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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 44 small balloon bouquets and 55 large balloon bouquets, which used a total of 116116 balloons. Then, for a Father's Day celebration, he used 225225 balloons to assemble 99 small balloon bouquets and 99 large balloon bouquets. 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 44 small balloon bouquets and 55 large balloon bouquets, which used a total of 116116 balloons. Then, for a Father's Day celebration, he used 225225 balloons to assemble 99 small balloon bouquets and 99 large balloon bouquets. 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. We can then write two equations based on the information given.\newlineFor the graduation party: 4x+5y=1164x + 5y = 116\newlineFor the Father's Day celebration: 9x+9y=2259x + 9y = 225
  2. Elimination Method: To solve this system using elimination, we want to eliminate one of the variables. We can do this by making the coefficients of one of the variables the same in both equations. In this case, we can multiply the first equation by 99 and the second equation by 44 to make the coefficients of xx the same.\newlineFirst equation multiplied by 99: (4x+5y)×9=116×9(4x + 5y) \times 9 = 116 \times 9\newlineSecond equation multiplied by 44: (9x+9y)×4=225×4(9x + 9y) \times 4 = 225 \times 4
  3. Perform Multiplication: Now let's perform the multiplication:\newlineFirst equation: 36x+45y=104436x + 45y = 1044\newlineSecond equation: 36x+36y=90036x + 36y = 900
  4. Subtract Equations: Next, we subtract the second equation from the first equation to eliminate xx:(36x+45y)−(36x+36y)=1044−900(36x + 45y) - (36x + 36y) = 1044 - 900This simplifies to:45y−36y=14445y - 36y = 144
  5. Solve for y: Now we solve for y:\newline9y=1449y = 144\newliney=1449y = \frac{144}{9}\newliney=16y = 16\newlineSo, each large balloon bouquet uses 1616 balloons.
  6. Substitute and Solve for xx: Now that we have the value for yy, we can substitute it back into one of the original equations to solve for xx. We'll use the first equation:\newline4x+5(16)=1164x + 5(16) = 116\newline4x+80=1164x + 80 = 116
  7. Substitute and Solve for x: Now that we have the value for y, we can substitute it back into one of the original equations to solve for x. We'll use the first equation:\newline4x+5(16)=1164x + 5(16) = 116\newline4x+80=1164x + 80 = 116 Subtract 8080 from both sides to solve for x:\newline4x=116−804x = 116 - 80\newline4x=364x = 36\newlinex=364x = \frac{36}{4}\newlinex=9x = 9\newlineSo, each small balloon bouquet uses 99 balloons.

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