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 $10$ small balloon bouquets and $7$ large balloon bouquets, which used a total of $220$ balloons. Then, for a Father's Day celebration, he used $84$ balloons to assemble $8$ small balloon bouquets and $1$ large balloon bouquet. How many balloons are in each bouquet? $\newline$ The small balloon bouquet uses $\_$ balloons and the large one uses $\_$ balloons.

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Grade 8

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

10

small balloon bouquets and

7

large balloon bouquets, which used a total of

220

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

84

balloons to assemble

8

small balloon bouquets and

1

large balloon bouquet. 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.
Write equations: We can write two equations based on the information given. For the graduation party, the equation is $10x + 7y = 220$ . For the Father's Day celebration, the equation is $8x + y = 84$ .
Form system of equations: The system of equations to describe the situation is: $\newline$ $1$ ) $10x + 7y = 220$ $\newline$ $2$ ) $8x + y = 84$
Eliminate variable: To solve the system using elimination, we need to eliminate one of the variables. We can multiply the second equation by $-7$ to eliminate $y$ .
Multiply and add equations: Multiplying the second equation by $-7$ gives us: $\newline$ $-7(8x + y) = -7(84)$ $\newline$ $-56x - 7y = -588$
Simplify equation: Now we add this new equation to the first equation to eliminate $y$ : $(10x + 7y) + (-56x - 7y) = 220 + (-588)$
Solve for x: Simplifying the equation gives us: $\newline$ $10x - 56x = 220 - 588$ $\newline$ $-46x = -368$
Substitute $x$ : Dividing both sides by $-46$ to solve for $x$ gives us: $\newline$ $x = \frac{-368}{-46}$ $\newline$ $x = 8$
Solve for y: Now that we have the value for $x$ , we can substitute it back into one of the original equations to solve for $y$ . We'll use the second equation: $8x + y = 84$ .
Final solution: Substituting $x = 8$ into the second equation gives us: $\newline$ $8(8) + y = 84$ $\newline$ $64 + y = 84$
Final solution: Substituting $x = 8$ into the second equation gives us: $\newline$ $8(8) + y = 84$ $\newline$ $64 + y = 84$ Subtracting $64$ from both sides to solve for $y$ gives us: $\newline$ $y = 84 - 64$ $\newline$ $y = 20$
Final solution: Substituting $x = 8$ into the second equation gives us: $\newline$ $8(8) + y = 84$ $\newline$ $64 + y = 84$ Subtracting $64$ from both sides to solve for $y$ gives us: $\newline$ $y = 84 - 64$ $\newline$ $y = 20$ We have found that a small balloon bouquet uses $8$ balloons and a large one uses $20$ balloons.