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:

Step1: Define total cost function

Let $x$ = number of games, $C(x)$ = total cost.
For $x=1$: $C(1)=14$
For $x=2$: $C(2)=14+10=24$
For $x\geq3$: $C(x)=24 + 5(x-2) = 5x + 14$

Step2: Check linearity

A linear relationship follows $C(x)=mx+b$. For $x\geq2$, the rate of change (slope $m=5$) is constant, but the cost changes by $10$ (not $5$) between $x=1$ and $x=2$. This means there is no single constant slope for all $x\geq1$, so it is not a linear relationship.

Step3: Check proportionality

A proportional relationship has $C(x)=mx$ (passes through origin). Here $C(1)=14
eq m(1)$ if $m=5$ or $10$, so it is not proportional.

Answer:

The description does not show a linear relationship or a proportional relationship.