Question

you are looking for an apartment. there are two final choices. apartment a has a $1000 security deposit and costs $1200 each month. apartment b has a $1500 security deposit and costs $1175 each month. how many months, m, will it take for the costs to be the same? let____=__________ equation:________________ the perimeter of a rectangle increased by 2 is the same as 18 less than 3 times the perimeter. what is the perimeter of the rectangle? let__=__________ equation:__________________

Explanation:

Step1: Define the variable for the first problem

Let \( m \) = the number of months.
The cost for Apartment A is \( 1000 + 1200m \).
The cost for Apartment B is \( 1500+1175m \).
Equation: \( 1000 + 1200m=1500 + 1175m \)

Step2: Solve the equation for the first problem

Subtract \( 1175m \) from both sides:
\( 1000+1200m - 1175m=1500+1175m-1175m \)
\( 1000 + 25m=1500 \)
Subtract \( 1000 \) from both sides:
\( 25m=1500 - 1000 \)
\( 25m=500 \)
Divide both sides by \( 25 \):
\( m=\frac{500}{25}=20 \)

Step3: Define the variable for the second problem

Let \( p \) = the perimeter of the rectangle.
The perimeter increased by \( 2 \) is \( p + 2 \).
\( 18 \) less than \( 3 \) times the perimeter is \( 3p-18 \).
Equation: \( p + 2=3p-18 \)

Step4: Solve the equation for the second problem

Subtract \( p \) from both sides:
\( p + 2-p=3p-18 - p \)
\( 2=2p-18 \)
Add \( 18 \) to both sides:
\( 2 + 18=2p-18+18 \)
\( 20=2p \)
Divide both sides by \( 2 \):
\( p = 10 \)

Answer:

For the first problem: \( m = 20 \)
For the second problem: \( p=10 \)