Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. 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? The florist can assemble a small arrangement in _____ minutes and a large one in _____ minutes.

Question

Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.  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?  The florist can assemble a small arrangement in _____ minutes and a large one in _____ minutes.

Bytelearn · Accepted Answer

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.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 6s = 12. Dividing both sides by 6 gives us s = 2. This means it takes 2 minutes to assemble a small arrangement.

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