- Identify the linear relationship between the total cost \(y\), the fixed cost of popcorn \(p=8\), the variable cost of tickets \(t\), and the number of tickets \(x\): \(y = 12x + 8\).
- Use the provided data to confirm \(p + 4t = 56\) and \(p + 6t = 80\), which results in a ticket price of \(t = 12\).
- Combine these values to construct the linear equation \(y = 12x + 8\).
- The final equation is \(\boxed{y = 12x + 8}\).
Explanation
- Problem Analysis
We are given the total cost for a bucket of popcorn and a certain number of movie tickets.
Let \(y\) be the total cost, \(x\) be the number of tickets, \(p\) be the cost of one bucket of popcorn, and \(t\) be the cost of one movie ticket.
The relationship is given by the linear equation:
\[y = p + x \times t\]
We are provided with three pieces of information:
- One bucket of popcorn plus 4 tickets costs \(\$56\): \(p + 4t = 56\).
- One bucket of popcorn plus 6 tickets costs \(\$80\): \(p + 6t = 80\).
- The cost of a bucket of popcorn is \(\$8\): \(p = 8\).
- Calculating the Ticket Cost
We substitute the cost of the popcorn \(p = 8\) into the first equation:
\[8 + 4t = 56\]
Subtract 8 from both sides:
\[4t = 48\]
Divide by 4:
\[t = 12\]
Thus, the cost of one movie ticket is \(\$12\). We can verify this with the second equation:
\[p + 6t = 8 + 6 \times 12 = 8 + 72 = 80\]
This matches the provided information. Therefore, the cost per ticket is indeed \(\$12\).
- Formulating the Equation
Now that we know the constant cost of the popcorn \(p = 8\) and the cost per ticket \(t = 12\), we can substitute these values into the linear equation \(y = p + x \times t\):
\[y = 8 + 12x\]
Rearranging into the standard form \(y = mx + c\), we get:
\[y = 12x + 8\]
Comparing this result to the given options, we find that the equation that represents the total cost is \(y = 12x + 8\).
Examples
Linear equations are used in daily life to determine total costs based on fixed and variable expenses. For instance, if you pay a flat monthly fee for a streaming service plus an extra charge for every premium movie you rent, you can model your total monthly bill using a linear equation \(y = mx + b\), where \(m\) is the cost per movie, \(x\) is the number of movies, and \(b\) is the fixed monthly fee. Understanding this allows you to effectively manage your budget and predict future expenses accurately.