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A family is going from Dallas, Texas, to Los Angeles, California. They have traveled 500 miles so far. The next day, their average speed is 70 miles per hour. Write an equation for the distance they traveled, y

, and the hours they have driven, x
. Indicate whether this is a proportional or a nonproportional situation.

1 answer

To write an equation for the distance traveled (\(y\)) as a function of the hours driven (\(x\)), we can start with the information provided.

  1. Distance Already Traveled: The family has already traveled 500 miles.
  2. Average Speed on the Next Day: The average speed is 70 miles per hour.

The total distance traveled after \(x\) hours can be expressed as:

\[ y = 500 + 70x \]

Here, \(y\) represents the total distance traveled, and \(x\) represents the additional hours driven at 70 miles per hour.

Proportional or Nonproportional

To determine whether this situation is proportional or nonproportional, we need to analyze the equation. A proportional relationship can be expressed in the form \(y = kx\) for some constant \(k\), where the ratio of \(y\) to \(x\) is constant and passes through the origin (0,0).

In the equation \(y = 500 + 70x\):

  • The presence of the constant \(500\) indicates that the relationship does not pass through the origin (when \(x = 0\), \(y = 500\)).
  • Therefore, the \(y\)-intercept of \(500\) means that there is an initial distance traveled before they start the next leg of the journey.

Since the relationship does not pass through the origin and includes a constant term, this is a nonproportional situation.

Summary

The equation for the distance traveled is:

\[ y = 500 + 70x \]

This is a nonproportional situation.