 three goes into 14 four times four times three is twelve I'm making this messy 12 4 14 minus 12 is to bring down the 4 3 goes into 24 eight times 8 times 3 is 24 no remainder so X is equal to zero point four eight so there you have it we figured out using elimination that the cost of a candy bar is equal to 48 cents and that the cost of a fruit roll-up is equal to 35 cents