You are choosing between two gyms. One gym offers membership for a fee of \$40 plus a monthly fee of \$25. The other offers membership for a fee of \$15 plus a monthly fee of \$30. After how many months will the total cost at each gym be the same? What will be the total cost for each gym?

검증된 단계별 안내

Define variables to represent the total cost for each gym after $ m $ months. Let $ C_1 $ be the total cost for the first gym and $ C_2 $ be the total cost for the second gym.

Write expressions for the total cost of each gym: For the first gym, the total cost is the initial fee plus the monthly fee times the number of months, so $ C_1 = 40 + 25m $. For the second gym, the total cost is $ C_2 = 15 + 30m $.

Set the two expressions equal to each other to find the number of months $ m $ when the costs are the same: $ 40 + 25m = 15 + 30m $.

Solve the equation for $ m $ by isolating the variable: subtract $ 25m $ and 15 from both sides to get $ 40 - 15 = 30m - 25m $, which simplifies to $ 25 = 5m $. Then divide both sides by 5 to find $ m $.

Once you have the value of $ m $, substitute it back into either $ C_1 $ or $ C_2 $ to find the total cost at that time.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Linear Equations

Linear equations represent relationships where the variables appear to the first power and graph as straight lines. In this problem, the total cost for each gym can be expressed as a linear equation in terms of months, combining fixed fees and monthly charges.

Setting Equations Equal to Find Intersection

To find when two costs are the same, set their linear expressions equal and solve for the variable. This represents the point where the two cost lines intersect, indicating the number of months when both gym costs match.

Substitution to Find Total Cost

After finding the number of months, substitute this value back into either linear equation to calculate the total cost at that time. This confirms the cost at which both gym memberships are equal.