To represent the situation algebraically, we can derive a linear equation based on the information given.
The total cost \( C \) can be represented as a function of the number of months \( m \) you have a gym membership.
From your description, we know:
- The y-intercept \( (0, 50) \) represents the one-time join fee. So, the join fee is \( 50 \).
- The graph also gives us another point \( (6, 140) \), which indicates that after 6 months, the total cost is \( 140 \).
We can use the point-slope form of a linear equation to derive the equation that models this situation.
Let’s denote:
- \( C \) as the total cost
- \( m \) as the number of months
- \( b \) as the one-time join fee (which we already know is \( 50 \))
- \( m \) as the monthly membership fee
Now, we can establish two points based on the information we have:
- The y-intercept (0, 50)
- The point (6, 140)
The general form of the linear equation is given by: \[ C = m \cdot T + b \] Where \( T \) is the number of months.
From the point (6, 140): \[ 140 = 6m + 50 \]
Now, we can solve for \( m \): \[ 140 - 50 = 6m \] \[ 90 = 6m \] \[ m = 15 \]
Now that we have \( m \), we can plug it back into the model: \[ C = 15T + 50 \]
Thus, the total cost \( C \) in terms of the number of months \( T \) is: \[ C = 15T + 50 \]
Where:
- \( 50 \) is the one-time join fee,
- \( 15 \) is the monthly membership fee.