- Identify the function \(y = 16 + 0.5x\) and test each provided ordered pair \((x, y)\) by substituting the \(x\)-coordinate.
- Calculate the \(y\)-value for \(x=5, 8, 0, 10\).
- Find that the only pair matching the equation is \((8, 20)\) because \(16 + 0.5(8) = 20\).
- Conclude that the correct option is \(\boxed{b}\).
Explanation
- Problem Analysis
The objective is to identify which of the given ordered pairs \((x, y)\) satisfies the linear equation \(y = 16 + 0.5x\). An ordered pair \((x, y)\) is a solution to an equation if substituting the given \(x\)-value into the equation results in the corresponding \(y\)-value. We will test each provided option one by one.
- Evaluating the Options
We evaluate each option by substituting the given \(x\)-coordinate into the function \(y = 16 + 0.5x\):
AFor the pair \((5, 19.5)\):
\[y = 16 + 0.5(5) = 16 + 2.5 = 18.5\]
Since \(18.5
eq 19.5\), this pair does not satisfy the function.
BFor the pair \((8, 20)\):
\[y = 16 + 0.5(8) = 16 + 4 = 20\]
Since the result matches the \(y\)-value of the pair (\(20 = 20\)), this pair satisfies the function.
CFor the pair \((0, 18)\):
\[y = 16 + 0.5(0) = 16 + 0 = 16\]
Since \(16
eq 18\), this pair does not satisfy the function.
DFor the pair \((10, 21.5)\):
\[y = 16 + 0.5(10) = 16 + 5 = 21\]
Since \(21
eq 21.5\), this pair does not satisfy the function.
- Conclusion
Based on our calculations, only the pair \((8, 20)\) satisfies the given equation \(y = 16 + 0.5x\). Therefore, the correct option is b).
Examples
In real-life scenarios, linear relationships are common, such as calculating the total cost of a service. For example, if a taxi charges a base fee of \(\$16\) plus \(\$0.50\) for every mile traveled, the total cost \(y\) for traveling \(x\) miles is modeled by \(y = 16 + 0.5x\). Checking ordered pairs ensures that the billing system is accurate and consistent with the defined rates. If you travel 8 miles, the total cost would be \(16 + 0.5(8) = \boxed{\$20}\).