Question

martin is on the boardwalk at the beach. his two favorite rides there are the ferris wheel and the roller coaster. if he rides them each once, it takes 14 minutes total. today, however, he rides the ferris wheel 6 times and the roller coaster 9 times, which means he’s riding those attractions for exactly an hour and a half. this system of equations can be used to represent the situation: $x + y = 14$ $6x + 9y = 90$ which statement is correct? in the system of equations, x represents the amount of time, in minutes, it takes to ride the ferris wheel once; and y represents the amount of time, in minutes, it takes to ride the roller coaster once. in the system of equations, x represents the amount of time, in minutes, it takes to ride the roller coaster once; and y represents the amount of time, in minutes, it takes to ride the ferris wheel once. in the system of equations, x represents the amount of time, in minutes, that martin spends riding the ferris wheel today; and y represents the amount of time, in minutes, he spends riding the roller coaster today. in the system of equations, x represents the amount of time, in minutes, that martin spends riding the roller coaster today; and y represents the amount of time, in minutes, he spends riding the ferris wheel today. how much time does martin spend riding the ferris wheel today? ____ minutes

Explanation:

Part 1: Which statement is correct?

• Analyze the first equation \(x + y = 14\): It represents riding Ferris wheel once (\(x\)) and roller coaster once (\(y\)) taking 14 minutes total.
• Analyze the second equation \(6x + 9y = 90\): \(6x\) is 6 times Ferris wheel ride time, \(9y\) is 9 times roller coaster ride time, summing to 90 minutes (1.5 hours).
• Check options:
• Option 1: Matches the meaning of \(x\) (Ferris wheel once) and \(y\) (roller coaster once).
• Option 2: Swaps \(x\) and \(y\), inconsistent with equations.
• Option 3: \(x\) here would be total today, but \(6x\) in second equation implies \(x\) is per ride, not total.
• Option 4: Similar to option 3, misinterprets \(x\) and \(y\) as total today, wrong.

Step 1: Solve the system of equations. We have:

\[

$$\begin{cases} x + y = 14 \\ 6x + 9y = 90 \end{cases}$$

\]
From the first equation, express \(y\) in terms of \(x\): \(y = 14 - x\).

Step 2: Substitute \(y = 14 - x\) into the second equation:

\[
6x + 9(14 - x) = 90
\]
Expand the equation:
\[
6x + 126 - 9x = 90
\]
Combine like terms:
\[
-3x + 126 = 90
\]
Subtract 126 from both sides:
\[
-3x = 90 - 126 = -36
\]
Divide both sides by \(-3\):
\[
x = \frac{-36}{-3} = 12
\]

Step 3: Find the total time for Ferris wheel today. Martin rides Ferris wheel 6 times, so total time is \(6x\). Substitute \(x = 12\):

\[
6x = 6 \times 12 = 72
\]

Answer:

In the system of equations, \(x\) represents the amount of time, in minutes, it takes to ride the Ferris wheel once; and \(y\) represents the amount of time, in minutes, it takes to ride the roller coaster once. (The first option among the given statements)

Part 2: How much time does Martin spend riding the Ferris wheel today?