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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineColleen loves riding Ferris wheels and roller coasters. While visiting the Butler County Fair, she first went on the Ferris wheel 44 times and the roller coaster 22 times, using a total of 1818 tickets. Then, after taking a break and having a snack, Colleen went on the Ferris wheel 44 times and the roller coaster 44 times, using a total of 2828 tickets. How many tickets does it take to ride each attraction?\newlineIt takes _\_ tickets to ride the Ferris wheel, and _\_ tickets to ride the roller coaster.

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineColleen loves riding Ferris wheels and roller coasters. While visiting the Butler County Fair, she first went on the Ferris wheel 44 times and the roller coaster 22 times, using a total of 1818 tickets. Then, after taking a break and having a snack, Colleen went on the Ferris wheel 44 times and the roller coaster 44 times, using a total of 2828 tickets. How many tickets does it take to ride each attraction?\newlineIt takes _\_ tickets to ride the Ferris wheel, and _\_ tickets to ride the roller coaster.
  1. Define Equations: Let's denote the number of tickets needed for one ride on the Ferris wheel as ff and for the roller coaster as rr. Colleen's first round of rides, where she went on the Ferris wheel 44 times and the roller coaster 22 times using 1818 tickets, gives us the equation 4f+2r=184f + 2r = 18.
  2. Elimination Method: Colleen's second round of rides, where she went on the Ferris wheel 44 times and the roller coaster 44 times using 2828 tickets, gives us the equation 4f+4r=284f + 4r = 28.
  3. Subtract and Solve: We now have a system of two equations. To use elimination, we need to make the coefficients of one of the variables the same in both equations. However, we notice that the coefficients of ff are already the same, so we can proceed to eliminate ff by subtracting the first equation from the second.
  4. Find Roller Coaster Tickets: Subtracting the first equation from the second, we get (4f+4r)−(4f+2r)=28−18(4f + 4r) - (4f + 2r) = 28 - 18, which simplifies to 2r=102r = 10. Dividing both sides by 22, we find r=5r = 5.
  5. Substitute and Solve: Now that we know r=5r = 5, we can substitute this value back into the first equation to find ff. Substituting into 4f+2r=184f + 2r = 18 gives us 4f+2(5)=184f + 2(5) = 18, which simplifies to 4f+10=184f + 10 = 18.
  6. Substitute and Solve: Now that we know r=5r = 5, we can substitute this value back into the first equation to find ff. Substituting into 4f+2r=184f + 2r = 18 gives us 4f+2(5)=184f + 2(5) = 18, which simplifies to 4f+10=184f + 10 = 18. Subtracting 1010 from both sides of the equation 4f+10=184f + 10 = 18 gives us 4f=84f = 8. Dividing both sides by 44, we find f=2f = 2.

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