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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineKaren has a home-based business making corsages and boutonnieres for school dances. Last year, she sold 3434 corsages and 3939 boutonnieres, which brought in a total of $1,669\$1,669. This year, she sold 3939 corsages and 3636 boutonnieres, for a total of $1,731\$1,731. How much does each item sell for?\newlineA corsage sells for $\$_____, and a boutonniere sells for $\$_____.

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineKaren has a home-based business making corsages and boutonnieres for school dances. Last year, she sold 3434 corsages and 3939 boutonnieres, which brought in a total of $1,669\$1,669. This year, she sold 3939 corsages and 3636 boutonnieres, for a total of $1,731\$1,731. How much does each item sell for?\newlineA corsage sells for $\$_____, and a boutonniere sells for $\$_____.
  1. Define variables: Define the variables for the cost of a corsage and a boutonniere.\newlineLet xx be the cost of a corsage and yy be the cost of a boutonniere.
  2. Write equations (last year): Write the system of equations based on last year's sales. \newline3434 corsages and 3939 boutonnieres brought in a total of $1,669\$1,669.\newlineThe equation is: 34x+39y=166934x + 39y = 1669.
  3. Write equations (this year): Write the system of equations based on this year's sales.\newline3939 corsages and 3636 boutonnieres brought in a total of $1,731\$1,731.\newlineThe equation is: 39x+36y=173139x + 36y = 1731.
  4. Eliminate variable: Choose which variable to eliminate.\newlineWe will eliminate yy by multiplying the first equation by 3636 and the second equation by 3939 to make the coefficients of yy equal.
  5. Multiply equations: Multiply the first equation by 3636 and the second equation by 3939.\newlineFirst equation: (34x+39y)×36=1669×36(34x + 39y) \times 36 = 1669 \times 36\newlineSecond equation: (39x+36y)×39=1731×39(39x + 36y) \times 39 = 1731 \times 39
  6. Write new equations: Write the new system of equations after multiplication.\newlineFirst equation: 1224x+1404y=600841224x + 1404y = 60084\newlineSecond equation: 1521x+1404y=674091521x + 1404y = 67409
  7. Subtract equations: Subtract the second equation from the first to eliminate yy.1224x+1404y−(1521x+1404y)=60084−674091224x + 1404y - (1521x + 1404y) = 60084 - 67409
  8. Perform subtraction: Perform the subtraction to solve for xx.1224x+1404y−1521x−1404y=60084−674091224x + 1404y - 1521x - 1404y = 60084 - 67409−297x=−7325-297x = -7325
  9. Solve for x: Solve for x.\newlinex=−7325−297x = \frac{-7325}{-297}\newlinex=24.66x = 24.66 (rounded to two decimal places)
  10. Substitute xx for yy: Substitute xx back into one of the original equations to solve for yy. Using the first equation: 34x+39y=166934x + 39y = 1669 34(24.66)+39y=166934(24.66) + 39y = 1669
  11. Perform multiplication: Perform the multiplication and solve for yy.\newline838.44+39y=1669838.44 + 39y = 1669\newline39y=1669−838.4439y = 1669 - 838.44\newline39y=830.5639y = 830.56
  12. Solve for y: Solve for y.\newliney=830.5639y = \frac{830.56}{39}\newliney=21.30y = 21.30 (rounded to two decimal places)

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