Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ An employee at a party store is assembling balloon bouquets. For a graduation party, he assembled $4$ small balloon bouquets and $5$ large balloon bouquets, which used a total of $116$ balloons. Then, for a Father's Day celebration, he used $225$ balloons to assemble $9$ small balloon bouquets and $9$ large balloon bouquets. How many balloons are in each bouquet? $\newline$ The small balloon bouquet uses $\_\_\_\_$ balloons and the large one uses $\_\_\_\_$ balloons.

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Solve a system of equations using elimination: word problems

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.

\newline

An employee at a party store is assembling balloon bouquets. For a graduation party, he assembled

4

small balloon bouquets and

5

large balloon bouquets, which used a total of

116

balloons. Then, for a Father's Day celebration, he used

225

balloons to assemble

9

small balloon bouquets and

9

large balloon bouquets. How many balloons are in each bouquet?

\newline

The small balloon bouquet uses

\_\_\_\_

balloons and the large one uses

\_\_\_\_

balloons.

Define Variables: Let's define two variables: let $x$ be the number of balloons in a small bouquet, and $y$ be the number of balloons in a large bouquet. We can then write two equations based on the information given. $\newline$ For the graduation party: $4x + 5y = 116$ $\newline$ For the Father's Day celebration: $9x + 9y = 225$
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 $9$ and the second equation by $4$ to make the coefficients of $x$ the same. $\newline$ First equation multiplied by $9$ : $(4x + 5y) \times 9 = 116 \times 9$ $\newline$ Second equation multiplied by $4$ : $(9x + 9y) \times 4 = 225 \times 4$
Perform Multiplication: Now let's perform the multiplication: $\newline$ First equation: $36x + 45y = 1044$ $\newline$ Second equation: $36x + 36y = 900$
Subtract Equations: Next, we subtract the second equation from the first equation to eliminate $x$ : $(36x + 45y) - (36x + 36y) = 1044 - 900$ This simplifies to: $45y - 36y = 144$
Solve for y: Now we solve for y: $\newline$ $9y = 144$ $\newline$ $y = \frac{144}{9}$ $\newline$ $y = 16$ $\newline$ So, each large balloon bouquet uses $16$ balloons.
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: $\newline$ $4x + 5(16) = 116$ $\newline$ $4x + 80 = 116$
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: $\newline$ $4x + 5(16) = 116$ $\newline$ $4x + 80 = 116$ Subtract $80$ from both sides to solve for x: $\newline$ $4x = 116 - 80$ $\newline$ $4x = 36$ $\newline$ $x = \frac{36}{4}$ $\newline$ $x = 9$ $\newline$ So, each small balloon bouquet uses $9$ balloons.