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Writing a System of Linear Equations

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Today we're going to talk about writing a system of linear equations. So we're going to go ahead and do a couple of examples of taking a real-world situation and then writing a system of equations that would represent that situation.

So for my first example, I'll give an example of a real-world situation. And I'll show you how to write a system of equations that would represent that situation. So let's say we have admission to a fair costs $1.50 for children and $4 for adults. And on a certain day, 2,200 people attended the fair and $5,050 was collected. So we can solve this system of equations to determine how many children and how many adults attended the fair that day.

So I'm going to start by defining a variable of x as the number children who attended the fair on that day. And my variable y is the number of adults that attended the fair on that day. So first I know that between the number of children and the number of adults, 2,200 people attended. So my first equation would say that the number of children plus the number of adults should equal 2,200.

My second equation in the system would show how much money was collected between the children and the adults. So if I know that it costs $1.50 for each child to get into the fair, the amount of money that they collected from the children would be equal to $1.50 times however many children attended, which is my x variable. So the first part of my second equation will be 1.50 times however many children attended, or x.

I'm going to add to that the amount of money that was collected from the adults who attended. And I know that each adults paid $4 to get into the fair. So to figure out how much money they got from all of the adults, I would multiply 4 times however many adults attended, which is going to my y variable, so plus 4y. And I know that they collected a total of $5,050 from the children and adults, so it's going to equal 5,050.

So I could solve this system of equations to find the number of children and the number of adults. And I can write these two equations as a system, because the corresponding variables in each equation have the same definition. x in both equations represents the number of children. And y in both equations represents the number of adults. So these two equations can be considered at the same time, which again makes them a system of equations.

And again, in each equation, x and y are the number of children and adults that attended and 2,200 was the total number of people who attended. And in my second equation, 5,050 was the total amount of money that was collected. And $1.50 and $4 were the amounts of money that the children and the adults each paid to get into the fair.

So let's see how we can take another real-world situation and write it as a system of equations. Let's say I go to the grocery store and I buy apples that are $0.80 each and cantaloupes that are $2.10 each. And at that grocery store, I spent a total of $8.20.

Then I go to another grocery store and buy the same number of apples that are $1 each and the same number of cantaloupes and they're $1.75 each. At that grocery store, I spent a total of $8.50. So I can write a system of equations and solve it to find the number of apples and cantaloupes that I bought at each store.

So I'm going to start by defining a couple of variables. I'm going to let x be equal to the number of apples that I bought. And I'm going to let y be equal to the number of cantaloupes that I bought. So I can write a system of equations using the relationships between x and y and the unit price of each fruit and the total amount that I spent at each store.

So at the first store, I know that apples cost $0.80 each. And so to find the total amount spent on apples, I'm going to multiply $0.80 by however many apples I bought, which is going to be my x variable. So I'll have 0.80 times x.

I'm going to add to that the amount that I spent on cantaloupes. And I know that cantaloupes were $2.10 each. So I'm going to multiply that by the number of cantaloupes that I bought, which is my y variable. And I know that that has to equal the total amount that I spent at the store, which is $8.20.

Then I can write an equation for what I bought at the second store. At the second store, I know that apples cost $1 each. I'm going to multiply that again by the number of apples that I bought, x.

And I need to add to that the amount that I spent on cantaloupes, so I'm going to multiply the unit price for cantaloupes at the second store, 1.75, by how many cantaloupes I bought. And that has to equal the total amount spent at that store, which was $8.50.

So again, when I want to solve this, I can consider this as a system of equations because x and y are defined the same in both equations. In both equations, x is the number of apples that I bought and y is the number of cantaloupes that I bought. And I know that they're the same, because I bought the same number of apples and the same number of cantaloupes at both stores.

So let's go over our key points from today. As usual, make sure you have them in your notes if you don't already so you can refer to them later. You can write a system of equations for real-world situations if the variables in each equation have the same definition. And the equations will be considered at the same time, meaning they will be solved together to determine the variables.

So I hope that these key points and examples helped you understand a little bit more about writing a system of linear equations. Keep using your notes. And keep on practicing. And soon you'll be a pro. Thanks for watching.