Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Reagan went to play miniature golf on Monday, when it cost $\$16$ to rent the club and ball, plus $\$4$ per game. Meg went Thursday, paying $\$5$ per game, plus rental fees of $\$8$ . By coincidence, they played the same number of games for the same total cost. How many games did each one play? How much did each one spend? $\newline$ Reagan and Meg each played $\_\_\_$ games and spent $\$\_\_\_$ .

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Solve a system of equations using substitution: word problems

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

\newline

Reagan went to play miniature golf on Monday, when it cost

\$16

to rent the club and ball, plus

\$4

per game. Meg went Thursday, paying

\$5

per game, plus rental fees of

\$8

. By coincidence, they played the same number of games for the same total cost. How many games did each one play? How much did each one spend?

\newline

Reagan and Meg each played

\_\_\_

games and spent

\$\_\_\_

Define Variables: Let's define the variables: $\newline$ Let $x$ be the number of games played by both Reagan and Meg. $\newline$ Let $y$ be the total cost for each person.
Reagan's Total Cost: We know Reagan's cost is $\$16$ for rental plus $\$4$ per game. So, the equation for Reagan's total cost is: $\newline$ $y = 4x + 16$
Meg's Total Cost: We know Meg's cost is $\$8$ for rental plus $\$5$ per game. So, the equation for Meg's total cost is: $\newline$ $y = 5x + 8$
Set Equations Equal: Now we have two equations: $\newline$ $1$ ) $y = 4x + 16$ (Reagan's cost) $\newline$ $2$ ) $y = 5x + 8$ (Meg's cost) $\newline$ Since they spent the same total cost, we can set the equations equal to each other to solve for $x$ : $\newline$ $4x + 16 = 5x + 8$
Solve for x: Subtract $4x$ from both sides to get: $\newline$ $16 = x + 8$
Substitute $x$ into Equation: Subtract $8$ from both sides to solve for $x$ :
$x = 16 - 8$
$x = 8$
Final Result: Now that we know $x$ (the number of games), we can substitute it back into either equation to find $y$ (the total cost). Let's use Reagan's equation: $\newline$ $y = 4x + 16$ $\newline$ $y = 4(8) + 16$ $\newline$ $y = 32 + 16$ $\newline$ $y = 48$
Final Result: Now that we know $x$ (the number of games), we can substitute it back into either equation to find $y$ (the total cost). Let's use Reagan's equation: $\newline$ $y = 4x + 16$ $\newline$ $y = 4(8) + 16$ $\newline$ $y = 32 + 16$ $\newline$ $y = 48$ Reagan and Meg each played $8$ games and spent $\$\(48$).