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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineCarson is a server at an all-you-can eat sushi restaurant. At one table, the customers ordered 11 child buffet and 44 adult buffets, which cost a total of $110\$110. At another table, the customers ordered 33 child buffets and 33 adult buffets, paying a total of $114\$114. How much does the buffet cost for each child and adult?\newlineThe cost for a child is $_\$\_, and the cost for an adult is $_\$\_.

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineCarson is a server at an all-you-can eat sushi restaurant. At one table, the customers ordered 11 child buffet and 44 adult buffets, which cost a total of $110\$110. At another table, the customers ordered 33 child buffets and 33 adult buffets, paying a total of $114\$114. How much does the buffet cost for each child and adult?\newlineThe cost for a child is $_\$\_, and the cost for an adult is $_\$\_.
  1. Equations Setup: Let's denote the cost of the child buffet as xx and the cost of the adult buffet as yy. We can write two equations based on the information given.\newlineFirst table: 11 child buffet + 44 adult buffets = $110\$110\newlineSecond table: 33 child buffets + 33 adult buffets = $114\$114\newlineTranslate this information into equations:\newline1x+4y=1101x + 4y = 110\newline3x+3y=1143x + 3y = 114
  2. Elimination Method: To use elimination, we need to make the coefficients of one of the variables the same in both equations. We can multiply the first equation by 33 to match the coefficient of xx in the second equation.\newlineMultiplying the first equation by 33 gives us:\newline3x+12y=3303x + 12y = 330\newlineWe now have the system:\newline3x+12y=3303x + 12y = 330\newline3x+3y=1143x + 3y = 114
  3. Subtract Equations: Subtract the second equation from the first equation to eliminate xx.\newline(3x+12y)−(3x+3y)=330−114(3x + 12y) - (3x + 3y) = 330 - 114\newlineThis simplifies to:\newline3x+12y−3x−3y=330−1143x + 12y - 3x - 3y = 330 - 114\newline9y=2169y = 216
  4. Solve for y: Divide both sides of the equation by 99 to solve for y.\newline9y9=2169\frac{9y}{9} = \frac{216}{9}\newliney=24y = 24
  5. Substitute and Solve: Now that we have the value of yy, we can substitute it back into one of the original equations to solve for xx. Let's use the first equation:\newline1x+4(24)=1101x + 4(24) = 110
  6. Final Cost Calculation: Simplify the equation and solve for xx.x+96=110x + 96 = 110x=110−96x = 110 - 96x=14x = 14
  7. Final Cost Calculation: Simplify the equation and solve for xx.x+96=110x + 96 = 110x=110−96x = 110 - 96x=14x = 14We have found the values of xx and yy, which represent the cost of the child buffet and the adult buffet, respectively.x=14x = 14y=24y = 24The cost for a child buffet is $14\$14, and the cost for an adult buffet is $24\$24.

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