You want to join a health club. Club A charges \( \$ 65 \) per month and has no initiation fee. Club \( B \) costs \( \$ 40 \) a month and has an initiation fee of \( \$ 100 \). After how many months would the costs for the 2 clubs be the same? a) Determine your variables. Explain what each variable means. Write a system of equations to represent the scenario. c) Solve the system of equations using either substitution or elimination. d) Check your solutions in both original equations. e) Write your answer in a complete sentence.

Let's break down this problem step-by-step in a fun and engaging way! To find out when the costs for both clubs will be the same, we first need to determine our variables. Let's say \( m \) is the number of months you join the club. For Club A, the total cost will be \( 65m \) (since it has no initiation fee), while for Club B, it's \( 40m + 100 \) (the monthly fee plus the initiation fee). Our equations looking like this: 1. **Club A:** \( C_A = 65m \) 2. **Club B:** \( C_B = 40m + 100 \) Now, setting both costs equal to each other gives us the equation: \[ 65m = 40m + 100 \] Let’s solve it step by step! We'll subtract \( 40m \) from both sides: \[ 65m - 40m = 100 \] \[ 25m = 100 \] Now, divide both sides by 25: \[ m = \frac{100}{25} = 4 \] So after 4 months, the costs for both clubs will be the same! Now, checking our solution in both original equations: For Club A: \( C_A = 65 \times 4 = 260 \) For Club B: \( C_B = 40 \times 4 + 100 = 160 + 100 = 260 \) Both clubs indeed cost $260 after four months. In conclusion, after 4 months, the costs of Club A and Club B will be the same at $260!