Lily sold 18 items at the street fair. She sold bracelets for $6 each and necklaces for $5 each for a total of $101. Which system of equations can be used to find $b$, the number of bracelets she sold, and $n$, the number of necklaces she sold?
A. [tex]
\begin{array}{l}
b+n=101 \\
6 b+5 n=18
\end{array}
[/tex]
B. [tex]
\begin{array}{l}
b+n=101 \\
5 b+6 n=18
\end{array}
[/tex]
C. [tex]
\begin{array}{l}
b+n=18 \\
6 b+5 n=101
\end{array}
[/tex]
D. [tex]
\begin{array}{l}
b+n=18 \\
5 b+6 n=101
\end{array}
[/tex]
Answer (2)
Establish the first equation based on the total number of items sold: b + n = 18 .
Establish the second equation based on the total revenue: 6 b + 5 n = 101 .
Combine the two equations to form the system of equations.
The correct system of equations is { b + n = 18 6 b + 5 n = 101 .
Explanation
Problem Analysis Let's analyze the given problem. We are given that Lily sold a total of 18 items, which include bracelets and necklaces. We are also given the price of each bracelet and each necklace, and the total revenue she made. We need to find the system of equations that represents this situation, where b represents the number of bracelets and n represents the number of necklaces.
Forming the Equations We know that the total number of items sold is 18. This gives us our first equation: b + n = 18 We also know that the total revenue is $101. Since bracelets sell for $6 each and necklaces sell for 5 e a c h , w ec an w r i t e t h eseco n d e q u a t i o na s : 6 b + 5 n = 101 T h u s , t h esys t e m o f e q u a t i o n s i s : { b + n = 18 6 b + 5 n = 101 $
Comparing with Options Now, let's compare our system of equations with the given options:
Option 1: { b + n = 101 6 b + 5 n = 18 This is incorrect because the total number of items is 18, not 101, and the total revenue is $101, not $18.
Option 2: { b + n = 101 5 b + 6 n = 18 This is also incorrect for the same reasons as Option 1, and the coefficients for b and n are switched.
Option 3: { b + n = 18 6 b + 5 n = 101 This matches our system of equations.
Option 4: { b + n = 18 5 b + 6 n = 101 This is incorrect because the coefficients for b and n in the second equation are switched.
Final Answer Therefore, the correct system of equations is: { b + n = 18 6 b + 5 n = 101
Examples
Imagine you're organizing a school bake sale. You sell cookies for $2 each and brownies for $3 each. If you sold a total of 50 items and made $120, you can use a system of equations to figure out exactly how many cookies and brownies you sold. This helps in inventory management and understanding sales patterns.
The system of equations that models Lily's sales is: b + n = 18 and 6 b + 5 n = 101 . The correct choice among the options given is Option C. This establishes the relationship between the total items sold and the total revenue from the sales.
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