Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Evan wants to take group fitness classes at a nearby gym, but needs to start by selecting a membership plan. With the first membership plan, Evan can pay $\$48$ per month, plus $\$2$ for each group class he attends. Alternately, he can get the second membership plan and pay $\$8$ per month plus $\$6$ per class. If Evan attends a certain number of classes in a month, the two membership plans end up costing the same total amount. What is that total amount? How many classes per month is that? $\newline$ Each membership plan costs $\$\_\_\_\_\_$ if Evan takes $\_\_\_\_\_$ classes per month.

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Solve a system of equations using any method: word problems

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Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

Evan wants to take group fitness classes at a nearby gym, but needs to start by selecting a membership plan. With the first membership plan, Evan can pay

\$48

per month, plus

\$2

for each group class he attends. Alternately, he can get the second membership plan and pay

\$8

per month plus

\$6

per class. If Evan attends a certain number of classes in a month, the two membership plans end up costing the same total amount. What is that total amount? How many classes per month is that?

\newline

Each membership plan costs

\$\_\_\_\_\_

if Evan takes

\_\_\_\_\_

classes per month.

Define Variables: Let's define the variables: $\newline$ Let $x$ be the number of classes Evan attends per month. $\newline$ Let $C$ be the total cost for each membership plan when they are equal. $\newline$ Now, we can write the equations for each membership plan based on the given information: $\newline$ First membership plan: $\$48$ per month + $\$2$ per class $\newline$ Second membership plan: $\$8$ per month + $\$6$ per class $\newline$ The equations representing the total cost for each plan are: $\newline$ First membership plan: $C = 48 + 2x$ $\newline$ Second membership plan: $C = 8 + 6x$ $\newline$ Since the total cost is the same for both plans, we can set the equations equal to each other: $\newline$ $48 + 2x = 8 + 6x$
Write Equations: Now, we will solve for $x$ by isolating the variable: $\newline$ Subtract $2x$ from both sides: $\newline$ $48 + 2x - 2x = 8 + 6x - 2x$ $\newline$ $48 = 8 + 4x$ $\newline$ Subtract $8$ from both sides: $\newline$ $48 - 8 = 8 + 4x - 8$ $\newline$ $40 = 4x$ $\newline$ Divide both sides by $4$ : $\newline$ $40 / 4 = 4x / 4$ $\newline$ $10 = x$ $\newline$ So, Evan attends $2x$ $0$ classes per month.
Solve for x: Now that we know Evan attends $10$ classes per month, we can find the total cost $C$ for each membership plan. $\newline$ Using the first membership plan's equation: $\newline$ $C = 48 + 2x$ $\newline$ $C = 48 + 2(10)$ $\newline$ $C = 48 + 20$ $\newline$ $C = 68$ $\newline$ So, the total cost for each membership plan is $\$68$ when Evan takes $10$ classes per month.