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Systems of equations with elimination: apples and oranges

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You've gone to a fruit stand to get some fresh produce. You notice that the person in front of you gets 5 apples and 4 oranges for $10. You get 5 apples and 5 oranges for $11. Can we solve for the price of an apple and an orange using this information in a system of linear equations in 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 o for orange because o looks too much like a zero. So I'll just say x for apples. Let's let x equal the price of apples. And let's let y equal the price of oranges. So let's describe what happened to the person in line in front of us. They bought 5 apples. So how much did they spend on apples? Well, they bought 5 apples times x dollars per apple, so they spent 5x dollars on their 5 apples. And they bought 4 oranges. They bought 4 oranges times y dollars per orange. So they spent 4y on oranges. So the total amount that they spent is 5x plus 4y. And they tell us that this is $10. This is equal to $10. Now, you get in line, and you buy 5 apples. So you buy 5 apples, just like the guy in front of you. And you paid x dollars per apple. So you're going to pay 5 apples times the price per apple. This is the amount that you spend on apples. And then you buy 5 oranges. So you're going to pay 5 oranges times the price per orange, which is y. So this is how much you spend on oranges. This is how much you spend on apples and oranges, the sum. And they tell us that this is $11. 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 the y's in 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, then 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 1. So it becomes negative 5x plus negative 5y is equal to negative 11. And then I'm going to essentially add both of these equations. And I could 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, I have 5x minus 5x. Well, those cancel out. 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 $1 per orange. So this is equal to 1. Now let's figure out the price of apples. So we can go back into either 1 of these equations. I'll go back into this first 1. So 5 times-- so let's go to the person in line in front of us. They bought 5 apples at x dollars per apple, plus 4 oranges at $1 per orange, and they spent a total of $10. So this of course is just 4. Let's subtract 4 from both sides, and 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. 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. Oh, sorry. We can divide both sides by 5 in order to solve for x. It's late in the day. Brain isn't working. Dividing by 6 wouldn't have done anything. We would have gotten 5/6x. We just want to get an x here. So dividing both sides by 5, we get x is equal to 6/5 dollars. Or you could say that 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 $1 per orange, and $1.20 per apple. So we absolutely could figure out the prices of apples and oranges using the information given.