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Naomi wants to take fitness classes at a nearby gym, but she needs to start by selecting a membership plan. With the first membership plan, Naomi can pay $54\$54 per month, plus $1\$1 for each group class she attends. With the second membership plan, she'd pay $12\$12 per month plus $3\$3 per class.\newlineIf Naomi attends a certain number of classes in a month, the two membership plans end up costing the same total amount. How many classes per month does that take? What is that total amount Naomi will pay per month for a membership and that many classes?\newlineIf Naomi attends ____\_\_\_\_ classes per month, she will pay $____\$\_\_\_\_ total each month for either membership plan.

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Q. Naomi wants to take fitness classes at a nearby gym, but she needs to start by selecting a membership plan. With the first membership plan, Naomi can pay $54\$54 per month, plus $1\$1 for each group class she attends. With the second membership plan, she'd pay $12\$12 per month plus $3\$3 per class.\newlineIf Naomi attends a certain number of classes in a month, the two membership plans end up costing the same total amount. How many classes per month does that take? What is that total amount Naomi will pay per month for a membership and that many classes?\newlineIf Naomi attends ____\_\_\_\_ classes per month, she will pay $____\$\_\_\_\_ total each month for either membership plan.
  1. Define Variables: Let xx represent the number of classes Naomi attends per month, and yy represent the total cost per month for either membership plan. For the first membership plan: Total cost = monthly fee + cost per class = $54+$1x\$54 + \$1x. Equation: y=54+xy = 54 + x
  2. Membership Plan Equations: For the second membership plan: Total cost = monthly fee + cost per class = $12+$3x\$12 + \$3x. Equation: y=12+3xy = 12 + 3x
  3. Set Equations Equal: Set the equations equal to find when the costs are the same: 54+x=12+3x54 + x = 12 + 3x.
  4. Solve for x: Solve for x: Subtract xx from both sides: 54+x−x=12+3x−x54 + x - x = 12 + 3x - x, Simplify: 54=12+2x54 = 12 + 2x, Subtract 1212 from both sides: 54−12=2x54 - 12 = 2x, 42=2x42 = 2x, Divide by 22: x=21x = 21.
  5. Substitute xx into Equation: Substitute x=21x = 21 back into either equation to find yy. Use the first equation: y=54+21y = 54 + 21, y=75y = 75.

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