Solving For Bracelets And Necklaces Using Systems Of Equations
Lily's entrepreneurial spirit shone brightly at the recent street fair, where she showcased her handcrafted jewelry. Her inventory consisted of two charming items: bracelets and necklaces. By the end of the day, Lily had successfully sold 18 items, a testament to her craftsmanship and salesmanship. The bracelets, priced at $6 each, and the necklaces, priced at $5 each, collectively brought in a total of $101. Now, the challenge lies in deciphering the exact number of bracelets and necklaces Lily sold individually. This is where the power of systems of equations comes into play, allowing us to translate this real-world scenario into a mathematical model.
To effectively tackle this problem, we need to define our variables. Let 'b' represent the number of bracelets Lily sold, and 'n' represent the number of necklaces she sold. With these variables in place, we can begin to construct the equations that capture the essence of Lily's sales. The first piece of information we have is the total number of items sold, which is 18. This translates directly into our first equation: b + n = 18. This equation establishes the fundamental relationship between the number of bracelets and necklaces sold.
Next, we need to incorporate the financial aspect of Lily's sales. We know that each bracelet sold for $6 and each necklace for $5, and the total revenue generated was $101. This information leads us to our second equation: 6b + 5n = 101. This equation represents the total revenue earned from the sales of bracelets and necklaces, taking into account their respective prices. Together, these two equations, b + n = 18 and 6b + 5n = 101, form a system of equations that accurately models Lily's sales at the street fair. This system provides a mathematical framework for determining the values of 'b' and 'n', thereby revealing the exact number of bracelets and necklaces Lily sold.
The beauty of using a system of equations lies in its ability to handle multiple unknowns simultaneously. In this case, we have two unknowns, 'b' and 'n', and two equations that relate them. This allows us to solve for both variables and gain a complete understanding of Lily's sales performance. By employing algebraic techniques such as substitution or elimination, we can find the values of 'b' and 'n' that satisfy both equations, giving us the solution to our problem. Understanding how to set up and solve systems of equations is a valuable skill in mathematics, with applications extending far beyond simple word problems. It is a fundamental tool used in various fields, including science, engineering, and economics, to model and solve real-world problems involving multiple variables and constraints.
H2: Constructing the Equations: Bracelets, Necklaces, and Revenue
To construct the system of equations that accurately represents Lily's sales, we need to carefully analyze the information provided and translate it into mathematical expressions. The core of the problem revolves around two key aspects: the total number of items sold and the total revenue generated. Let's delve deeper into how each of these aspects contributes to the formation of our equations. As established earlier, 'b' represents the number of bracelets sold, and 'n' represents the number of necklaces sold. The first piece of information we have is that Lily sold a total of 18 items. This means that the sum of the bracelets and necklaces sold must equal 18. This directly translates into the equation b + n = 18. This equation serves as the foundation of our system, establishing the relationship between the two variables based on the total quantity of items sold. It's a simple yet crucial equation that captures a fundamental aspect of the problem.
The second piece of information we need to incorporate is the revenue generated from the sales. We know that each bracelet sold for $6 and each necklace for $5, and the total revenue amounted to $101. To express this information mathematically, we need to consider the revenue generated from each type of item separately. The revenue from bracelets is calculated by multiplying the number of bracelets sold (b) by the price per bracelet ($6), which gives us 6b. Similarly, the revenue from necklaces is calculated by multiplying the number of necklaces sold (n) by the price per necklace ($5), resulting in 5n. The total revenue is the sum of these two amounts, which is given as $101. Therefore, we can write the second equation as 6b + 5n = 101. This equation represents the total revenue earned from the sales of bracelets and necklaces, taking into account their individual prices and the quantities sold.
Together, the two equations, b + n = 18 and 6b + 5n = 101, form a system of linear equations. This system provides a mathematical representation of the problem, allowing us to use algebraic techniques to solve for the unknowns, 'b' and 'n'. The system captures the two essential constraints of the problem: the total number of items sold and the total revenue generated. By solving this system, we can determine the exact number of bracelets and necklaces Lily sold at the street fair. It's important to note that the coefficients in the equations (1, 1, 6, and 5) represent the quantities and prices associated with each item. The constants (18 and 101) represent the total number of items sold and the total revenue, respectively. These values are crucial in defining the specific relationships between the variables and in determining the solution to the system. Understanding how to translate real-world scenarios into mathematical equations is a fundamental skill in algebra and problem-solving. It allows us to model complex situations and use mathematical tools to find solutions.
H2: The System of Equations: A Clear Representation
The system of equations that models Lily's sales can be clearly represented as follows:
- b + n = 18
- 6b + 5n = 101
This concise representation encapsulates the core mathematical relationships within the problem. The first equation, b + n = 18, highlights the relationship between the number of bracelets (b) and necklaces (n) sold, stating that their sum equals the total number of items sold, which is 18. This equation provides a foundational constraint, limiting the possible combinations of bracelets and necklaces that Lily could have sold.
The second equation, 6b + 5n = 101, focuses on the financial aspect of Lily's sales. It represents the total revenue generated from selling bracelets and necklaces. The term 6b represents the revenue from bracelet sales, where 6 is the price of each bracelet, and b is the number of bracelets sold. Similarly, 5n represents the revenue from necklace sales, with 5 being the price per necklace and n being the number of necklaces sold. The sum of these two terms, 6b + 5n, equals the total revenue of $101. This equation provides another crucial constraint, linking the number of bracelets and necklaces sold to the total revenue earned.
The system of equations, consisting of these two equations, provides a complete mathematical model of Lily's sales scenario. It captures both the quantity constraint (total items sold) and the financial constraint (total revenue). To solve this system, we can employ various algebraic techniques, such as substitution or elimination. The goal is to find the values of b and n that simultaneously satisfy both equations. These values will represent the exact number of bracelets and necklaces Lily sold at the street fair.
The clarity of this system of equations allows for a systematic approach to solving the problem. It provides a structured framework for understanding the relationships between the variables and for applying algebraic methods to find the solution. By representing the problem in this mathematical form, we can leverage the power of algebra to unravel the unknowns and gain insights into Lily's sales performance. This ability to translate real-world situations into mathematical models is a fundamental skill in mathematics and its applications across various fields.
H2: Solving the System: Finding the Values of b and n
Now that we have established the system of equations:
- b + n = 18
- 6b + 5n = 101
We can proceed to solve it to find the values of 'b' (number of bracelets) and 'n' (number of necklaces). There are several methods we can use to solve a system of equations, including substitution, elimination, and graphing. For this problem, let's use the substitution method, as it can be particularly efficient in this case.
The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, which can then be easily solved. From the first equation, b + n = 18, we can solve for 'b' by subtracting 'n' from both sides: b = 18 - n. Now we have an expression for 'b' in terms of 'n'. We can substitute this expression into the second equation, 6b + 5n = 101, replacing 'b' with (18 - n): 6(18 - n) + 5n = 101. This substitution eliminates 'b' from the second equation, leaving us with an equation in terms of 'n' only.
Next, we need to simplify and solve the resulting equation for 'n': 6(18 - n) + 5n = 101. Distribute the 6: 108 - 6n + 5n = 101. Combine like terms: 108 - n = 101. Subtract 108 from both sides: -n = -7. Divide both sides by -1: n = 7. So, we have found that n = 7, meaning Lily sold 7 necklaces. Now that we know the value of 'n', we can substitute it back into either of the original equations to solve for 'b'. Let's use the first equation, b + n = 18: b + 7 = 18. Subtract 7 from both sides: b = 11. Therefore, Lily sold 11 bracelets. We have now successfully solved the system of equations and found the values of 'b' and 'n'. To verify our solution, we can substitute these values back into both original equations to ensure they hold true. For the first equation, 11 + 7 = 18, which is correct. For the second equation, 6(11) + 5(7) = 66 + 35 = 101, which is also correct. This confirms that our solution, b = 11 and n = 7, is accurate.
H2: Conclusion: Lily's Success in Numbers
In conclusion, by utilizing a system of equations, we have successfully determined the number of bracelets and necklaces Lily sold at the street fair. The system of equations, consisting of b + n = 18 and 6b + 5n = 101, provided a mathematical framework for representing the problem and finding the solution. Through the application of the substitution method, we found that Lily sold 11 bracelets and 7 necklaces. This solution satisfies both equations in the system, confirming its accuracy.
This problem exemplifies the power of systems of equations in solving real-world scenarios involving multiple variables and constraints. By translating the given information into mathematical expressions, we were able to create a model that allowed us to find the unknowns. The process of setting up and solving a system of equations involves several key steps, including defining variables, constructing equations, choosing a solution method, and verifying the solution. Each of these steps is crucial for ensuring the accuracy and validity of the results. Understanding systems of equations is a fundamental skill in mathematics, with applications extending across various disciplines, including science, engineering, economics, and computer science. The ability to model and solve problems using systems of equations is essential for making informed decisions and solving complex challenges in a wide range of contexts.
Lily's success at the street fair can now be quantified in numbers: 11 bracelets and 7 necklaces. This demonstrates the value of her craftsmanship and her ability to connect with customers. The mathematical analysis provides a clear and concise picture of her sales performance, highlighting the individual contributions of each type of item to her overall revenue. This type of analysis can be valuable for businesses and individuals alike, allowing them to track their performance, identify trends, and make strategic decisions for future success. The combination of entrepreneurial spirit and mathematical problem-solving is a powerful force, enabling individuals like Lily to achieve their goals and make a positive impact in their communities.
