- Define the variable \(x\) as the number of 30-minute sessions and \(x-2\) as the number of 60-minute sessions.
- Set up the equation \(15x + 25(x-2) = 230\) representing total earnings.
- Simplify and solve the equation: \(40x = 280\), which results in \(x = 7\).
- Final value is \(\boxed{7}\).
Explanation
- Analysis and Setting Up the Equation
The problem asks for the number of thirty-minute sessions, denoted by \(x\). We are given that Carly earns \(\$15\) for each thirty-minute session and \(\$25\) for each sixty-minute session. The number of sixty-minute sessions is given as \(x-2\). We know the total earnings for the weekend are \(\$230\). We can represent this situation with an algebraic equation, where the total earnings are the sum of the earnings from both types of sessions. The earnings from the thirty-minute sessions are \(15 \times x\) and the earnings from the sixty-minute sessions are \(25 \times (x-2)\). Therefore, the equation for the total earnings is:
\[15x + 25(x - 2) = 230\]
- Solving the Equation for x
To solve for \(x\), we first distribute the \(\$25\) to the terms inside the parentheses:
\[15x + 25x - 50 = 230\]
Next, combine the like terms involving \(x\):
\[40x - 50 = 230\]
To isolate the term with \(x\), add \(50\) to both sides of the equation:
\[40x = 230 + 50\]
\[40x = 280\]
Finally, divide by \(40\) to find the value of \(x\):
\[x = \frac{280}{40}\]
\[x = 7\]
- Verification and Final Answer
We found that \(x = 7\). This means Carly had \(7\) thirty-minute sessions. We can verify this by checking the number of sixty-minute sessions, which is \(x - 2 = 7 - 2 = 5\). The total earnings would be:
\[15 \times 7 + 25 \times 5 = 105 + 125 = 230\]
This matches the total earnings of \(\$230\). Comparing our result to the given options, we select 7. Therefore, the value of \(x\) is \(\boxed{7}\).
Examples
This problem demonstrates how to model real-world scenarios using linear equations. Similar techniques are used in business to calculate production costs, manage budgets, or determine profitability when different products have different pricing structures. By breaking down costs and quantities into variables, you can effectively solve for unknown values and optimize your decision-making process.