Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Stanley loves riding Ferris wheels and roller coasters. While visiting the Harrison County Fair, he first went on the Ferris wheel $5$ times and the roller coaster $2$ times, using a total of $20$ tickets. Then, after taking a break and having a snack, Stanley went on the Ferris wheel $2$ times and the roller coaster $5$ times, using a total of $29$ 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 any method: word problems

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

\newline

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

5

times and the roller coaster

2

times, using a total of

20

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

2

times and the roller coaster

5

times, using a total of

29

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: Let $x$ be the number of tickets needed for one ride on the Ferris wheel, and $y$ be the number of tickets needed for one ride on the roller coaster. Stanley first went on the Ferris wheel $5$ times and the roller coaster $2$ times, using a total of $20$ tickets. This gives us the first equation: $\newline$ $5x + 2y = 20$
First Equation: Then, Stanley went on the Ferris wheel $2$ times and the roller coaster $5$ times, using a total of $29$ tickets. This gives us the second equation: $\newline$ $2x + 5y = 29$
Second Equation: We now have a system of equations: $\newline$ $5x + 2y = 20$ $\newline$ $2x + 5y = 29$ $\newline$ We can solve this system using either substitution or elimination. Let's use the elimination method.
Elimination Method: Multiply the first equation by $2$ and the second equation by $5$ to make the coefficients of $x$ the same: $\newline$ $(5x + 2y) \times 2 = 20 \times 2$ $\newline$ $(2x + 5y) \times 5 = 29 \times 5$ $\newline$ This gives us: $\newline$ $10x + 4y = 40$ $\newline$ $10x + 25y = 145$
Multiply Equations: Subtract the second new equation from the first new equation to eliminate $x$ : $(10x + 4y) - (10x + 25y) = 40 - 145$ $10x + 4y - 10x - 25y = -105$ $-21y = -105$
Eliminate $x$ : Solve for $y$ :$\newline$ $-21$ $y$ = $-105$ $\newline$ $y$ = $-105$ / $-21$ $\newline$ $y$ = $5$
Solve for y: Now that we have the value for $y$ , we can substitute it back into one of the original equations to solve for $x$ . Let's use the first equation: $\newline$ $5x + 2y = 20$ $\newline$ $5x + 2(5) = 20$ $\newline$ $5x + 10 = 20$
Substitute into Equation: Solve for $x$ : $5x + 10 = 20$ $5x = 20 - 10$ $5x = 10$ $x = \frac{10}{5}$ $x = 2$