In this video, you will learn how to use variables to set up equations and inequalities that solve real-world problems. You will learn what a variable is and see how the variable is used in setting up the equations and inequalities.
How is a variable used? A variable is usually represented by the letter 𝑥, although it can actually be represented by any letter. You might not wanna use the letter 𝑙, because that looks a lot like a one.
Here we have an algebraic equation: two 𝑥 plus five equals thirteen. And we have 𝑥. What we would do to solve this equation is first subtract five from both sides. So we now have two 𝑥 equals eight. And then we would divide both sides by two. And we have 𝑥 equals four. So in an algebraic equation, we just solve for the variable to get a value. But that value doesn’t represent anything. This could be four soccer balls, four toasters, four pieces of bread. We just don’t know.
It’s good that we don’t always have to use the letter 𝑥 as our variable. Because when we solve a real-world problem using a variable, we usually use a letter that helps us understand what we’re solving for. For example, if we want to know the number of apples in a basket, we might assign the letter 𝑎 as our variable.
So using our example about apples, a group of three sisters go apple picking in a local orchard. When they bring the bag of apples home, mom takes five to make some apple sauce, and the remaining apples are shared equally by the sisters. If there were fifty-six apples in the bag at the end of apple picking, set up an equation to determine how many apples each sister gets.
First, we want to define our variable. We will use 𝑎 since the problem is about apples. We want to know how many apples each sister gets. So 𝑎 equals the number of apples each sister gets. Now we can set up our equation. There are three sisters, so the first part of the equation is gonna be three times 𝑎, three times the number of apples each sister gets. And then mom takes five apples from the bag. And you might think we would want to subtract, but these five apples are in addition to what each sister gets. So we add five and this equals the total number of apples that they had in the bag, which is fifty-six. So here is the equation that we would use to solve this problem. And then you would just use your algebraic techniques to solve it and find out how many apples each sister gets. We will do that in a future video, but right now we’re just learning how to set up the equations and how to use the variables.
Here’s another problem. Daniel buys lunch every day for two dollars and fifty cents. His father gives him twenty dollars, but one day he also buys a bottle of water for two dollars. How many lunches will Daniel be able to buy if he can’t spend more than twenty dollars? And since we have a limit here, it has to be less than or less than or equal to twenty dollars. This is gonna be an inequality. So again, the first step is to define our variable. And you might wanna use 𝑙 because it’s lunches. But remember, I said that 𝑙 can sometimes look like a one, so we will use 𝑥. And 𝑥 will equal the number of lunches Daniel can buy. Now let’s set up the inequality. And just like the last problem, we have the number of lunches Daniel can buy. But we know each one is two dollars and fifty cents. So we have two point five zero 𝑥. This could also be written as just two point five. And then he also buys a bottle of water with that twenty dollars and that cost two dollars. So we’re going to add two here, because of eventually that will be subtracted. The lunches plus the water, we want that to be less than the twenty dollars. So here’s an inequality that we would write to set up this problem. And again in a future video, we’ll be doing more work with solving these inequalities and understanding why the variable has the solution that it does and what it represents.
So now you should have some understanding of how to set up equations and inequalities to solve real-world problems and how we define our variables, so that we know what we’re solving for. So watch for other videos in which we solve equations and inequalities that represent real-world problems.