Questions: 7. Pretend you're installing a security system in your home and are trying to decide whether to lease or purchase the system. Here are your options: Lease: 125 to install and 31 a month for monitoring fee Purchase: 573 for the equipment and installation and 10 a month for monitoring fee b) Write an equation for each situation in y=mx+b format (label which equation is which) c) Use your equations to estimate how long it will take for the two plans to cost the same.

7. Pretend you're installing a security system in your home and are trying to decide whether to lease or purchase the system. Here are your options:

Lease: 125 to install and 31 a month for monitoring fee Purchase: 573 for the equipment and installation and 10 a month for monitoring fee

b) Write an equation for each situation in y=mx+b format (label which equation is which)

c) Use your equations to estimate how long it will take for the two plans to cost the same.

Transcript text: 7. Pretend you're installing a security system in your home and are trying to decide whether to lease or purchase the system. Here are your options: Lease: $\$ 125$ to install and \$31 a month for monitoring fee Purchase: $\$ 573$ for the equipment and installation and $\$ 10$ a month for monitoring fee b) Write an equation for each situation in $y=m x+b$ format (label which equation is which) c) Use your equations to estimate how long it will take for the two plans to cost the same.

Solution

Solution Steps

To solve this problem, we need to create two linear equations representing the total cost over time for each option. The equations will be in the form $ y = mx + b $, where $ y $ is the total cost, $ m $ is the monthly fee, $ x $ is the number of months, and $ b $ is the initial cost. For the lease option, the equation will be $ y = 31x + 125 $. For the purchase option, the equation will be $ y = 10x + 573 $. To find when the costs are equal, we set the two equations equal to each other and solve for $ x $.

Step 1: Define the Equations

We define the total cost equations for both options. For the lease option, the total cost $ y $ after $ x $ months is given by: \[ y = 31x + 125 \] For the purchase option, the total cost $ y $ after $ x $ months is: \[ y = 10x + 573 \]

Step 2: Set the Equations Equal

To find the point at which the costs are the same, we set the two equations equal to each other: \[ 31x + 125 = 10x + 573 \]

Step 3: Solve for $ x $

Rearranging the equation gives: \[ 31x - 10x = 573 - 125 \] This simplifies to: \[ 21x = 448 \] Dividing both sides by 21, we find: \[ x = \frac{448}{21} \approx 21.3333 \]