Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ The owner of Summerfield Florist is assembling flower arrangements for Valentine's Day. This morning, she assembled $6$ small arrangements and $8$ large arrangements, which took her a total of $92$ minutes. After lunch, she arranged $5$ small arrangements and $12$ large arrangements, which took $130$ minutes. How long does it take to assemble each type? $\newline$ The florist can assemble a small arrangement in $\_$ minutes and a large one in $\_$ minutes.

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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

The owner of Summerfield Florist is assembling flower arrangements for Valentine's Day. This morning, she assembled

6

small arrangements and

8

large arrangements, which took her a total of

92

minutes. After lunch, she arranged

5

small arrangements and

12

large arrangements, which took

130

minutes. How long does it take to assemble each type?

\newline

The florist can assemble a small arrangement in

\_

minutes and a large one in

\_

minutes.

Define Variables: Let's denote the time it takes to assemble a small arrangement as $s$ minutes and the time it takes to assemble a large arrangement as $l$ minutes. The first scenario gives us the equation $6s + 8l = 92$ because the florist assembled $6$ small and $8$ large arrangements in $92$ minutes.
Form Equations: The second scenario gives us the equation $5s + 12l = 130$ because the florist assembled $5$ small and $12$ large arrangements in $130$ minutes.
Eliminate Variable: We now have a system of two equations. We need to eliminate one of the variables, $s$ or $l$ . We choose to eliminate $s$ because its coefficients ( $6$ and $5$ ) are close in value, which might make the calculations simpler.
Multiply Equations: To eliminate $s$ , we can multiply the first equation by $5$ and the second equation by $6$ , so that the coefficients of $s$ in both equations are the same ( $30$ ). This gives us the new equations $30s + 40l = 460$ and $30s + 72l = 780$ .
Solve for Large: We now subtract the first new equation from the second new equation to eliminate $s$ . This gives us $32l = 320$ .
Substitute and Solve for Small: Solving for $l$ , we divide both sides of the equation by $32$ to get $l = 10$ . This means it takes $10$ minutes to assemble a large arrangement.
Substitute and Solve for Small: Solving for $l$ , we divide both sides of the equation by $32$ to get $l = 10$ . This means it takes $10$ minutes to assemble a large arrangement.We substitute $l = 10$ into the first original equation $6s + 8l = 92$ and solve for $s$ . This gives us $6s + 8(10) = 92$ , which simplifies to $6s + 80 = 92$ .
Substitute and Solve for Small: Solving for $l$ , we divide both sides of the equation by $32$ to get $l = 10$ . This means it takes $10$ minutes to assemble a large arrangement.We substitute $l = 10$ into the first original equation $6s + 8l = 92$ and solve for $s$ . This gives us $6s + 8(10) = 92$ , which simplifies to $6s + 80 = 92$ .Subtracting $80$ from both sides of the equation gives us $32$ $0$ . Dividing both sides by $32$ $1$ gives us $32$ $2$ . This means it takes $32$ $3$ minutes to assemble a small arrangement.