Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Colleen loves riding Ferris wheels and roller coasters. While visiting the Butler County Fair, she first went on the Ferris wheel $4$ times and the roller coaster $2$ times, using a total of $18$ tickets. Then, after taking a break and having a snack, Colleen went on the Ferris wheel $4$ times and the roller coaster $4$ times, using a total of $28$ 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.

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Solve a system of equations using elimination: word problems

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

\newline

Colleen loves riding Ferris wheels and roller coasters. While visiting the Butler County Fair, she first went on the Ferris wheel

4

times and the roller coaster

2

times, using a total of

18

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

4

times and the roller coaster

4

times, using a total of

28

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 Equations: Let's denote the number of tickets needed for one ride on the Ferris wheel as $f$ and for the roller coaster as $r$ . Colleen's first round of rides, where she went on the Ferris wheel $4$ times and the roller coaster $2$ times using $18$ tickets, gives us the equation $4f + 2r = 18$ .
Elimination Method: Colleen's second round of rides, where she went on the Ferris wheel $4$ times and the roller coaster $4$ times using $28$ tickets, gives us the equation $4f + 4r = 28$ .
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 $f$ are already the same, so we can proceed to eliminate $f$ by subtracting the first equation from the second.
Find Roller Coaster Tickets: Subtracting the first equation from the second, we get $(4f + 4r) - (4f + 2r) = 28 - 18$ , which simplifies to $2r = 10$ . Dividing both sides by $2$ , we find $r = 5$ .
Substitute and Solve: Now that we know $r = 5$ , we can substitute this value back into the first equation to find $f$ . Substituting into $4f + 2r = 18$ gives us $4f + 2(5) = 18$ , which simplifies to $4f + 10 = 18$ .
Substitute and Solve: Now that we know $r = 5$ , we can substitute this value back into the first equation to find $f$ . Substituting into $4f + 2r = 18$ gives us $4f + 2(5) = 18$ , which simplifies to $4f + 10 = 18$ . Subtracting $10$ from both sides of the equation $4f + 10 = 18$ gives us $4f = 8$ . Dividing both sides by $4$ , we find $f = 2$ .