Question

a bowling alley charges $14 for the first game, $10 for the second game, and $5 per game for every game after that.

which statement is true, based on the given description?

• the description shows a linear relationship and a proportional relationship.
• the description does not show a linear relationship or a proportional relationship.
• the description shows a proportional relationship, but not a linear relationship.
• the description shows a linear relationship, but not a proportional relationship.

Explanation:

Calculate total cost for games

Let \(x\) be the number of games played, and \(y\) be the total cost in dollars.

• For 1 game: \(y = 14\)
• For 2 games: \(y = 14 + 10 = 24\)
• For 3 games: \(y = 24 + 5 = 29\)
• For 4 games: \(y = 29 + 5 = 34\)

Analyze rate of change

A linear relationship requires a constant rate of change (slope) between any two points.

• Rate of change from 1 to 2 games:

$$\frac{24 - 14}{2 - 1} = 10$$

• Rate of change from 2 to 3 games:

$$\frac{29 - 24}{3 - 2} = 5$$

Since the rate of change is not constant (\(10
eq 5\)), the relationship is not linear.

Analyze proportionality

A proportional relationship must be linear and pass through the origin \((0,0)\), meaning the ratio \(\frac{y}{x}\) is constant.

• For 1 game: \(\frac{14}{1} = 14\)
• For 2 games: \(\frac{24}{2} = 12\)

Since the ratio \(\frac{y}{x}\) is not constant, the relationship is not proportional.

Determine the correct statement

Since the relationship is neither linear nor proportional, the correct statement is:
"The description does not show a linear relationship or a proportional relationship."

Answer:

• The description shows a linear relationship and a proportional relationship.
• The description does not show a linear relationship or a proportional relationship. (Correct answer)
• The description shows a proportional relationship, but not a linear relationship.
• The description shows a linear relationship, but not a proportional relationship.