Write an equation to model the situation. Then solve the equation.

One gym membership charges [tex]$10 a month with a sign-up fee of $[/tex]45. Another gym does not charge a sign-up fee but costs $25 a month. At what month is the total cost of both memberships the same?



Answer :

Sure, let's break this problem down step by step.

1. Define the Costs:
- Let [tex]\( M_1 \)[/tex] represent the total cost for the first gym membership.
- Let [tex]\( M_2 \)[/tex] represent the total cost for the second gym membership.

2. Write Out the Expressions:
- For the first gym membership:
- Monthly cost: [tex]$10 - Sign-up fee: $[/tex]45
- After [tex]\( m \)[/tex] months, the total cost will be:
[tex]\[ M_1 = 10m + 45 \][/tex]

- For the second gym membership:
- Monthly cost: $25
- No sign-up fee
- After [tex]\( m \)[/tex] months, the total cost will be:
[tex]\[ M_2 = 25m \][/tex]

3. Set Up the Equation to Find When the Costs Are Equal:
We need to find the month [tex]\( m \)[/tex] when [tex]\( M_1 = M_2 \)[/tex]. So, we set the two expressions equal to each other:
[tex]\[ 10m + 45 = 25m \][/tex]

4. Solve the Equation:
- First, move all terms involving [tex]\( m \)[/tex] to one side of the equation and constants to the other:
[tex]\[ 10m + 45 = 25m \][/tex]
Subtract [tex]\( 10m \)[/tex] from both sides:
[tex]\[ 45 = 25m - 10m \][/tex]
Simplify the equation:
[tex]\[ 45 = 15m \][/tex]

- Next, solve for [tex]\( m \)[/tex] by dividing both sides by 15:
[tex]\[ m = \frac{45}{15} \][/tex]
[tex]\[ m = 3 \][/tex]

Therefore, the total cost of both gym memberships will be the same after 3 months.