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Analyzing One-Variable Relationships in Context

Equations can be used to represent real-world relationships. When the quantity a variable represents is known, solving the equation makes it possible to determine unknown information. To create an equation, use the relationship between given quantities.
Method

Problem-Solving using Modeling

Equations that represent real relationships are called mathematical models. What follows is one method of using mathematical models to solve problems.

Suppose a taxi ride from the airport to downtown costs $ Suppose also that it costs $ to ride in the taxi, and an additional per mile traveled. Calculate the distance of the ride using the following method.

1

Make sense of given information

First, it can be helpful to highlight the information given about the situation.

  • The total cost for the taxi ride is $
  • The cost per mile traveled is $
  • There is a starting fee of $

2

Define variable

A variable can be used to represent the unknown quantity in the situation.
Here, the unknown quantity is the length of the ride. Thus, the variable will be used to represent the number of miles traveled.

3

Relate quantities

Next, it is necessary to understand how the different quantities in the problem relate.
The includes the and the cost of the miles traveled. Additionally, the cost of the miles traveled can be found by multiplying the by the As a verbal equation, this relationship can be expressed as follows.

4

Create equation

Creating the equation involves translating the relationship from Step 3 into symbols. To do this, replace each quantity with the corresponding value.
For this situation, the following equation can be written.

5

Solve equation
Solve the created equation to determine the unknown quantity.
The equation has the solution Thus, the distance traveled was miles.
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Exercise

Given a perimeter of feet, what is the measure of the longest side of the triangle?

Show Solution
Solution

To begin, let's make sense of the given information. The perimeter of the triangle is feet, and the side lengths of the triangle are The perimeter of a polygon is the sum of all its side lengths. Therefore, we can equate the sum of the given lengths with feet. This gives the following equation. Solving this equation gives us the value of which will help us find the longest side. We'll start by combining like terms.

From here, inverse operations can be used to isolate

Thus, feet. By substituting for in the expressions for the unknown side lengths we can find their measures.

The side lengths of the triangle are and feet.

Therefore, the longest side in the triangle is feet long.

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