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# Systems of equations with elimination: apples and oranges

CCSS.Math:

## Video transcript

you've gone to a fruit stand to get some fresh produce you notice that the person in front of you gets five apples and four oranges for ten dollars you get five apples and five oranges for $11 can we solve for the price of an apple and an orange using this information in a system of linear equations and two variables if yes what is the solution if no what is the reason we cannot so we're trying to figure out the price of an apple and the price of an orange so I would use a for Apple but I don't like using Oh for orange because Oh looks too much like a zero so I'll just say X for apples let's let X equals a let's let X equal the price of Apple's price of apples and let's let Y let's let y equal the price of oranges price of oranges so let's describe what happened to the person in line in front of us they bought five apples so how much did they spend on Apple's well they bought five apples times X dollars per Apple so they spent five x dollars on their five apples and they bought four oranges so they spent four they bought four oranges times y dollars per orange so they spent four Y and oranges so the total amount that they spent is five x plus four y and they tell us that this is ten dollars this is equal to ten dollars now you get in line and you buy five apples so you buy five apples just like the guy in front of you and you paid X dollars per Apple so you're going to pay five time five apples times the price per Apple this is the amount that you spend on apples and then you buy five oranges you buy five oranges so you're going to pay five oranges times the price per orange which is y so this is how much you spend on oranges this is how much you spent on apples and oranges this sum and they tell us that this is going to be this is$11 console' so can we solve for an x and a y and it looks like we can and a big giveaway right over here is the ratio between the X's and why's these two equations are different so we're getting some information here if the ratios were exactly the same if this was 5x plus 4y right over here and we got a different number then we would be in trouble because we bought the same combination but we got a different price but the good thing is is that we have a different combination here so let's see if we can work it out now the most obvious thing that jumps out at me is that I have a 5x here and I have a 5x right over here so if I could subtract this 5x from that 5x and I would cancel out all of the X terms so what I'm going to do is I'm going to multiply this bottom equation by negative one so it becomes negative 5x minus 5 or plus negative 5y is equal to negative 11 and then I'm going to essentially add both of these equations and I can do that because I'm doing the same thing to both sides I already know that this thing is equal to this thing so I'm just adding those things to either side so on the left-hand side on the left-hand side I have 5x minus 5x will those cancel out and then I have and then I have 4y minus 5y well that's negative Y and that's going to be equal to 10 minus 11 which is negative 1 and then if we multiply both sides of this times negative 1 or divide both sides by negative 1 we're going to get Y is equal to 1 so just like that we were able to figure out the price of oranges it's one dollar per orange now let's figure out so this is equal to one now let's figure out the price of Apple's so we can go back in to either one of these equations I'll go back into this first one so five times so let's go to the person in line in front of us they bought five apples at X dollars per Apple plus four oranges at $1 per orange and they spent a total of ten dollars they spent a total of ten dollars so this of course is just 4 let's subtract 4 from both sides and we get we get well 4 times 1 minus 4 that just cancels out we're just going to be left with 5x on the left hand side we're just going to have 5x on the left hand side and on the right hand side we have 10 minus 4 which is equal to 6 and we can just divide both sides by 6 now in order to solve for X and so we get sorry we can divide both sides by five in order to solve for x my brain is it's late in the day brain isn't working so dividing by six wouldn't have done anything we would've gotten five 6x we just want to get an X here so dividing both sides by five we get X is equal to 6/5 dollars or you could say that X X is equal to 6/5 which is the same thing as 1 and 1/5 which is the same thing as$1 20 so it's a dollar per it's a dollar per orange and a dollar 20 per Apple so we absolutely could figure out the prices of apples and oranges using the information given