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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineHerman loves riding Ferris wheels and roller coasters. While visiting the Union County Fair, he first went on the Ferris wheel 11 time and the roller coaster 22 times, using a total of 1111 tickets. Then, after taking a break and having a snack, Herman went on the Ferris wheel 44 times and the roller coaster 22 times, using a total of 2020 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.\newlineHerman loves riding Ferris wheels and roller coasters. While visiting the Union County Fair, he first went on the Ferris wheel 11 time and the roller coaster 22 times, using a total of 1111 tickets. Then, after taking a break and having a snack, Herman went on the Ferris wheel 44 times and the roller coaster 22 times, using a total of 2020 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 variables: Define the variables for the number of tickets needed for each ride.\newlineLet xx represent the number of tickets needed for the Ferris wheel.\newlineLet yy represent the number of tickets needed for the roller coaster.
  2. Write equations: Write the system of equations based on the given information.\newlineFrom the first visit: 1x+2y=111x + 2y = 11 (Herman rode the Ferris wheel once and the roller coaster twice)\newlineFrom the second visit: 4x+2y=204x + 2y = 20 (Herman rode the Ferris wheel four times and the roller coaster twice)
  3. Elimination method: Use the elimination method to solve the system of equations.\newlineTo eliminate yy, we can subtract the first equation from the second equation.\newline(4x+2y)−(1x+2y)=20−11(4x + 2y) - (1x + 2y) = 20 - 11\newline4x+2y−1x−2y=20−114x + 2y - 1x - 2y = 20 - 11\newline3x=93x = 9
  4. Solve for x: Solve for x.\newline3x=93x = 9\newlinex=93x = \frac{9}{3}\newlinex=3x = 3
  5. Substitute and solve for yy: Substitute the value of xx into one of the original equations to solve for yy.\newlineUsing the first equation: 1x+2y=111x + 2y = 11\newline1(3)+2y=111(3) + 2y = 11\newline3+2y=113 + 2y = 11\newline2y=11−32y = 11 - 3\newline2y=82y = 8\newliney=82y = \frac{8}{2}\newliney=4y = 4
  6. Check solution: Check the solution by substituting both values into the second equation.\newline4x+2y=204x + 2y = 20\newline4(3)+2(4)=204(3) + 2(4) = 20\newline12+8=2012 + 8 = 20\newline20=2020 = 20\newlineThe solution checks out.

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