- Variables: \(x\) is the count of small bottles, \(y\) is the count of large bottles.
- Intercepts: The x-intercept \((180, 0)\) means \(180\) small bottles, and the y-intercept \((0, 90)\) means \(90\) large bottles.
- Reasonable solutions: Any pair \((x, y)\) where \(x\) and \(y\) are non-negative integers and \(y = 90 - 0.5x\), which requires \(x\) to be even.
- Final answer: The line connects \((0, 90)\) and \((180, 0)\), with integer points as feasible solutions. \(\boxed{x=180, y=90}\)
Explanation
- Defining the Variables
The equation \(0.05x + 0.1y = 9\) represents Issay's earnings from recycling. Here, \(x\) is the number of bottles less than \(24\) fluid ounces, and \(y\) is the number of bottles \(24\) fluid ounces or greater. Each bottle represented by \(x\) contributes \(\$0.05\) to the total, and each bottle represented by \(y\) contributes \(\$0.10\). Together, these amounts sum to \(\$9\).
- Calculating and Interpreting Intercepts
To find the x-intercept, we set \(y = 0\) in the equation \(0.05x + 0.1(0) = 9\), which gives \(0.05x = 9\), so \(x = 180\). This point, \((180, 0)\), represents the scenario where Issay recycles only small bottles; in this case, the maximum is \(180\) small bottles.
To find the y-intercept, we set \(x = 0\) in the equation \(0.05(0) + 0.1y = 9\), which gives \(0.1y = 9\), so \(y = 90\). This point, \((0, 90)\), represents the scenario where Issay recycles only large bottles; in this case, the maximum is \(90\) large bottles. The intercepts mark the boundaries of our possible recycling combinations.
- Identifying Reasonable Solutions
A point \((x, y)\) is a reasonable solution if \(x\) and \(y\) are both non-negative integers. This is because it is impossible to recycle a fractional part of a bottle. Rearranging the equation \(0.05x + 0.1y = 9\) gives \(y = \frac{9 - 0.05x}{0.1} = 90 - 0.5x\). For \(y\) to be an integer, \(x\) must be an even number. Some examples of reasonable solutions include:
- If \(x = 0\), \(y = 90\).
- If \(x = 20\), \(y = 90 - 0.5(20) = 80\).
- If \(x = 100\), \(y = 90 - 0.5(100) = 40\).
- If \(x = 180\), \(y = 0\).
These coordinate pairs represent the different combinations of small and large bottles that yield total earnings of \(\$9\).
Examples
Understanding linear relationships helps in real-world resource management and planning. For example, if you are planning a budget for a school event with a limited total, you can model the relationship between two types of items (e.g., snacks and drinks) using a linear equation. By finding the intercepts, you determine the maximum possible quantity of one item if the other is not purchased, and the integer solutions reveal all the feasible combinations that stick exactly to your budget. This approach ensures your resources are distributed optimally and your constraints are met.