Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Zane went to play miniature golf on Monday, when it cost $\$16$ to rent the club and ball, plus $\$1$ per game. Omar went Thursday, paying $\$2$ per game, plus rental fees of $\$6$ . 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? $\newline$ Zane and Omar each spent $\$$ _ and played _ games.

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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

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

\$16

to rent the club and ball, plus

\$1

per game. Omar went Thursday, paying

\$2

per game, plus rental fees of

\$6

. 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

Zane and Omar each spent

\$

_____ and played _____ games.

Define variables: Let's define the variables for the system of equations. Let $x$ represent the number of games played by each, and let $y$ represent the total cost for each person.
Zane's equation: For Zane, the cost is $\$16$ for the rental plus $\$1$ per game. So, the equation for Zane is: $\newline$ $y = 16 + 1x$
Omar's equation: For Omar, the cost is $\$6$ for the rental plus $\$2$ per game. So, the equation for Omar is: $\newline$ $y = 6 + 2x$
Set equations equal: Now we have a system of two equations: $\newline$ $1$ ) $y = 16 + x$ $\newline$ $2$ ) $y = 6 + 2x$ $\newline$ We can solve this system using substitution. Since both equations equal $y$ , we can set them equal to each other: $\newline$ $16 + x = 6 + 2x$
Solve for x: Next, we solve for x by subtracting $x$ from both sides of the equation: $\newline$ $16 + x - x = 6 + 2x - x$ $\newline$ $16 = 6 + x$
Isolate $x$ : Now, subtract $6$ from both sides to isolate $x$ : $16 - 6 = 6 + x - 6$ $10 = x$
Substitute $x$ into equation: We found that $x$ , the number of games played, is $10$ . Now we can substitute $x$ back into either of the original equations to find $y$ , the total cost. Let's use Zane's equation: $\newline$ $y = 16 + 1x$ $\newline$ $y = 16 + 1(10)$
Calculate Zane's cost: Calculate the total cost for Zane: $\newline$ $y = 16 + 10$ $\newline$ $y = 26$
Calculate Omar's cost: Since Zane and Omar spent the same amount and played the same number of games, Omar also spent $\$26$ and played $10$ games.