Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Candice loves riding Ferris wheels and roller coasters. While visiting the Lincoln County Fair, she first went on the Ferris wheel $4$ times and the roller coaster $4$ times, using a total of $32$ tickets. Then, after taking a break and having a snack, Candice went on the Ferris wheel $5$ times and the roller coaster $2$ times, using a total of $25$ 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

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

4

times and the roller coaster

4

times, using a total of

32

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

5

times and the roller coaster

2

times, using a total of

25

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.

Equations Setup: Let's denote the number of tickets for one ride on the Ferris wheel as $F$ and the number of tickets for one ride on the roller coaster as $R$ . We can then write two equations based on the information given: $\newline$ $1$ ) For the first round of rides: $4F + 4R = 32$ $\newline$ $2$ ) For the second round of rides: $5F + 2R = 25$
Elimination Method: To solve this system using elimination, we want to eliminate one of the variables. We can do this by multiplying the first equation by a number that will make the coefficient of $R$ the same in both equations. If we multiply the first equation by $\frac{1}{2}$ , we get: $\newline$ $\left(\frac{1}{2}\right)(4F + 4R) = \left(\frac{1}{2}\right)(32)$ $\newline$ $2F + 2R = 16$
New System of Equations: Now we have a new system of equations: $\newline$ $1$ ) $2F + 2R = 16$ $\newline$ $2$ ) $5F + 2R = 25$ $\newline$ We can eliminate $R$ by subtracting the first equation from the second: $\newline$ $(5F + 2R) - (2F + 2R) = 25 - 16$ $\newline$ $3F = 9$
Solving for F: Divide both sides of the equation by $3$ to solve for F: $\newline$ $\frac{3F}{3} = \frac{9}{3}$ $\newline$ $F = 3$ $\newline$ So, it takes $3$ tickets to ride the Ferris wheel.
Substitute F into Equation: Now that we know the value of F, we can substitute it back into one of the original equations to find R. Let's use the first equation: $\newline$ $4F + 4R = 32$ $\newline$ $4(3) + 4R = 32$ $\newline$ $12 + 4R = 32$
Solving for R: Subtract $12$ from both sides of the equation to solve for R: $\newline$ $12 + 4R - 12 = 32 - 12$ $\newline$ $4R = 20$
Solving for R: Subtract $12$ from both sides of the equation to solve for R: $\newline$ $12 + 4R - 12 = 32 - 12$ $\newline$ $4R = 20$ Divide both sides of the equation by $4$ to solve for R: $\newline$ $\frac{4R}{4} = \frac{20}{4}$ $\newline$ $R = 5$ $\newline$ So, it takes $5$ tickets to ride the roller coaster.