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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Karen has a home-based business making corsages and boutonnieres for school dances. Last year, she sold $34$ corsages and $39$ boutonnieres, which brought in a total of $\$1,669$ . This year, she sold $39$ corsages and $36$ boutonnieres, for a total of $\$1,731$ . How much does each item sell for? $\newline$ A corsage sells for $\$$ _, and a boutonniere sells for $\$$ _.

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Solve a system of equations using elimination: word problems

Full solution

Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.

\newline

Karen has a home-based business making corsages and boutonnieres for school dances. Last year, she sold

34

corsages and

39

boutonnieres, which brought in a total of

\$1,669

. This year, she sold

39

corsages and

36

boutonnieres, for a total of

\$1,731

. How much does each item sell for?

\newline

A corsage sells for

\$

_____, and a boutonniere sells for

\$

_____.

Define variables: Define the variables for the cost of a corsage and a boutonniere. $\newline$ Let $x$ be the cost of a corsage and $y$ be the cost of a boutonniere.
Write equations (last year): Write the system of equations based on last year's sales. $\newline$ $34$ corsages and $39$ boutonnieres brought in a total of $\$1,669$ . $\newline$ The equation is: $34x + 39y = 1669$ .
Write equations (this year): Write the system of equations based on this year's sales. $\newline$ $39$ corsages and $36$ boutonnieres brought in a total of $\$1,731$ . $\newline$ The equation is: $39x + 36y = 1731$ .
Eliminate variable: Choose which variable to eliminate. $\newline$ We will eliminate $y$ by multiplying the first equation by $36$ and the second equation by $39$ to make the coefficients of $y$ equal.
Multiply equations: Multiply the first equation by $36$ and the second equation by $39$ . $\newline$ First equation: $(34x + 39y) \times 36 = 1669 \times 36$ $\newline$ Second equation: $(39x + 36y) \times 39 = 1731 \times 39$
Write new equations: Write the new system of equations after multiplication. $\newline$ First equation: $1224x + 1404y = 60084$ $\newline$ Second equation: $1521x + 1404y = 67409$
Subtract equations: Subtract the second equation from the first to eliminate $y$ . $1224x + 1404y - (1521x + 1404y) = 60084 - 67409$
Perform subtraction: Perform the subtraction to solve for $x$ . $1224x + 1404y - 1521x - 1404y = 60084 - 67409$ $-297x = -7325$
Solve for x: Solve for x. $\newline$ $x = \frac{-7325}{-297}$ $\newline$ $x = 24.66$ (rounded to two decimal places)
Substitute $x$ for $y$ : Substitute $x$ back into one of the original equations to solve for $y$ . Using the first equation: $34x + 39y = 1669$ $34(24.66) + 39y = 1669$
Perform multiplication: Perform the multiplication and solve for $y$ . $\newline$ $838.44 + 39y = 1669$ $\newline$ $39y = 1669 - 838.44$ $\newline$ $39y = 830.56$
Solve for y: Solve for y. $\newline$ $y = \frac{830.56}{39}$ $\newline$ $y = 21.30$ (rounded to two decimal places)

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