In this video, we will be using two variables to represent quantities in real-world problems, and we will show independent variables and dependent variables. And in order to show the relationship, we will use tables, equations, and graphs.
An independent variable is a value that does not depend on that of another variable. Sometimes we call this the input or the 𝑥-value in an equation. And the dependent variable is a value that does depend on that of another variable. Sometimes we call this the output or the 𝑦-value.
Now we’ll look at some examples and show how we use tables and equations and graphs along with the independent and dependent variables. So the first question, a hen lays three eggs per day. How many eggs will 𝑥 hens lay per day?
So first let’s write an equation and we know three eggs every day means that we have an equation 𝑦 equals three times 𝑥, and here 𝑥 is our independent variable because we can put any number we want in for 𝑥, and 𝑦 is going to be the dependent variable and that is because its value depends on what value we put in for 𝑥.
So now let’s use this equation to fill in the table here. We have a table, number of hens and the number of eggs. So if we have one hen, three times one equals three; if we have two hens, we put two in for 𝑥; three times two equals six; three hens, three times three equals nine; and four hens, four times three equals twelve.
So we see here these are all multiples of three and all we need to do is put in the number of hens and we can calculate the number of eggs we’ll get per day. Here is another problem. Josh wants to increase the number of points he scores at his basketball games, so he spends time shooting baskets every day.
The following table shows the total amount of baskets based on the number of days he has practiced. Identify the dependent and independent variables. So here’s the table and we have the number of days practiced and the number of baskets. So at one day, he shoots sixty baskets. So after two days of practice, he has shot one hundred twenty baskets, three days is one hundred eighty, four is two hundred forty, and five days is three hundred baskets.
So now we want to identify the dependent and independent variables. Remember dependent depends on what is input into the activity, the equation, and so that is going to be the number of baskets because the number of baskets depend on the number of days he has practiced.
So that makes the independent variable the number of days, because we can select any number of days and we just multiply by sixty because in one day he does sixty and we see that the difference between each value here is sixty; sixty to one twenty is sixty; one twenty to one eighty is also sixty. So here are the dependent and independent variables based on this table.
In this problem, we’re going to use a graph. Maria opens a new bank account. She earns five dollars each week in allowance for helping around the house, which she puts in the bank. Draw a graph which shows how much money she has after five weeks. So first let’s identify the independent and dependent variables.
So the independent variable, remember, is the input, so the input would be the number of weeks because each week she adds more money to her account, and you can see in the graph that’s been set up the number of weeks is on the 𝑥-axis. The 𝑥-axis is always for the independent variables.
Now the dependent variable would be the total amount of money in the bank account after each week, because that depends on a deposit into the account which she only gets when she gets her allowance. So using this information, first let’s create an equation. So if we use our output 𝑦, and that equals each week she puts five dollars in the bank, so that’s gonna be five 𝑥. We have 𝑦 equals five 𝑥 as our equation.
Now use this to draw the graph. So when 𝑥 is zero at the beginning, five times zero is zero, she has no money in the account. After one week, one times five is five, so we go over to one and up to five here on the 𝑦-axis. When 𝑥 is two, two times five is ten, so we go over to two and up to ten. Five times three is fifteen; we go over to three and up to fifteen. Five times four is twenty, so we go over to four and up to twenty. And five times five is twenty-five, so we go over to the five weeks and up to twenty-five.
So we can actually draw a line to show how this keeps increasing. So after five weeks, she has twenty-five dollars, so this would be twenty-five dollars after five weeks.
In this problem, we could’ve also used a table by taking the equation and creating a table of the input and output, the independent and dependent variables. So here we just use a graph to show another of way to display the data.
In this last problem, we’re going to use a graph to identify the independent and dependent variables and write an equation that shows the relationship between the two variables. In a local pet store, bird seed is sold by the bag. The graph shows how many bags are sold each week. Identify the independent and dependent variables and use the graph to write an equation that shows the relationship of the number of bags sold over time.
So we look at the graph; we have on the 𝑥-axis the number of weeks and on the 𝑦-axis the number of bags sold. And here is a line showing it’s constant and the independent variable, our input, is going to be the number of weeks, and the dependent variable which is on the 𝑦-axis here will be the number of bags sold. And this makes sense because we need to know how many weeks the bags have been sold to determine the number of bags sold.
So the input is the number weeks; the output is the number of bags sold. So now we go over to the graph and we look here at week number one. If we draw a little line here and across, this is about in the middle of ten and twenty. So we’ll say that it’s fifteen. And if we look at two, we go up here across and that’s thirty, so the difference between fifteen and thirty is fifteen, so we can say that the constant is fifteen, and the output we can just make 𝑦 and that equals fifteen times 𝑥.
So this would be the equation that you use to generate the graph, but here we used the graph to generate the equation. So now you should be able to take a table and generate a graph or generate an equation or take an equation and generate a table and a graph and just use all those to create the relationship between the independent and dependent variables.