Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Denise plans to attend the Boone County Fair and is trying to decide what would be a better deal. She can pay $\$36$ for unlimited rides, or she can pay $\$15$ for admission plus $\$3$ per ride. If Denise goes on a certain number of rides, the two options wind up costing her the same amount. How many rides is that? What is that cost? $\newline$ If Denise goes on _ rides, the two options will both cost $\$$ _.

View step-by-step help

Home

Math Problems

Algebra 1

Solve a system of equations using any method: word problems

Full solution

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

\newline

Denise plans to attend the Boone County Fair and is trying to decide what would be a better deal. She can pay

\$36

for unlimited rides, or she can pay

\$15

for admission plus

\$3

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

\newline

If Denise goes on _____ rides, the two options will both cost

\$

_____.

Define 'r' as number of rides: Let's define the number of rides Denise goes on as 'r ext{'). The cost for unlimited rides is a flat rate of }$\$\(36 ext{. The cost for the pay-per-ride option is }\$\(15\text{ for admission plus }\$\(3\text{ per ride. We need to set up two equations that equal each other since the problem states that the two options end up costing the same amount.} $\newline$ Unlimited rides cost = \$\(36 ext{.} $\newline$ Pay-per-ride cost = \$\(15 + \$\(3$ \times r ext{.}$\newline$So, the equation is:}$\newline$$\$\(36$ = \$$15$ + \$$3$ \times r ext{.}
Solve for 'r': Now, we will solve the equation for 'r' to find out how many rides Denise needs to go on for the costs to be equal.$\newline$$\$36 = \$15 + \$3 \times r$$\newline$Subtract $\$15$ from both sides to isolate the term with 'r':$\newline$$\$36 - \$15 = \$3 \times r$$\newline$$\$21 = \$3 \times r$$\newline$Now, divide both sides by $\$3$ to solve for 'r':$\newline$$\$21 / \$3 = r$$\newline$r = $7$
Calculate cost for $7$ rides: Since we have found that 'r' equals $7$, Denise needs to go on $7$ rides for the costs to be the same. Now we need to calculate what that cost is. We can use either the unlimited rides cost or the pay-per-ride cost equation since they are equal when Denise goes on $7$ rides.$\newline$Using the unlimited rides cost:$\newline$Cost = $\$36$
Confirm cost calculation: Alternatively, using the pay-per-ride cost equation to check our work:$\newline$Cost = $\$15$ + $\$3$ * $r$$\newline$Cost = $\$15$ + $\$3$ * $7$$\newline$Cost = $\$15$ + $\$21$$\newline$Cost = $\$36$$\newline$This confirms our previous cost calculation.