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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 $$46.37.$ Suppose also that it costs$$4.85$ to ride in the taxi, and an additional $\1.73$ 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 $$46.37.$ • The cost per mile traveled is$$1.73.$
• There is a starting fee of \$$4.85.$

### 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 $m$ 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 ${\color{#0000FF}{\text{total cost}}}$ includes the ${\color{#009600}{\text{starting fee}}}$ and the cost of the miles traveled. Additionally, the cost of the miles traveled can be found by multiplying the ${\color{#FF0000}{\text{cost per mile}}}$ by the ${\textcolor{purple}{\text{distance traveled}}}.$ As a verbal equation, this relationship can be expressed as follows. ${\color{#0000FF}{\text{total cost}}} = \text{{\color{#009600}{starting fee}}} + {\color{#FF0000}{\text{cost per mile}}} \cdot \textcolor{purple}{\text{distance}}$

### 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. \begin{aligned} {\color{#0000FF}{\text{total cost}}}\ =& \ \ \text{{\color{#009600}{starting fee}}} + {\color{#FF0000}{\text{cost per mile}}} \cdot \textcolor{purple}{\text{distance}}\\ &{\color{#0000FF}{46.37}}={\color{#009600}{4.85}}+{\color{#FF0000}{1.73}}\cdot \textcolor{purple}{m} \end{aligned}

### 5

Solve equation
Solve the created equation to determine the unknown quantity.
$46.37=4.85+1.73m$
$46.37-4.85=4.85+1.73m-4.85$
$41.52=1.73m$
$\dfrac{41.52}{1.73}=\dfrac{1.73m}{1.73}$
$24=m$
$m=24$
The equation has the solution $x=24.$ Thus, the distance traveled was $24$ miles.
Exercise

Given a perimeter of $23$ feet, what is the measure of the longest side of the triangle?

Solution

To begin, let's make sense of the given information. The perimeter of the triangle is $23$ feet, and the side lengths of the triangle are $5, \quad (x+3), \quad \text{and} \quad (3x-1).$ The perimeter of a polygon is the sum of all its side lengths. Therefore, we can equate the sum of the given lengths with $23$ feet. This gives the following equation. $5 + (x + 3) + (3x - 1) = 23$ Solving this equation gives us the value of $x,$ which will help us find the longest side. We'll start by combining like terms.

$5 + x + 3 + 3x - 1 = 23$
$x + 3x + 5 + 3 - 1 = 23$
$4x + 7 = 23$

From here, inverse operations can be used to isolate $x.$

$4x + 7 = 23$
$4x + 7 - 7 = 23 - 7$
$4x = 16$
$\dfrac{4x}{4} = \dfrac{16}{4}$
$x = 4$

Thus, $x=4$ feet. By substituting $x$ for $4$ in the expressions for the unknown side lengths we can find their measures.

\begin{aligned} x+3 &\Leftrightarrow {\color{#0000FF}{4}}+3 = 7 \\ 3x-1 &\Leftrightarrow 3 \cdot {\color{#0000FF}{4}}-1 = 11 \end{aligned}

The side lengths of the triangle are $5,$ $7,$ and $11$ feet.

Therefore, the longest side in the triangle is $11$ feet long.

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