Rewriting Equations As Functions A Step-by-Step Guide For 56x + 7y + 21 = 0

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In the realm of mathematics, the ability to manipulate equations and express them in different forms is a fundamental skill. This article delves into the process of rewriting an equation as a function of x, using the example equation 56x + 7y + 21 = 0. We will explore the steps involved, the underlying concepts, and the importance of this skill in various mathematical contexts.

Understanding Functions and Equations

Before we dive into the specifics of rewriting the equation, it's crucial to understand the core concepts of functions and equations. An equation is a mathematical statement that asserts the equality of two expressions. It typically involves variables, constants, and mathematical operations. A function, on the other hand, is a special type of relation that maps each input value (often denoted as x) to a unique output value (often denoted as y or f(x)). In essence, a function describes a relationship between two sets of values, where each input has only one corresponding output.

When we rewrite an equation as a function of x, we are essentially isolating the variable y on one side of the equation and expressing it in terms of x. This allows us to easily determine the value of y for any given value of x, effectively defining a function that represents the relationship described by the original equation.

Step-by-Step Guide to Rewriting 56x + 7y + 21 = 0 as a Function of x

Let's now embark on the journey of rewriting the equation 56x + 7y + 21 = 0 as a function of x. We'll break down the process into manageable steps, ensuring a clear understanding of each stage.

1. Isolate the Term Containing y

The initial step involves isolating the term containing y (which is 7y) on one side of the equation. To achieve this, we need to eliminate the other terms (56x and 21) from the left side. We can accomplish this by subtracting 56x and 21 from both sides of the equation.

56x + 7y + 21 - 56x - 21 = 0 - 56x - 21

This simplifies to:

7y = -56x - 21

2. Solve for y

Now that we have isolated the term 7y, our next goal is to solve for y itself. To do this, we need to get rid of the coefficient 7 that is multiplying y. We can achieve this by dividing both sides of the equation by 7.

(7y) / 7 = (-56x - 21) / 7

This simplifies to:

y = -8x - 3

3. Express as a Function of x

The final step is to express the equation in function notation. We replace y with f(x) to indicate that y is a function of x. This gives us:

f(x) = -8x - 3

Therefore, the equation 56x + 7y + 21 = 0, when rewritten as a function of x, becomes f(x) = -8x - 3. This corresponds to option D in the original question.

Why is Rewriting Equations as Functions Important?

Rewriting equations as functions is a crucial skill in mathematics for several reasons:

  • Function Analysis: Expressing an equation as a function allows us to analyze its behavior, such as its slope, intercepts, and domain. This information is invaluable in understanding the relationship between the variables involved.
  • Graphing: Functions can be easily graphed, providing a visual representation of the relationship between x and y. This visual representation can aid in understanding the function's properties and behavior.
  • Solving Systems of Equations: Rewriting equations as functions is a key step in solving systems of equations. By expressing each equation as a function, we can use various techniques, such as substitution or elimination, to find the values of x and y that satisfy all equations in the system.
  • Modeling Real-World Phenomena: Functions are used extensively to model real-world phenomena. By expressing relationships between variables as functions, we can make predictions and gain insights into the behavior of these phenomena.

Common Mistakes to Avoid

While the process of rewriting equations as functions is relatively straightforward, there are some common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate results.

  • Incorrectly Applying the Distributive Property: When dividing both sides of the equation by a constant, it's crucial to apply the distributive property correctly. Make sure to divide every term on the right side of the equation by the constant.
  • Forgetting the Sign: Pay close attention to the signs of the terms when moving them from one side of the equation to the other. Remember to change the sign when moving a term across the equals sign.
  • Not Simplifying Completely: After solving for y, make sure to simplify the expression as much as possible. This will ensure that the function is in its simplest form and easier to work with.

Practice Problems

To solidify your understanding of rewriting equations as functions, let's work through a few practice problems.

Problem 1: Rewrite the equation 2x - 3y + 6 = 0 as a function of x.

Solution:

  1. Isolate the term containing y: -3y = -2x - 6
  2. Solve for y: y = (2/3)x + 2
  3. Express as a function of x: f(x) = (2/3)x + 2

Problem 2: Rewrite the equation 4x + 2y - 8 = 0 as a function of x.

Solution:

  1. Isolate the term containing y: 2y = -4x + 8
  2. Solve for y: y = -2x + 4
  3. Express as a function of x: f(x) = -2x + 4

Conclusion

Rewriting equations as functions of x is a fundamental skill in mathematics that has wide-ranging applications. By understanding the steps involved and practicing regularly, you can master this skill and confidently tackle more complex mathematical problems. This article has provided a comprehensive guide to rewriting the equation 56x + 7y + 21 = 0 as a function of x, along with explanations of the underlying concepts, common mistakes to avoid, and practice problems to reinforce your understanding. With dedication and practice, you can become proficient in this essential mathematical skill.

In summary, to rewrite the equation 56x + 7y + 21 = 0 as a function of x, we follow these steps:

  1. Isolate the term containing y: 7y = -56x - 21
  2. Solve for y: y = -8x - 3
  3. Express as a function of x: f(x) = -8x - 3

Therefore, the correct answer is D. f(x) = -8x - 3. This skill is essential for understanding and manipulating mathematical relationships, and mastering it will undoubtedly benefit you in your mathematical journey.