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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineAn employee at a party store is assembling balloon bouquets. For a graduation party, he assembled 1010 small balloon bouquets and 77 large balloon bouquets, which used a total of 220220 balloons. Then, for a Father's Day celebration, he used 8484 balloons to assemble 88 small balloon bouquets and 11 large balloon bouquet. How many balloons are in each bouquet?\newlineThe small balloon bouquet uses _\_ balloons and the large one uses _\_ balloons.

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineAn employee at a party store is assembling balloon bouquets. For a graduation party, he assembled 1010 small balloon bouquets and 77 large balloon bouquets, which used a total of 220220 balloons. Then, for a Father's Day celebration, he used 8484 balloons to assemble 88 small balloon bouquets and 11 large balloon bouquet. How many balloons are in each bouquet?\newlineThe small balloon bouquet uses _\_ balloons and the large one uses _\_ balloons.
  1. Define variables: Let's define two variables: let xx be the number of balloons in a small bouquet, and yy be the number of balloons in a large bouquet.
  2. Write equations: We can write two equations based on the information given. For the graduation party, the equation is 10x+7y=22010x + 7y = 220. For the Father's Day celebration, the equation is 8x+y=848x + y = 84.
  3. Form system of equations: The system of equations to describe the situation is:\newline11) 10x+7y=22010x + 7y = 220\newline22) 8x+y=848x + y = 84
  4. Eliminate variable: To solve the system using elimination, we need to eliminate one of the variables. We can multiply the second equation by −7-7 to eliminate yy.
  5. Multiply and add equations: Multiplying the second equation by −7-7 gives us:\newline−7(8x+y)=−7(84)-7(8x + y) = -7(84)\newline−56x−7y=−588-56x - 7y = -588
  6. Simplify equation: Now we add this new equation to the first equation to eliminate yy:(10x+7y)+(−56x−7y)=220+(−588)(10x + 7y) + (-56x - 7y) = 220 + (-588)
  7. Solve for x: Simplifying the equation gives us:\newline10x−56x=220−58810x - 56x = 220 - 588\newline−46x=−368-46x = -368
  8. Substitute xx: Dividing both sides by −46-46 to solve for xx gives us:\newlinex=−368−46x = \frac{-368}{-46}\newlinex=8x = 8
  9. Solve for y: Now that we have the value for xx, we can substitute it back into one of the original equations to solve for yy. We'll use the second equation: 8x+y=848x + y = 84.
  10. Final solution: Substituting x=8x = 8 into the second equation gives us:\newline8(8)+y=848(8) + y = 84\newline64+y=8464 + y = 84
  11. Final solution: Substituting x=8x = 8 into the second equation gives us:\newline8(8)+y=848(8) + y = 84\newline64+y=8464 + y = 84 Subtracting 6464 from both sides to solve for yy gives us:\newliney=84−64y = 84 - 64\newliney=20y = 20
  12. Final solution: Substituting x=8x = 8 into the second equation gives us:\newline8(8)+y=848(8) + y = 84\newline64+y=8464 + y = 84 Subtracting 6464 from both sides to solve for yy gives us:\newliney=84−64y = 84 - 64\newliney=20y = 20 We have found that a small balloon bouquet uses 88 balloons and a large one uses 2020 balloons.

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