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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineHannah went to play miniature golf on Monday, when it cost $16\$16 to rent the club and ball, plus $4\$4 per game. Alana went Thursday, paying $5\$5 per game, plus rental fees of $2\$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?\newlineHannah and Alana each spent $\$_____ and played _____ games.

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Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineHannah went to play miniature golf on Monday, when it cost $16\$16 to rent the club and ball, plus $4\$4 per game. Alana went Thursday, paying $5\$5 per game, plus rental fees of $2\$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?\newlineHannah and Alana each spent $\$_____ and played _____ games.
  1. Define Variables: Let's define the variables.\newlineLet xx be the number of games each person played.\newlineLet yy be the total cost for each person.
  2. Hannah's Total Cost: Write the equation for Hannah's total cost.\newlineHannah's cost per game: $4\$4\newlineHannah's rental cost: $16\$16\newlineHannah's total cost equation: y=4x+16y = 4x + 16
  3. Alana's Total Cost: Write the equation for Alana's total cost.\newlineAlana's cost per game: $5\$5\newlineAlana's rental cost: $2\$2\newlineAlana's total cost equation: y=5x+2y = 5x + 2
  4. 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 xx.4x+16=5x+24x + 16 = 5x + 2
  5. Solve for x: Solve for x by subtracting 4x4x from both sides and then subtracting 22 from both sides.\newline4x+16−4x=5x+2−4x4x + 16 - 4x = 5x + 2 - 4x\newline16=x+216 = x + 2\newline16−2=x16 - 2 = x\newlinex=14x = 14
  6. Substitute xx for yy: Substitute xx back into one of the original equations to find yy, the total cost.\newlineUsing Hannah's equation: y=4x+16y = 4x + 16\newliney=4(14)+16y = 4(14) + 16\newliney=56+16y = 56 + 16\newliney=72y = 72
  7. Verify Solution: Verify the solution by substituting xx into Alana's equation to ensure it gives the same yy value.\newlineUsing Alana's equation: y=5x+2y = 5x + 2\newliney=5(14)+2y = 5(14) + 2\newliney=70+2y = 70 + 2\newliney=72y = 72

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