Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Gabrielle has a home-based business making corsages and boutonnieres for school dances. Last year, she sold $18$ corsages and $37$ boutonnieres, which brought in a total of $\$1,121$ . This year, she sold $28$ corsages and $19$ boutonnieres, for a total of $\$857$ . 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 any method: word problems

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

\newline

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

18

corsages and

37

boutonnieres, which brought in a total of

\$1,121

. This year, she sold

28

corsages and

19

boutonnieres, for a total of

\$857

. How much does each item sell for?

\newline

A corsage sells for

\$

_____, and a boutonniere sells for

\$

_____.

Define Prices: Let's denote the price of a corsage as $c$ dollars and the price of a boutonniere as $b$ dollars.
Formulate Equations: According to the problem, Gabrielle sold $18$ corsages and $37$ boutonnieres for a total of $\$1,121$ last year. This gives us the first equation: $\newline$ $18c + 37b = 1121$
Solve Using Elimination: This year, she sold $28$ corsages and $19$ boutonnieres for a total of $\$857$ . This gives us the second equation: $\newline$ $28c + 19b = 857$
Multiply Equations: We now have a system of two equations with two variables: $\newline$ $18c + 37b = 1121$ $\newline$ $28c + 19b = 857$ $\newline$ We can solve this system using either substitution or elimination. Let's use the elimination method.
Subtract Equations: To eliminate one of the variables, we can multiply the first equation by $19$ and the second equation by $37$ to make the coefficients of $b$ equal: $\newline$ $(18c + 37b) \times 19 = 1121 \times 19$ $\newline$ $(28c + 19b) \times 37 = 857 \times 37$
Solve for c: Multiplying out, we get: $\newline$ $342c + 703b = 21319$ $\newline$ $1036c + 703b = 31699$
Substitute to Solve for b: Now, we subtract the second equation from the first to eliminate b: $\newline$ $(1036c + 703b) - (342c + 703b) = 31699 - 21319$ $\newline$ $1036c - 342c = 31699 - 21319$ $\newline$ $694c = 10380$
Substitute to Solve for b: Now, we subtract the second equation from the first to eliminate b: $\newline$ $(1036c + 703b) - (342c + 703b) = 31699 - 21319$ $\newline$ $1036c - 342c = 31699 - 21319$ $\newline$ $694c = 10380$ Dividing both sides by $694$ to solve for c: $\newline$ $c = \frac{10380}{694}$ $\newline$ $c = 15$
Substitute to Solve for b: Now, we subtract the second equation from the first to eliminate b: $\newline$ $(1036c + 703b) - (342c + 703b) = 31699 - 21319$ $\newline$ $1036c - 342c = 31699 - 21319$ $\newline$ $694c = 10380$ Dividing both sides by $694$ to solve for c: $\newline$ $c = \frac{10380}{694}$ $\newline$ $c = 15$ Now that we have the value of c, we can substitute it back into one of the original equations to solve for b. Let's use the first equation: $\newline$ $18c + 37b = 1121$ $\newline$ $18(15) + 37b = 1121$
Substitute to Solve for b: Now, we subtract the second equation from the first to eliminate b: $\newline$ $(1036c + 703b) - (342c + 703b) = 31699 - 21319$ $\newline$ $1036c - 342c = 31699 - 21319$ $\newline$ $694c = 10380$ Dividing both sides by $694$ to solve for c: $\newline$ $c = \frac{10380}{694}$ $\newline$ $c = 15$ Now that we have the value of c, we can substitute it back into one of the original equations to solve for b. Let's use the first equation: $\newline$ $18c + 37b = 1121$ $\newline$ $18(15) + 37b = 1121$ Multiplying out and solving for b: $\newline$ $270 + 37b = 1121$ $\newline$ $37b = 1121 - 270$ $\newline$ $1036c - 342c = 31699 - 21319$ $0$
Substitute to Solve for b: Now, we subtract the second equation from the first to eliminate b: $\newline$ $(1036c + 703b) - (342c + 703b) = 31699 - 21319$ $\newline$ $1036c - 342c = 31699 - 21319$ $\newline$ $694c = 10380$ Dividing both sides by $694$ to solve for c: $\newline$ $c = \frac{10380}{694}$ $\newline$ $c = 15$ Now that we have the value of c, we can substitute it back into one of the original equations to solve for b. Let's use the first equation: $\newline$ $18c + 37b = 1121$ $\newline$ $18(15) + 37b = 1121$ Multiplying out and solving for b: $\newline$ $270 + 37b = 1121$ $\newline$ $37b = 1121 - 270$ $\newline$ $1036c - 342c = 31699 - 21319$ $0$ Dividing both sides by $1036c - 342c = 31699 - 21319$ $1$ to solve for b: $\newline$ $1036c - 342c = 31699 - 21319$ $2$ $\newline$ $1036c - 342c = 31699 - 21319$ $3$