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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 $46.37.46.37. Suppose also that it costs $4.854.85 to ride in the taxi, and an additional $1.73\$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 $46.37.46.37.
  • The cost per mile traveled is $1.73.1.73.
  • There is a starting fee of $4.85.4.85.

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 mm 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 total cost{\color{#0000FF}{\text{total cost}}} includes the starting fee{\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 cost per mile{\color{#FF0000}{\text{cost per mile}}} by the distance traveled.{\textcolor{purple}{\text{distance traveled}}}. As a verbal equation, this relationship can be expressed as follows. total cost=starting fee+cost per miledistance {\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. total cost =  starting fee+cost per miledistance46.37=4.85+1.73m\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+1.73m46.37=4.85+1.73m
46.374.85=4.85+1.73m4.8546.37-4.85=4.85+1.73m-4.85
41.52=1.73m41.52=1.73m
41.521.73=1.73m1.73\dfrac{41.52}{1.73}=\dfrac{1.73m}{1.73}
24=m24=m
m=24m=24
The equation has the solution x=24.x=24. Thus, the distance traveled was 2424 miles.
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Exercise

Given a perimeter of 2323 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 2323 feet, and the side lengths of the triangle are 5,(x+3),and(3x1). 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 2323 feet. This gives the following equation. 5+(x+3)+(3x1)=23 5 + (x + 3) + (3x - 1) = 23 Solving this equation gives us the value of x,x, which will help us find the longest side. We'll start by combining like terms.

5+x+3+3x1=235 + x + 3 + 3x - 1 = 23
x+3x+5+31=23x + 3x + 5 + 3 - 1 = 23
4x+7=234x + 7 = 23

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

4x+7=234x + 7 = 23
4x+77=2374x + 7 - 7 = 23 - 7
4x=164x = 16
4x4=164\dfrac{4x}{4} = \dfrac{16}{4}
x=4x = 4

Thus, x=4x=4 feet. By substituting xx for 44 in the expressions for the unknown side lengths we can find their measures.

x+34+3=73x1341=11\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 5,5, 7,7, and 1111 feet.

Therefore, the longest side in the triangle is 1111 feet long.

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