- To find the miles traveled in part 1b, substitute \(y = 35\) into \(y = 2.75x + 7.50\), which results in \(x = \boxed{10}\) miles.
- For part 9, the table represents a non-linear relationship because the first differences are not constant (\(3, 12, 48, 192\)); specifically, it is an exponential relationship defined by \(y = 4^x\).
- For part 11, the equation \(y = 3x\) is a linear relationship because it maintains a constant rate of change (slope) of 3, fitting the form \(y = mx + b\).
Explanation
- Analysis and Problem Breakdown
The problem asks us to solve three distinct tasks involving linear and non-linear relationships.
First, for the cab fare equation \(y = 2.75x + 7.50\), where \(y\) is the cost in dollars and \(x\) is the distance in miles, we need to find the number of miles traveled when the cost is \(\$35\).
Second, for the table provided in problem 9 (with inputs \(x = 0, 1, 2, 3, 4\) and outputs \(y = 1, 4, 16, 64, 256\)), we need to determine if the relationship is linear or non-linear and explain the reason.
Third, for the equation in problem 11, \(y = 3x\), we need to determine if it is linear or non-linear and explain why.
For problem 10, no information is provided, so it will be omitted from the analysis.
- Solving for Cab Miles (Part 1b)
We are given the linear equation \(y = 2.75x + 7.50\). To find the number of miles \(x\) for a total cost of \(y = 35\), we set up the equation:
\[35 = 2.75x + 7.50\]
Subtract 7.50 from both sides:
\[35 - 7.50 = 2.75x\]
\[27.50 = 2.75x\]
Divide both sides by 2.75:
\[x = \frac{27.50}{2.75}\]
\[x = 10\]
So, the customer traveled 10 miles.
- Analyzing the Table (Part 9)
The table provides:
\(x=0, y=1\)
\(x=1, y=4\)
\(x=2, y=16\)
\(x=3, y=64\)
\(x=4, y=256\)
A relationship is linear if the rate of change \(\Delta y / \Delta x\) is constant.
\(\Delta y_1 = 4 - 1 = 3\)
\(\Delta y_2 = 16 - 4 = 12\)
Since \(3
eq 12\), the rate of change is not constant. Therefore, the relationship is non-linear. Specifically, notice that each output is a power of 4 (\(y = 4^x\)), making it an exponential relationship.
- Analyzing the Equation (Part 11)
The equation is \(y = 3x\). This equation is in the form \(y = mx + b\), where \(m = 3\) and \(b = 0\). Since the slope \(m\) is a constant value of 3, the rate of change is constant. Thus, the relationship is linear.
Examples
Understanding linear and non-linear relationships is essential in everyday life. For instance, when budgeting for travel, fixed costs plus a per-mile fee (like a taxi fare) follow a linear pattern, allowing you to easily predict expenses. Conversely, many natural processes, such as the spread of a virus or the growth of a bank account with compound interest, follow exponential (non-linear) patterns, where the rate of growth itself increases over time. Recognizing these patterns helps in better planning and decision-making in finance and science.