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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Kenny plans to attend the Marshall County Fair and is trying to decide what would be a better deal. He can pay $\$30$ for unlimited rides, or he can pay $\$14$ for admission plus $\$1$ per ride. If Kenny goes on a certain number of rides, the two options wind up costing him the same amount. What is that cost? How many rides is that? $\newline$ It will cost Kenny $\$$ _ for both options if he goes on _ rides.

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Solve a system of equations using substitution: word problems

Full solution

Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.

\newline

Kenny plans to attend the Marshall County Fair and is trying to decide what would be a better deal. He can pay

\$30

for unlimited rides, or he can pay

\$14

for admission plus

\$1

per ride. If Kenny goes on a certain number of rides, the two options wind up costing him the same amount. What is that cost? How many rides is that?

\newline

It will cost Kenny

\$

_____ for both options if he goes on _____ rides.

Define Variables: Let's define the variables: $\newline$ Let $x$ be the number of rides Kenny goes on. $\newline$ Let $y$ be the total cost for Kenny. $\newline$ We have two options for Kenny: $\newline$ Option $1$ : Unlimited rides for $\$30$ . $\newline$ Option $2$ : Admission for $\$14$ plus $\$1$ per ride. $\newline$ We can write two equations to represent these options: $\newline$ For Option $1$ , the cost is always $\$30$ , regardless of the number of rides: $\newline$ $y = 30$ $\newline$ For Option $2$ , the cost is $\$14$ plus $\$1$ for each ride: $\newline$ $y = 14 + x$ $\newline$ Now we have our system of equations: $\newline$ $y = 30$ $\newline$ $y = 14 + x$ $\newline$ We will use substitution to solve for $x$ .
Options for Kenny: Since $y$ is the same in both equations, we can set them equal to each other to find the number of rides where the cost is the same: $\newline$ $30 = 14 + x$ $\newline$ Now we solve for $x$ : $\newline$ $30 - 14 = x$ $\newline$ $16 = x$ $\newline$ Kenny will go on $16$ rides for the costs to be the same.
Equations Representation: Now we need to find the total cost $y$ when Kenny goes on $16$ rides. We can use either equation, but let's use the second one since it includes $x$ :
$y = 14 + x$
$y = 14 + 16$
$y = 30$

So, the total cost for both options is $\$30$ when Kenny goes on $16$ rides.

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