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27. Modeling with Mathematics

Write and solve an equation to find the month when you would pay the same total amount for each Internet service.

[tex]\[
\begin{tabular}{|l|c|c|}
\cline{2-3}
& \textbf{Installation fee} & \textbf{Price per month} \\
\hline
\textbf{Company A} & \$60.00 & \$42.95 \\
\textbf{Company B} & \$25.00 & \$49.95 \\
\hline
\end{tabular}
\][/tex]

Solve the equation with variables on both sides.



Answer :

GinnyAnswer
To find the month when you would have paid the same total amount for each internet service, we need to create an equation that models the total cost for each company and then solve for the number of months, [tex]\( m \)[/tex], where the total costs are equal.

1. Define the costs:
- Company A:
- Installation fee: [tex]$60.00 - Monthly price: $[/tex]42.95

- Company B:
- Installation fee: [tex]$25.00 - Monthly price: $[/tex]49.95

2. Create the total cost equations:
- Company A: Total cost after [tex]\( m \)[/tex] months = [tex]\( 60.00 + 42.95m \)[/tex]
- Company B: Total cost after [tex]\( m \)[/tex] months = [tex]\( 25.00 + 49.95m \)[/tex]

3. Set the total costs equal to each other:

[tex]\[ 60.00 + 42.95m = 25.00 + 49.95m \][/tex]

4. Solve for [tex]\( m \)[/tex] (the number of months):
- Subtract [tex]\( 42.95m \)[/tex] from both sides to isolate the variable terms:

[tex]\[ 60.00 = 25.00 + (49.95m - 42.95m) \][/tex]

- Simplify inside the parentheses:

[tex]\[ 60.00 = 25.00 + 7.00m \][/tex]

- Subtract [tex]\( 25.00 \)[/tex] from both sides to isolate the term with [tex]\( m \)[/tex]:

[tex]\[ 60.00 - 25.00 = 7.00m \][/tex]

[tex]\[ 35.00 = 7.00m \][/tex]

- Divide both sides by 7.00 to solve for [tex]\( m \)[/tex]:

[tex]\[ m = \frac{35.00}{7.00} \][/tex]

[tex]\[ m = 5 \][/tex]

Therefore, after 5 months, you would have paid the same total amount for each internet service.

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