Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Herman loves riding Ferris wheels and roller coasters. While visiting the Union County Fair, he first went on the Ferris wheel $1$ time and the roller coaster $2$ times, using a total of $11$ tickets. Then, after taking a break and having a snack, Herman went on the Ferris wheel $4$ times and the roller coaster $2$ times, using a total of $20$ tickets. How many tickets does it take to ride each attraction? $\newline$ It takes $\_$ tickets to ride the Ferris wheel, and $\_$ tickets to ride the roller coaster.

Full solution

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

\newline

Herman loves riding Ferris wheels and roller coasters. While visiting the Union County Fair, he first went on the Ferris wheel

1

time and the roller coaster

2

times, using a total of

11

tickets. Then, after taking a break and having a snack, Herman went on the Ferris wheel

4

times and the roller coaster

2

times, using a total of

20

tickets. How many tickets does it take to ride each attraction?

\newline

It takes

\_

tickets to ride the Ferris wheel, and

\_

tickets to ride the roller coaster.

Define variables: Define the variables for the number of tickets needed for each ride. $\newline$ Let $x$ represent the number of tickets needed for the Ferris wheel. $\newline$ Let $y$ represent the number of tickets needed for the roller coaster.
Write equations: Write the system of equations based on the given information. $\newline$ From the first visit: $1x + 2y = 11$ (Herman rode the Ferris wheel once and the roller coaster twice) $\newline$ From the second visit: $4x + 2y = 20$ (Herman rode the Ferris wheel four times and the roller coaster twice)
Elimination method: Use the elimination method to solve the system of equations. $\newline$ To eliminate $y$ , we can subtract the first equation from the second equation. $\newline$ $(4x + 2y) - (1x + 2y) = 20 - 11$ $\newline$ $4x + 2y - 1x - 2y = 20 - 11$ $\newline$ $3x = 9$
Solve for x: Solve for x. $\newline$ $3x = 9$ $\newline$ $x = \frac{9}{3}$ $\newline$ $x = 3$
Substitute and solve for $y$ : Substitute the value of $x$ into one of the original equations to solve for $y$ . $\newline$ Using the first equation: $1x + 2y = 11$ $\newline$ $1(3) + 2y = 11$ $\newline$ $3 + 2y = 11$ $\newline$ $2y = 11 - 3$ $\newline$ $2y = 8$ $\newline$ $y = \frac{8}{2}$ $\newline$ $y = 4$
Check solution: Check the solution by substituting both values into the second equation. $\newline$ $4x + 2y = 20$ $\newline$ $4(3) + 2(4) = 20$ $\newline$ $12 + 8 = 20$ $\newline$ $20 = 20$ $\newline$ The solution checks out.