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 \( \boxed{100} \).