When a formula or other equation has two variables and one variable is by itself on a side, the variable by itself is generally called the dependent variable. Its value depends on the value of the other variable, which is called the independent variable. Basically, the value of the independent variable precedes and determines the value of the dependent variable. Consider the equation y = 3x + 4. In that equation, x is the independent variable, and y is the dependent variable. The value of y depends on what value is put in for x.
Relationships between variables can be determined and represented by equations. An algebraic equation can be a formula that represents a pattern. Consider an example of someone who makes $25 for babysitting plus another $10 for each hour of babysitting. A question could ask how much money the babysitter would make for 5 hours on the job. The student could answer the question by making a table.
The table shows that the money earned for 5 hours of babysitting would be $75. Another question that could be asked about the pay pattern is what formula could represent it. Two major components of the formula are the numbers 25 and 10 because they are part of what determines the amount of money earned. Also, the number of hours worked is part of what determines an amount of money earned, so it is part of the formula and must be represented by a variable, since it varies and is unknown until a specific number is given for a particular situation.
So, three parts of the formula are 25, 10, and a variable representing hours worked, which could be h. The amounts of money are in terms of dollars earned, so d would make a good variable for representing that quantity. The other terms come together and determine its value. Since 25 is always added, the formula involves + 25. To what is it added? It is always added to the product of 10 and the number of hours. All of that together equals the value of d. Therefore, a formula that can represent the number of dollars earned for a number of hours of babysitting is d = 10h + 25. You can check every example in the table to see that the formula works.
Practice question
- A teacher gave her students the following problem: Janet has 4 more than twice the number of oranges Mark has. The total number of oranges Janet and Mark have together is 13. How many oranges does Janet have? If a student uses the variable g to represent the number of oranges Mark has, what should the student be instructed to use to represent the number of oranges Janet has, in order to write an equation based on the situation? A. 4g + 2 B. 2g + 4 C. 13 – 2g D. 6g
Answer and explanation
- The correct answer is Choice (B). 2g represents twice the number of oranges Mark has, and 2g + 4 represents 4 more than that, which is what Janet has. Choice (A) represents 2 more than 4 times the number of oranges Mark has, Choice (C) symbolizes twice the number of oranges Mark has less than 13, and Choice (D) represents (4 + 2) times Mark's number of oranges.