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Determine an Equation in Context

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Hi. My name is Anthony Varela. And in this tutorial, we're going to be determining an equation in context. So we're going to be drawing information and defining variables. We will develop equations to represent some scenarios. And then, we'll review some language to look out for as you develop these equations.

So here is our first scenario. A couple is looking at food options for their wedding. And their favorite option costs $11.50 per person. So we'd like to write an equation to show the cost of food for the guests at their wedding. So the first step that I like to follow is define variables for things that we do not know.

So I don't know how many people are going to show up to this wedding. So we're going to say x equals the number of guests at the wedding. I also don't know what the cost is going to be all together. So let's assign a variable to the cost. We'll say that y equals the cost of food for the guests.

So now, next, I'd like to write down what I do know. I know that the cost is going to be $11.50 per person. So we're going to use this information to develop our equations. The next step, then, is to think about what operations are involved here, addition, subtraction, multiplication, division, et cetera. What operations are going to go into our equation?

Well, I know that for every person that comes, that's going to be adding $11.50 to the cost. And repeated addition is multiplication. So I'm guessing there's going to be multiplication in my equation. So the total cost is going to equal $11.50 for every person that comes.

So I can just multiply that by x. If x equals 1, it's going to be $11.50. If x equals 100, I need to multiply 100 by $11.50 to get my cost. So there's my equation. y equals $11.50x describes the total cost of food for everyone person that's going to come to the wedding.

Here's scenario number two. So I need to take a cab to the airport. And the cab charges an initial $4 fee plus an additional $1.25 per mile. So I'm going to write an equation for the cost of the cab ride in terms of miles traveled. So let's define variables for what I don't know. So x is going to be the miles that I travel. y is going to be the cost of the cab ride.

So now, let's write down things that we know. Well, no that's no matter how far I ride I'm going to be charged at least $4. That's the initial fee. And then, there's going to be $1.25 for every mile. So now, what operations are there going to be? Well, there's going to be multiplication, because I see per mile. And there's going to be addition, because that's going to be added on to my initial fee of $4.

So y equals, so the cost of the cab ride equals, an automatic $4 initial fee plus my $1.25 per mile, which is $1.25x. And remember, x equals the miles that I travel in the cab. So that's my equation for the cost of my cab ride to the airport.

We're going to go through another scenario. And this is a bit of a fun riddle. So we have that Ray is four years older than Judy. And 6 years ago, Ray's age was twice Judy's age. So how old are Ray and Judy. So we're going to define variables for what we don't know. So we're going to say r is Ray's age and j is Judy's age.

Now, write down what we do know. So this is also going to be some equation. So I know that Ray is four years older than Judy. So if I take Judy's age and add 4, that will give me Ray's age. So I have r equals j plus 4. And I know that 6 years ago, Ray's age was twice Judy's age. And this is a bit tricky here.

So 6 years ago, that would be r minus 6 is Ray's age 6 years ago. That would be twice what Judy's age was. And remember, this is 6 years ago. So I'm going to subtract 6 from j as well. It's a bit tricky there. So that's the equation to show that 6 years ago, Ray was twice as old as Judy was.

So now, we want to then develop an equation and figure out how old Ray and Judy are. So first, I'm going to distribute the 2. I'm taking this equation here and distributing the 2 into Jane's 6. So we have r minus 6 equals 2j minus 12. Next, I'm going to use substitution. I know that r equals j plus 4. So I'm going to put j plus 4 in for r here. Now, we can just simplify the left side of our equation. I have that j minus 2 equals 2j minus 12.

So there is that same equation here. And now, what I want to do is add 2 to both sides, so I get j on one side of my equation. So j equals 2j minus 10. Now, what I'm going to do is subtract 2j from both sides. So I'm getting my j term over here. And then, I have no variables on this side. And j minus 2j is negative j. So negative j equals negative 10.

So what I can do is multiply or divide by a negative 1. So I get positive j equals positive 10. And remember, j is Judy's age. So I have determined that Judy is 10 years old. So great. Judy is 10 years old. Now, how old is Ray? Remember, Ray equals j plus 4. So now, I can substitute, then, 10 in for j. So r equals 10 plus 4. And this gives me then that Ray is 14 years old. So Ray is 4 years older than Judy. That makes sense. Judy's 10. Ray is 14.

So let's quickly review, then, some buzzwords, as I like to call them, words or phrases to look out for, which would indicate certain operations when developing these equations. So if you see words like sum or more than or received, that's all going to indicate addition in your equation.

For subtraction, look for words or phrases such as difference or less then or gave away. Those would all indicate subtraction. For multiplication, you might see words like product, twice as much, or three times as much. You might also see per. In our examples before, $11.50 per person indicated multiplication.

And for division, you'll be looking for words or phrases such as quotients, half of, or a third of, or a fourth of, and then, oftentimes, sharing. So if you need to share your pizza with your friends, that would all be dividing.

So let's review determining an equation in context. We talked about defining your variables for things you don't know. Then, you want to write down things that you do know. And then, what words mean what operations? And you can refer to this table of common words and phrases, this certainly doesn't represent all of them, that would indicate certain operations in your equation. Well, thanks for watching this tutorial on determining an equation in a context. Thanks for watching and hope to see you next time.