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Determine an Equation in Context

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Today we're going to talk about how you can determine an equation in the context of a real-world problem. So we'll go through some examples of real-world problems. We'll talk about how you can identify what you're trying to solve for, how you can come up with a variable for what you're trying to solve for, use that to write an equation, and then solve to solve the problem.

So here's my first example. The time in New York is 3 hours later than the time in California. It's 7:00 PM in California. What time is it in New York?

So I'm going to develop an equation to help me figure out what I'm trying to find for this problem. So what I'm trying to find is what time it is in New York. So I'm going to define a variable, n, to be equal to the time in New York.

So if I know that the time in New York is 3 hours later than the time in California, if I want to find the time in California, let's define the time in California as a variable, c.

Since the time in New York is later or more than the time in California, I know I'm going to be adding. So the time in California plus 3 will be equal to my time in New York.

Now if it tells me that the time in California is 7:00 or 7:00 PM, then I can simply substitute 7 in for my c variable, add three to that, and that will equal my time in New York. 7 plus 3 gives me 10. So the time in New York is going to be 10:00 PM.

So here's my second example. It cost $10 for admission to an amusement park. Rides cost an additional $1.50. So if you have $25, how many rides can you go on?

So we're trying to find the number of rides that we can go on, so we're going to define that with a variable. We'll use r. And so we'll say r is the number of rides that you can go on.

So to develop my equation so that I can solve for the number of rides, I know that the amount of money that I have, $25, is going to be equal to how much I have to pay.

I have to pay for sure the $10 to get in. And then I have to pay $1.50 for every ride that I need to go on. So every ride that I'm going to go on, I'm going to have to pay another $1.50. So I can express that as 1.50 times the number of rides that I'm going on are multiplied by r. Now all I need to do is add my two costs together and I have my equation.

So now to solve my equation I'm going to start by subtracting $10 from both sides. This will give me $15. Here this will cancel out. So $15 will be equal to $1.50 times r.

Now to get r by itself I'm just going to divide both sides by $1.50. That will cancel. And $15 divided by $1.50 is going to give me 10. So the number of rides that I can go on at the amusement park is 10.

So here's my third example. There are 35 students in a class. If the girls number the boys by 5, how many boys are in the class? So I want to know how many boys are in the class. So I'm going to start by defining my first variable, b, as the number of boys in the class.

Now I need to relate the number of boys to two things. First, to the fact that there are 35 students in the class, boys and girls combined. And second, that the girls outnumber the boys by 5. So I also am going to define a second variable, g, as the number of girls.

So now I can write two equations for those two pieces of information. So first I know that there are 35 students in a class, which is made up of boys and girls. So that means the number boys plus the number of girls has to be equal to 35.

Second, I know that the girls outnumber the boys by 5, which means that there are 5 more girls in the class than there are boys. So that means that my number of girls is going to be equal to 5 plus the number of boys.

So now if I want to find the number of boys, I can combine these two equations by using substitution. So if I know that my g variable, or the number of girls, is equal to 5 plus b, or the number of boys, I can substitute this whole expression into my first equation for my g variable.

So combining those two, this first equation becomes b plus 5 plus b is equal to 35. Now I can combine some like terms. I have b and another b, or 1b plus 1b, that will give me 2b plus my 5 is equal to 35.

Solving this equation, I'm going to subtract 5 from both sides. This will cancel. And I'll have 2b is equal to 35 minus 5, or 30. Then I just need to divide both sides by 2. I'll have b by itself, and 30 divided by 2 will give me 15. So I found that the number of boys in my class is equal to 15.

So let's go over our key points from today. Make sure you have them in your notes so you can refer to them later. When writing equations from a problem in context, look for key words to help you choose different math operations.

Addition could mean the word increased, more than, combined, total, or sum. Subtraction could be decrease, less, difference, fewer than, or change. Multiplication could mean of, times, product, double, or triple. And division could be indicated by per, out of, ratio, quotient, or split.

So I hope that these key points and examples helped you understand a little bit more about how you can determine and solve an equation in the context of a word problem. Keep using your notes. And keep on practicing, and soon you'll be a pro. Thanks for watching.