Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Keenan and Darell went to an arcade where the machines took tokens. Keenan played $10$ games of skee ball and $6$ games of pinball, using a total of $28$ tokens. At the same time, Darell played $8$ games of skee ball and $4$ games of pinball, using up $20$ tokens. How many tokens does each game require? $\newline$ Every game of skee ball requires $\_$ tokens, and every game of pinball requires $\_$ tokens.

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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

Keenan and Darell went to an arcade where the machines took tokens. Keenan played

10

games of skee ball and

6

games of pinball, using a total of

28

tokens. At the same time, Darell played

8

games of skee ball and

4

games of pinball, using up

20

tokens. How many tokens does each game require?

\newline

Every game of skee ball requires

\_

tokens, and every game of pinball requires

\_

tokens.

Define variables: Let's define two variables: let $x$ be the number of tokens required for a game of skee ball, and $y$ be the number of tokens required for a game of pinball. We can then write two equations based on the information given: $\newline$ For Keenan: $10x + 6y = 28$ $\newline$ For Darell: $8x + 4y = 20$
Use elimination method: To use elimination, we want the coefficients of one of the variables to be the same (or opposites) in both equations. We can achieve this by multiplying the second equation by $1.5$ to match the coefficients of $y$ in the first equation: $\newline$ $1.5(8x + 4y) = 1.5(20)$ $\newline$ This gives us: $12x + 6y = 30$
Eliminate y: Now we have a new system of equations: $\newline$ $10x + 6y = 28$ $\newline$ $12x + 6y = 30$ $\newline$ We can eliminate y by subtracting the first equation from the second: $\newline$ $(12x + 6y) - (10x + 6y) = 30 - 28$ $\newline$ This simplifies to: $\newline$ $2x = 2$
Solve for x: Dividing both sides of the equation by $2$ gives us the value of $x$ : $\frac{2x}{2} = \frac{2}{2}$ $x = 1$ So, each game of skee ball requires $1$ token.
Substitute $x$ into equation: Now that we know the value of $x$ , we can substitute it back into one of the original equations to find the value of $y$ . We'll use the first equation: $\newline$ $10(1) + 6y = 28$ $\newline$ $10 + 6y = 28$
Solve for $y$ : Subtracting $10$ from both sides gives us: $\newline$ $6y = 28 - 10$ $\newline$ $6y = 18$
Solve for y: Subtracting $10$ from both sides gives us: $\newline$ $6y = 28 - 10$ $\newline$ $6y = 18$ Dividing both sides by $6$ gives us the value of $y$ : $\newline$ $\frac{6y}{6} = \frac{18}{6}$ $\newline$ $y = 3$ $\newline$ So, each game of pinball requires $3$ tokens.