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 $\$54$ per month, plus $\$1$ for each group class she attends. With the second membership plan, she'd pay $\$12$ per month plus $\$3$ per class. $\newline$ If 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? $\newline$ If Naomi attends $\_\_\_\_$ classes per month, she will pay $\$\_\_\_\_$ total each month for either membership plan.

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Solve a system of equations using substitution: word problems

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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

per month, plus

\$1

for each group class she attends. With the second membership plan, she'd pay

\$12

per month plus

\$3

per class.

\newline

If 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?

\newline

If Naomi attends

\_\_\_\_

classes per month, she will pay

\$\_\_\_\_

total each month for either membership plan.

Define Variables: Let $x$ represent the number of classes Naomi attends per month, and $y$ represent the total cost per month for either membership plan. For the first membership plan: Total cost = monthly fee + cost per class = $\$54 + \$1x$ . Equation: $y = 54 + x$
Membership Plan Equations: For the second membership plan: Total cost = monthly fee + cost per class = $\$12 + \$3x$ . Equation: $y = 12 + 3x$
Set Equations Equal: Set the equations equal to find when the costs are the same: $54 + x = 12 + 3x$ .
Solve for x: Solve for x: Subtract $x$ from both sides: $54 + x - x = 12 + 3x - x$ , Simplify: $54 = 12 + 2x$ , Subtract $12$ from both sides: $54 - 12 = 2x$ , $42 = 2x$ , Divide by $2$ : $x = 21$ .
Substitute $x$ into Equation: Substitute $x = 21$ back into either equation to find $y$ . Use the first equation: $y = 54 + 21$ , $y = 75$ .