Answered


Number of cars		Maximum number of passengers and crew

3		174

5		284

10		559






The table shows the linear relationship between the number of cars, $c$, on a commuter train and the maximum number of passengers and crew, $p$, that the train can carry. Which equation represents the linear relationship between $c$ and $p$?


A$55 c-p=-9$
B$55 c-p=9$
C$55 p-c=-9$
D$55 p-c=9$

Verified Answer :

  • Examine the provided data points \((3, 174)\), \((5, 284)\), and \((10, 559)\) for a linear relationship between the number of cars \(c\) and passengers \(p\).
  • Test the candidate equations against these points to check for validity.
  • Observe that the equation \(55c - p = -9\) correctly models all data points.
  • The correct option is A, \(\boxed{A}\).

Explanation

  1. Problem Analysis and Data Review

The problem provides a table showing the relationship between the number of cars (\(c\)) on a commuter train and the maximum number of passengers and crew (\(p\)). We are given the following data points \((c, p)\): \((3, 174)\), \((5, 284)\), and \((10, 559)\). We are told that there is a linear relationship between \(c\) and \(p\), and we must identify which of the given linear equations correctly models this relationship.

  1. Testing the Linear Equations with Data Points

To determine which equation represents the relationship, we test each of the provided options against the given data points:

  • Option A: \(55c - p = -9\)
  • Option B: \(55c - p = 9\)
  • Option C: \(55p - c = -9\)
  • Option D: \(55p - c = 9\)

For the first data point \((3, 174)\):

  • Option A: \(55(3) - 174 = 165 - 174 = -9\). This is true.
  • Option B: \(55(3) - 174 = 165 - 174 = -9 eq 9\). This is false.
  • Option C: \(55(174) - 3 = 9570 - 3 = 9567 eq -9\). This is false.
  • Option D: \(55(174) - 3 = 9570 - 3 = 9567 eq 9\). This is false.

Only Option A holds true for the first point.

  1. Verifying the Correct Equation

We must ensure that Option A holds true for the remaining data points to confirm it is the correct relationship.

For the second data point \((5, 284)\):

  • \(55(5) - 284 = 275 - 284 = -9\). This is true.

For the third data point \((10, 559)\):

  • \(55(10) - 559 = 550 - 559 = -9\). This is true.

Since the equation \(55c - p = -9\) satisfies all three data points, it is the correct model for the relationship between the number of cars and the number of passengers and crew.

Examples

Linear relationships are used in everyday life for planning and budgeting. For instance, if you are organizing a school trip, you might find that the total number of seats required is linearly related to the number of buses rented. By creating a table of data points and determining the linear equation, you can predict the total capacity for any number of buses, helping you allocate resources efficiently and ensure that every student and teacher has a seat. This is the same logic used to design transit systems and manage fleet logistics.

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