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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Hannah went to play miniature golf on Monday, when it cost $\$16$ to rent the club and ball, plus $\$4$ per game. Alana went Thursday, paying $\$5$ per game, plus rental fees of $\$2$ . By coincidence, they played the same number of games for the same total cost. How much did each one spend? How many games did each one play? $\newline$ Hannah and Alana each spent $\$$ _ and played _ games.

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Solve a system of equations using substitution: word problems

Full solution

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

\newline

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

\$16

to rent the club and ball, plus

\$4

per game. Alana went Thursday, paying

\$5

per game, plus rental fees of

\$2

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

\newline

Hannah and Alana each spent

\$

_____ and played _____ games.

Define Variables: Let's define the variables. $\newline$ Let $x$ be the number of games each person played. $\newline$ Let $y$ be the total cost for each person.
Hannah's Total Cost: Write the equation for Hannah's total cost. $\newline$ Hannah's cost per game: $\$4$ $\newline$ Hannah's rental cost: $\$16$ $\newline$ Hannah's total cost equation: $y = 4x + 16$
Alana's Total Cost: Write the equation for Alana's total cost. $\newline$ Alana's cost per game: $\$5$ $\newline$ Alana's rental cost: $\$2$ $\newline$ Alana's total cost equation: $y = 5x + 2$
Set Equations Equal: Since both spent the same total cost for the same number of games, set the equations equal to each other to solve for $x$ . $4x + 16 = 5x + 2$
Solve for x: Solve for x by subtracting $4x$ from both sides and then subtracting $2$ from both sides. $\newline$ $4x + 16 - 4x = 5x + 2 - 4x$ $\newline$ $16 = x + 2$ $\newline$ $16 - 2 = x$ $\newline$ $x = 14$
Substitute $x$ for $y$ : Substitute $x$ back into one of the original equations to find $y$ , the total cost. $\newline$ Using Hannah's equation: $y = 4x + 16$ $\newline$ $y = 4(14) + 16$ $\newline$ $y = 56 + 16$ $\newline$ $y = 72$
Verify Solution: Verify the solution by substituting $x$ into Alana's equation to ensure it gives the same $y$ value. $\newline$ Using Alana's equation: $y = 5x + 2$ $\newline$ $y = 5(14) + 2$ $\newline$ $y = 70 + 2$ $\newline$ $y = 72$

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