Question

You are planning on installing a security system at your home. For Safety First Security, you can get monitoring for $2 a day with a one-time set-up fee of $150. For Home Safety Security, you can get monitoring for $1.50 a day with a one-time set-up fee of $200. Using substitution, find the number of days for which both companies would cost the same.

Answer 1

To find the number of days for which both security companies would cost the same, let's set up equations representing the total cost for each company.

Let \( d \) be the number of days of service.

1. **For Safety First Security**:
- Daily cost: $2
- One-time setup fee: $150
- Total cost: \( C_1 = 2d + 150 \)

2. **For Home Safety Security**:
- Daily cost: $1.50
- One-time setup fee: $200
- Total cost: \( C_2 = 1.5d + 200 \)

We want to find the number of days \( d \) where both costs are equal:

\[
C_1 = C_2
\]

Substituting the expressions for \( C_1 \) and \( C_2 \):

\[
2d + 150 = 1.5d + 200
\]

Now, we will solve for \( d \).

First, subtract \( 1.5d \) from both sides:

\[
2d - 1.5d + 150 = 200
\]

This simplifies to:

\[
0.5d + 150 = 200
\]

Next, subtract 150 from both sides:

\[
0.5d = 50
\]

Now, divide by 0.5:

\[
d = \frac{50}{0.5} = 100
\]

Thus, the number of days for which both companies would cost the same is \( \boxed{100} \).