Linear Function Basketball Ticket Costs A Step-by-Step Solution

Introduction

Understanding linear functions is crucial in many real-world applications, and one common example is calculating costs. In this article, we'll break down a problem involving the cost of basketball tickets ordered online. We aim to determine the linear function that represents the total cost ($c$) when $x$ tickets are ordered, given a set price per ticket and a service fee. This article will delve deep into how to construct this function step-by-step, ensuring you grasp the underlying concepts and can apply them to similar scenarios. This is not just about solving a math problem; it's about understanding how mathematical models reflect everyday financial transactions. Let's begin by analyzing the information provided and laying the groundwork for our solution. We will carefully dissect the components of the cost structure, identify the variables and constants at play, and then piece them together to form the linear function. The ability to translate real-world situations into mathematical expressions is a fundamental skill, and this example provides an excellent opportunity to hone that skill. Whether you're a student learning about linear functions or simply someone interested in the practical applications of math, this article will offer valuable insights and a clear methodology for tackling such problems.

Problem Statement

Tickets to a basketball game can be ordered online for a set price per ticket, plus a $5.50 service fee. The total cost for ordering 5 tickets is $108.00. Our goal is to find the linear function that represents $c$, the total cost, when $x$ tickets are ordered. This problem encapsulates the essence of linear functions in a practical context. We are given a fixed cost (the service fee) and a variable cost (the price per ticket multiplied by the number of tickets). The combination of these costs gives us the total cost, which we can express as a linear function. To solve this, we need to first determine the price per ticket. Once we have that, we can construct the linear function in the form of $c = mx + b$, where $m$ is the slope (price per ticket) and $b$ is the y-intercept (service fee). This approach highlights the power of algebra in modeling real-world scenarios. By breaking down the problem into smaller, manageable steps, we can systematically arrive at the solution. This example underscores the importance of identifying key information and using it to build a mathematical representation of the situation. The problem provides a concrete scenario that allows us to apply our understanding of linear functions and their components.

Setting Up the Equation

To find the linear function, we first need to determine the price per ticket. Let's denote the price per ticket as $p$. The total cost for 5 tickets can be represented as: 5p + 5.50 = 108.00. This equation is the cornerstone of our solution. It translates the problem statement into a mathematical form, allowing us to use algebraic techniques to find the unknown variable, $p$. The equation reflects the structure of the cost calculation: the cost of the tickets (5 times the price per ticket) plus the fixed service fee equals the total cost. By setting up the equation correctly, we ensure that our subsequent steps will lead us to the accurate solution. This process of translating words into equations is a fundamental skill in mathematics and is essential for problem-solving in various contexts. The equation serves as a model of the situation, capturing the relationships between the variables and constants involved. It allows us to apply the tools of algebra to extract the information we need. In this case, the equation allows us to isolate the variable $p$ and solve for the price per ticket, which is a crucial step in determining the linear function.

Solving for the Price per Ticket

Now, let's solve for $p$ in the equation 5p + 5.50 = 108.00. First, subtract 5.50 from both sides: 5p = 108.00 - 5.50 which simplifies to 5p = 102.50. Next, divide both sides by 5: p = 102.50 / 5 which gives us p = 20.50. Therefore, the price per ticket is $20.50. This calculation is a straightforward application of algebraic principles. We use the properties of equality to isolate the variable $p$ and determine its value. Each step in the process is designed to simplify the equation while maintaining its balance. The result, $p = 20.50$, is a key piece of information that we will use to construct the linear function. Knowing the price per ticket allows us to express the variable cost component of the total cost. This step demonstrates the power of algebra in solving for unknowns and extracting valuable information from equations. The calculated price per ticket provides a concrete value that we can incorporate into our model of the total cost.

Constructing the Linear Function

Now that we know the price per ticket is $20.50, we can construct the linear function that represents the total cost $c$ when $x$ tickets are ordered. The linear function will have the form c = mx + b, where m is the price per ticket and b is the service fee. In this case, m = 20.50 and b = 5.50. Therefore, the linear function is: c = 20.50x + 5.50. This function is the culmination of our problem-solving process. It encapsulates the relationship between the number of tickets ordered and the total cost. The function is linear because the cost increases at a constant rate ($20.50 per ticket) plus a fixed service fee. The slope of the line, 20.50, represents the price per ticket, and the y-intercept, 5.50, represents the service fee. This linear function allows us to calculate the total cost for any number of tickets ordered. By substituting different values for $x$, we can determine the corresponding total cost $c$. This demonstrates the utility of mathematical models in making predictions and understanding relationships. The linear function provides a concise and accurate representation of the cost structure for ordering basketball tickets online.

Final Answer

The linear function that represents $c$, the total cost, when $x$ tickets are ordered is: c = 20.50x + 5.50. This is our final answer, and it represents a comprehensive solution to the problem. The linear function is a powerful tool for understanding and predicting costs in this scenario. It clearly shows the relationship between the number of tickets ordered and the total cost, taking into account both the variable cost (price per ticket) and the fixed cost (service fee). The function is expressed in slope-intercept form, which makes it easy to interpret the meaning of the coefficients. The slope of 20.50 represents the cost per ticket, and the y-intercept of 5.50 represents the service fee. This solution demonstrates the practical application of linear functions in everyday situations. By breaking down the problem into smaller steps and using algebraic techniques, we were able to derive a mathematical model that accurately represents the cost structure for ordering basketball tickets online. This example highlights the importance of mathematical literacy in navigating financial transactions and making informed decisions.

Conclusion

In conclusion, we have successfully determined the linear function that represents the total cost of ordering basketball tickets online. By carefully analyzing the problem statement, setting up an equation, solving for the unknown variable, and constructing the function, we have demonstrated a systematic approach to solving real-world mathematical problems. This exercise underscores the importance of understanding linear functions and their applications in various contexts. The linear function c = 20.50x + 5.50 provides a clear and concise representation of the cost structure, allowing us to easily calculate the total cost for any number of tickets ordered. This example highlights the power of mathematics in modeling real-world scenarios and making predictions. The ability to translate verbal descriptions into mathematical equations is a crucial skill, and this problem provides a valuable opportunity to practice that skill. Furthermore, this exercise reinforces the importance of understanding the components of a linear function, such as the slope and y-intercept, and their meanings in the context of the problem. By mastering these concepts, you will be well-equipped to tackle similar problems and apply mathematical principles to solve real-world challenges. The process of solving this problem demonstrates the interconnectedness of mathematical concepts and their relevance to everyday situations.