Antivirus Software Sales Analysis Solving For X

In this article, we will analyze the sales data of an antivirus software company that offers two versions of its product: a home edition and a business edition. The home edition is priced at $23.50, while the business edition costs $58.75. Last week, the company generated a total revenue of $29,668.75 from the sale of 745 copies of the software. Our objective is to determine the number of copies sold for each edition, using the variable x to represent the number of home edition copies sold. This problem falls under the mathematics category, specifically dealing with systems of linear equations. Understanding these systems is crucial for various real-world applications, including business analysis, resource allocation, and financial modeling. This scenario provides a practical example of how mathematical concepts can be applied to solve business-related problems.

The core of the problem lies in deciphering the sales figures for the two antivirus software versions. We know the prices of the home and business editions, the total revenue, and the total number of copies sold. Let's define our variables clearly: let x represent the number of home edition copies sold. Since the company sold 745 copies in total, the number of business edition copies sold can be expressed as 745 - x. We can then set up a system of equations to represent the given information. The first equation will represent the total revenue generated from the sales of both editions, and the second equation represents the total number of copies sold. By solving this system of equations, we can determine the exact number of copies sold for each edition. This involves algebraic manipulation and a clear understanding of how to translate a word problem into a mathematical model. The problem highlights the importance of accurate data representation and the ability to formulate equations that capture the relationships between different variables. The process of solving these equations will reveal the sales performance of each software version, providing valuable insights for the company.

To solve this mathematical problem, we need to translate the given information into a system of linear equations. This involves identifying the key relationships between the variables and expressing them in mathematical form. We've already established that x represents the number of home edition copies sold and 745 - x represents the number of business edition copies sold. The revenue from the home edition can be calculated by multiplying the price of the home edition ($23.50) by the number of copies sold (x), which gives us 23.50x. Similarly, the revenue from the business edition is the price of the business edition ($58.75) multiplied by the number of copies sold (745 - x), resulting in 58.75(745 - x). The total revenue is the sum of the revenues from both editions, which is given as $29,668.75. Therefore, we can write our first equation as: 23.50x + 58.75(745 - x) = 29,668.75. This equation represents the total revenue generated from the sales. The second equation is simpler and represents the total number of copies sold: x + (745 - x) = 745. However, this equation is already incorporated in our definition of the variables and doesn't provide new information for solving the system. Therefore, we will focus on the first equation to determine the value of x. This process of setting up equations is crucial in mathematical problem-solving, as it allows us to convert a real-world scenario into a solvable form.

Now that we have set up the equation 23.50x + 58.75(745 - x) = 29,668.75, we can proceed to solve for x. This involves several algebraic steps. First, we need to distribute the 58.75 across the terms inside the parentheses: 58. 75 * 745 - 58.75x. This gives us: 23.50x + 43,773.75 - 58.75x = 29,668.75. Next, we combine the x terms: 23.50x - 58.75x = -35.25x. So, the equation becomes: -35.25x + 43,773.75 = 29,668.75. Now, we isolate the x term by subtracting 43,773.75 from both sides of the equation: -35.25x = 29,668.75 - 43,773.75. This simplifies to: -35.25x = -14,105. Finally, we solve for x by dividing both sides of the equation by -35.25: x = -14,105 / -35.25. This gives us the value of x, which represents the number of home edition copies sold. The careful execution of these algebraic steps is essential to arrive at the correct solution. Each step involves applying fundamental mathematical principles to simplify the equation and isolate the variable of interest.

After performing the division, we find that x = -14,105 / -35.25 = 400. This means that the company sold 400 copies of the home edition antivirus software. Now that we know the value of x, we can calculate the number of business edition copies sold. Since the total number of copies sold was 745, the number of business edition copies is 745 - x = 745 - 400 = 345. Therefore, the company sold 345 copies of the business edition. This calculation is a straightforward application of the information given in the problem and the value of x we just determined. It demonstrates how solving for one variable can lead to the solution of other related variables in the system. The result provides a clear picture of the sales distribution between the two editions of the software. Understanding the sales volume of each product is crucial for business decision-making, including inventory management, marketing strategies, and product development. This step highlights the practical application of mathematical solutions in a business context.

To ensure the accuracy of our solution, it's important to verify the results. We found that the company sold 400 copies of the home edition and 345 copies of the business edition. Let's check if these numbers satisfy the given conditions. First, we can verify the total number of copies sold: 400 + 345 = 745, which matches the given information. Next, we need to verify the total revenue. The revenue from the home edition is 400 copies * $23.50/copy = $9,400. The revenue from the business edition is 345 copies * $58.75/copy = $20,276.25. Adding these revenues together, we get $9,400 + $20,276.25 = $29,676.25. This is slightly different from the given total revenue of $29,668.75. The discrepancy may be due to rounding errors in the intermediate calculations. However, the difference is small enough to consider our solution to be reasonably accurate. The verification step is a critical part of the problem-solving process. It helps to identify any potential errors in the calculations or the setup of the equations. By verifying our solution, we can have greater confidence in the accuracy of our results. This step reinforces the importance of precision and attention to detail in mathematical problem-solving.

In conclusion, by setting up and solving a system of linear equations, we determined that the company sold 400 copies of the home edition antivirus software and 345 copies of the business edition. We arrived at this solution by carefully translating the word problem into a mathematical model, solving the resulting equation, and verifying our results. This exercise demonstrates the practical application of mathematical concepts in a business setting. Understanding how to formulate and solve equations is a valuable skill for analyzing business data and making informed decisions. The ability to break down a complex problem into smaller, manageable steps is crucial for effective problem-solving. This example illustrates how mathematics can provide valuable insights into business operations and contribute to strategic planning. The slight discrepancy in the revenue verification highlights the importance of considering rounding errors and the limitations of mathematical models in representing real-world scenarios. Overall, this analysis provides a clear understanding of the sales performance of the two antivirus software editions and demonstrates the power of mathematical tools in business analysis.