Expressing 5x + 7y = 17 As A Function Y = F(x)

In the realm of mathematics, expressing relationships between variables is fundamental. A common way to represent such relationships is through functions, where one variable is defined in terms of another. In this article, we will delve into the process of expressing the linear equation 5x + 7y = 17 as a function in the form y = f(x). This transformation allows us to clearly see how the value of y depends on the value of x, providing a powerful tool for analysis and problem-solving. Understanding how to manipulate equations and isolate variables is a crucial skill in algebra and beyond, and this example will provide a step-by-step guide to achieving this.

Understanding the Basics: Equations and Functions

Before diving into the specifics of our equation, let's lay the groundwork by understanding the core concepts involved: equations and functions. An equation is a mathematical statement that asserts the equality of two expressions. It typically involves variables (symbols representing unknown values) and constants (fixed numerical values), connected by mathematical operations such as addition, subtraction, multiplication, and division. The goal when working with equations is often to find the values of the variables that make the equation true.

A function, on the other hand, describes a relationship between two sets of values: the input (often denoted by x) and the output (often denoted by y). For each input value, there is exactly one output value. This relationship is often expressed as y = f(x), where f is the function's name and f(x) represents the output value when x is the input. The beauty of a function lies in its ability to model real-world phenomena and make predictions based on the relationship it defines. For instance, a function could describe the trajectory of a projectile, the growth of a population, or the cost of producing a certain number of items.

The equation 5x + 7y = 17 represents a linear relationship between x and y. To express this relationship as a function y = f(x), we need to isolate y on one side of the equation. This involves using algebraic manipulations to rearrange the terms and solve for y. Let's embark on this journey step by step.

Step-by-Step Solution: Isolating y

Our starting point is the equation 5x + 7y = 17. The objective is to isolate y on one side of the equation, which means getting y by itself. To achieve this, we will employ a series of algebraic operations, ensuring that we perform the same operation on both sides of the equation to maintain balance. This principle is fundamental to solving equations and guarantees that the equality remains valid throughout the process.

1. Subtract 5x from both sides: The first step is to eliminate the term involving x from the left side of the equation. To do this, we subtract 5x from both sides: 5x + 7y - 5x = 17 - 5x. This simplifies to 7y = 17 - 5x. This step isolates the term containing y on the left side, bringing us closer to our goal of expressing y as a function of x. By subtracting 5x from both sides, we maintain the equality of the equation while strategically rearranging the terms.
2. Divide both sides by 7: Now that we have 7y on the left side, we need to isolate y completely. To do this, we divide both sides of the equation by 7: (7y) / 7 = (17 - 5x) / 7. This simplifies to y = (17 - 5x) / 7. This step completes the isolation of y, expressing it in terms of x. Dividing both sides by 7 ensures that the coefficient of y becomes 1, leaving us with the desired form. Remember, any operation performed on one side of the equation must be mirrored on the other side to maintain the balance and validity of the equation.
3. Rewrite the expression: To express the result in the standard function notation y = f(x) and to match the answer choices provided, we can rewrite the expression. We have y = (17 - 5x) / 7, which can be separated into two fractions: y = 17/7 - (5x)/7. This can further be written as y = 17/7 - (5/7)x. This final step presents the equation in a clear and concise form, making it easy to identify the slope and y-intercept of the linear function. The expression now clearly shows how y depends on x, fulfilling the requirement of expressing the original equation as a function.

By following these steps, we have successfully transformed the equation 5x + 7y = 17 into the function y = 17/7 - (5/7)x. This transformation allows us to easily analyze the relationship between x and y, and to use this relationship for further calculations and applications.

Identifying the Correct Option

Now that we have derived the function y = 17/7 - (5/7)x, we can compare it to the given options to identify the correct answer. Let's examine each option:

• A. y = f(x) = 7/17 - (5/7)x: This option has the correct coefficient for the x term (-5/7), but the constant term is incorrect (7/17 instead of 17/7). Therefore, this option is incorrect.
• B. y = f(x) = 7/17 - (7/5)x: This option has both the constant term and the coefficient for the x term incorrect. The constant term should be 17/7, and the coefficient for x should be -5/7. Thus, this option is also incorrect.
• C. y = f(x) = 17/7 - (7/5)x: This option has the correct constant term (17/7), but the coefficient for the x term is incorrect (-7/5 instead of -5/7). Therefore, this option is incorrect.
• D. y = f(x) = 17/7 - (5/7)x: This option perfectly matches the function we derived, y = 17/7 - (5/7)x. It has the correct constant term (17/7) and the correct coefficient for the x term (-5/7). Therefore, this is the correct option.

Therefore, the correct answer is D. y = f(x) = 17/7 - (5/7)x. This process of comparing our derived solution with the provided options highlights the importance of accurate algebraic manipulation and careful attention to detail.

Key Concepts and Takeaways

This exercise in expressing 5x + 7y = 17 as a function y = f(x) provides several key takeaways for understanding and working with mathematical equations and functions:

• Isolating Variables: The core technique in this problem is isolating the variable y. This involves performing inverse operations on both sides of the equation to move terms around and ultimately get y by itself. Understanding how to isolate variables is a fundamental skill in algebra and is crucial for solving equations and manipulating formulas.
• Maintaining Equality: When manipulating equations, it's essential to maintain equality. This means performing the same operation on both sides of the equation. Any operation performed on one side must be mirrored on the other side to ensure that the equation remains balanced and the solution remains valid. This principle is the cornerstone of algebraic manipulation.
• Function Notation: Expressing a relationship as a function, y = f(x), provides a clear way to see how the output (y) depends on the input (x). Function notation is a powerful tool for modeling relationships and making predictions. The f(x) notation clearly indicates that the value of y is determined by the function f when applied to the input x.
• Linear Functions: The equation 5x + 7y = 17 represents a linear relationship. Linear functions have a constant rate of change, and their graphs are straight lines. The slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept, provides valuable insights into the behavior of the line. In our solution, y = 17/7 - (5/7)x, we can identify the slope as -5/7 and the y-intercept as 17/7.
• Step-by-Step Approach: Breaking down the problem into smaller, manageable steps makes the solution process clearer and less prone to errors. Each step should be carefully executed, ensuring that the operations are performed correctly and the equality is maintained. This methodical approach is a valuable strategy for tackling complex mathematical problems.

By mastering these concepts, you will be well-equipped to tackle a wide range of algebraic problems and gain a deeper understanding of the relationships between variables.

Practical Applications and Further Exploration

The ability to express equations as functions is not just a theoretical exercise; it has practical applications in various fields. For instance, in physics, understanding the relationship between distance, time, and velocity often involves expressing one variable as a function of the others. In economics, supply and demand curves can be represented as functions, allowing economists to analyze market equilibrium. In computer science, functions are the building blocks of programs, allowing programmers to create reusable blocks of code.

To further explore this topic, consider the following avenues:

• Graphing Linear Functions: Graphing the function y = 17/7 - (5/7)x can provide a visual representation of the relationship between x and y. You can plot points, identify the slope and y-intercept, and observe the line's behavior. Tools like graphing calculators or online graphing utilities can be helpful in this process. Visualizing the function can enhance your understanding of its properties and how it relates to the original equation.
• Solving Systems of Equations: The technique of isolating variables is also crucial for solving systems of equations, where you have multiple equations with multiple unknowns. By expressing one variable in terms of others, you can substitute and eliminate variables to find the solution. This skill is essential for many applications in mathematics, science, and engineering.
• Non-Linear Functions: While this article focused on a linear equation, the same principles can be applied to non-linear equations. However, the algebraic manipulations may be more complex. Exploring quadratic, exponential, and trigonometric functions can broaden your understanding of different types of relationships between variables. Each type of function has its unique characteristics and applications, offering a rich landscape for mathematical exploration.

By continuing to explore these concepts and practice your problem-solving skills, you can build a strong foundation in algebra and its applications.

In conclusion, transforming the equation 5x + 7y = 17 into the function y = f(x) = 17/7 - (5/7)x demonstrates a fundamental algebraic skill with broad applications. By understanding the principles of isolating variables, maintaining equality, and using function notation, you can confidently tackle similar problems and deepen your understanding of mathematical relationships.