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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineThe owner of Arlington Florist is assembling flower arrangements for Valentine's Day. This morning, she assembled 1111 small arrangements and 1111 large arrangements, which took her a total of 8888 minutes. After lunch, she arranged 88 small arrangements and 1414 large arrangements, which took 100100 minutes. How long does it take to assemble each type?\newlineThe florist can assemble a small arrangement in _____ minutes and a large one in _____ minutes.

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Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineThe owner of Arlington Florist is assembling flower arrangements for Valentine's Day. This morning, she assembled 1111 small arrangements and 1111 large arrangements, which took her a total of 8888 minutes. After lunch, she arranged 88 small arrangements and 1414 large arrangements, which took 100100 minutes. How long does it take to assemble each type?\newlineThe florist can assemble a small arrangement in _____ minutes and a large one in _____ minutes.
  1. Define Equations: Let's denote the time it takes to assemble a small arrangement as xx minutes and a large arrangement as yy minutes. The first equation is based on the morning session where the florist assembled 1111 small arrangements and 1111 large arrangements in 8888 minutes.\newlineFirst equation: 11x+11y=8811x + 11y = 88
  2. Morning Session: The second equation is based on the afternoon session where the florist assembled 88 small arrangements and 1414 large arrangements in 100100 minutes.\newlineSecond equation: 8x+14y=1008x + 14y = 100
  3. Afternoon Session: We have a system of equations:\newline11x+11y=8811x + 11y = 88\newline8x+14y=1008x + 14y = 100\newlineTo solve the system, we can simplify the first equation by dividing all terms by 1111 to make the coefficients easier to work with.\newlinex+y=8x + y = 8
  4. Simplify Equations: Now we have a simplified system of equations:\newlinex+y=8x + y = 8\newline8x+14y=1008x + 14y = 100\newlineWe can use the substitution or elimination method to solve this system. Let's use the elimination method by multiplying the first equation by −8-8 to eliminate xx.\newline−8(x+y)=−8(8)-8(x + y) = -8(8)\newline−8x−8y=−64-8x - 8y = -64
  5. Elimination Method: Add the new equation to the second equation to eliminate xx:−8x−8y=−64-8x - 8y = -648x+14y=1008x + 14y = 100Adding these equations gives us:6y=366y = 36
  6. Solve for y: Divide both sides of the equation by 66 to solve for y:\newline6y6=366\frac{6y}{6} = \frac{36}{6}\newliney=6y = 6\newlineThis means it takes 66 minutes to assemble a large arrangement.
  7. Solve for x: Substitute the value of yy back into the simplified first equation to solve for xx:x+6=8x + 6 = 8x=8−6x = 8 - 6x=2x = 2This means it takes 22 minutes to assemble a small arrangement.
  8. Final Values: We have found the values for xx and yy: \newlinex=2x = 2 (time to assemble a small arrangement) \newliney=6y = 6 (time to assemble a large arrangement) \newlineThese values answer the question prompt.

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