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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 speeds 70 miles per hour. Write an equation for the distance they traveled, y, and the hours they have driven. Indicate whether this is a proportional or nonproportional situation.

answers:

y = 500x + 70, nonproportional

y = 70x + 500, nonproportional

y = 70x + 500, proportional

y = 70x, proportional

3 answers

To write the equation for the distance traveled (y) based on the hours driven (x) after they have already traveled 500 miles, we can express it as follows:

  1. They have already traveled 500 miles.
  2. Each hour after that, they will travel 70 miles.

So, the equation can be represented as: \[ y = 70x + 500 \]

In this equation:

  • \( y \) is the total distance traveled,
  • \( x \) is the number of hours driven after the first 500 miles,
  • 70 is the speed in miles per hour,
  • 500 is the distance already traveled.

Now, we evaluate if this situation is proportional or nonproportional:

  • A proportional relationship would mean that the distance \( y \) is directly proportional to \( x \) (in the form \( y = kx \), where \( k \) is a constant). In this case, because of the +500 (the initial distance), it is not in that form.

Thus, the situation is nonproportional.

The correct answer is: y = 70x + 500, nonproportional.

you sure

Yes, I'm sure. Let's break it down again to ensure clarity:

  1. The family has already traveled 500 miles. This means that any distance they cover afterward adds to that initial distance.

  2. They will travel at a speed of 70 miles per hour after the first day.

  3. If we let \( x \) represent the number of hours driven after they have reached 500 miles, the distance they will cover after that can be expressed as \( 70x \). However, since they have already covered 500 miles before this component starts, we need to add that to the equation.

So the total distance traveled after the initial 500 miles can be modeled by the equation: \[ y = 70x + 500 \]

Proportional vs. Nonproportional

  • A proportional relationship is one that can be represented by an equation of the form \( y = kx \) (where \( k \) is a constant). This means that if \( x = 0 \), then \( y \) would also be 0.
  • A nonproportional relationship means that there is a constant addition or subtraction. In this case, the relationship is nonproportional because when \( x = 0 \) (which means no additional hours have been driven), the distance \( y \) is already at 500 miles, not 0.

So, the equation \( y = 70x + 500 \) is indeed nonproportional because it has a y-intercept (the 500 miles already traveled).

Therefore, the answer y = 70x + 500, nonproportional is correct.