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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.

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.

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.$

Define variable

Here, the unknown quantity is the length of the ride. Thus, the variable $m$ will be used to represent the number of miles traveled.

Relate quantities

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.
${\color{#0000FF}{\text{total cost}}} = \text{{\color{#009600}{starting fee}}} + {\color{#FF0000}{\text{cost per mile}}} \cdot \textcolor{purple}{\text{distance}}$

Create equation

For this situation, the following equation can be written. $\begin{aligned}
{\color{#0000FF}{\text{total cost}}}\ =& \ \ \text{{\color{#009600}{starting fee}}} + {\color{#FF0000}{\text{cost per mile}}} \cdot \textcolor{purple}{\text{distance}}\\
&{\color{#0000FF}{46.37}}={\color{#009600}{4.85}}+{\color{#FF0000}{1.73}}\cdot \textcolor{purple}{m}
\end{aligned}$

Solve equation

$46.37=4.85+1.73m$

$46.37-4.85=4.85+1.73m-4.85$

$41.52=1.73m$

$\dfrac{41.52}{1.73}=\dfrac{1.73m}{1.73}$

$24=m$

$m=24$

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

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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