Solving -5y + 6 = 16 A Step-by-Step Guide

Introduction

In the realm of mathematics, particularly in algebra, one of the fundamental skills is solving equations. Solving equations involves isolating the unknown variable to determine its value. This article will provide a comprehensive, step-by-step guide on how to solve linear equations, focusing on the specific example of -5y + 6 = 16. Understanding these steps will empower you to tackle a wide array of algebraic problems with confidence.

Linear equations, at their core, are algebraic expressions where the highest power of the variable is one. They represent straight lines when graphed and are ubiquitous in various fields, from physics and engineering to economics and computer science. Mastering the techniques to solve them is, therefore, essential for anyone pursuing these disciplines or simply seeking to enhance their analytical abilities. Solving such equations is a cornerstone of mathematical literacy and an important skill to develop. The process is not merely about finding the answer; it is also about understanding the underlying principles of equality and how mathematical operations can be used to manipulate equations while maintaining their balance. This understanding forms the basis for more advanced mathematical concepts and problem-solving techniques.

Before diving into the specific problem, let's first clarify some basic algebraic principles that underpin the process of solving equations. The most important of these is the principle of equality, which states that performing the same operation on both sides of an equation maintains the equality. This principle is the bedrock upon which we build our equation-solving strategy. We can add, subtract, multiply, or divide both sides of an equation by the same number without disturbing the balance, as long as we apply the operation consistently. Another key concept is the idea of inverse operations. Every mathematical operation has an inverse that undoes it. Addition and subtraction are inverse operations, as are multiplication and division. When solving equations, we use inverse operations to isolate the variable. For example, if a variable is being added to a number, we subtract that number from both sides of the equation. If a variable is being multiplied by a number, we divide both sides by that number. By strategically applying inverse operations, we gradually strip away the terms surrounding the variable until we are left with the variable alone on one side of the equation, equal to its value on the other side. With these basic principles in mind, we are well-equipped to approach the task of solving the equation -5y + 6 = 16. The following sections will guide you through each step of the solution process, providing clear explanations and insights along the way.

Step 1: Isolate the Term with the Variable

In the equation -5y + 6 = 16, our primary goal is to isolate the term containing the variable, which in this case is -5y. To achieve this, we need to eliminate the constant term, +6, that is present on the same side of the equation. The mathematical operation that undoes addition is subtraction. Therefore, to remove the +6, we will subtract 6 from both sides of the equation. This is a crucial step that directly applies the principle of equality, ensuring that the equation remains balanced. By performing the same operation on both sides, we maintain the integrity of the relationship between the expressions. It's not just about getting rid of the +6; it's about methodically moving towards isolating the y term while adhering to the fundamental rules of algebra. The subtraction operation is the key here, but the underlying principle of equality is the guiding force. After subtracting 6 from both sides, the equation will be transformed, bringing us one step closer to the solution. This methodical approach is essential for solving equations accurately and efficiently. The goal is not merely to arrive at the correct answer, but also to understand the process and the reasoning behind each step. This understanding is what allows us to apply these techniques to more complex problems in the future.

Subtracting 6 from both sides, we get:

-5y + 6 - 6 = 16 - 6

This simplifies to:

-5y = 10

Step 2: Isolate the Variable

Now that we have isolated the term -5y, our next step is to isolate the variable y itself. Currently, y is being multiplied by -5. To undo this multiplication, we need to perform the inverse operation, which is division. We will divide both sides of the equation by -5. This is another critical application of the principle of equality. Just as we subtracted the same value from both sides in the previous step, we now divide both sides by the same value. This ensures that the equation remains balanced and that we are progressing towards an accurate solution. The division operation is the key to unlocking the value of y, but the principle of equality is the fundamental rule that governs our actions. By dividing by -5, we effectively separate y from its coefficient, bringing us closer to the final answer. This process is not just about following a mechanical procedure; it is about strategically manipulating the equation to reveal the hidden value of the variable. Each step is a deliberate move, guided by the principles of algebra and the goal of isolation. After performing the division, we will have y standing alone on one side of the equation, equal to its numerical value on the other side. This will complete the process of solving for y.

Dividing both sides by -5, we have:

(-5y) / -5 = 10 / -5

This simplifies to:

y = -2

Step 3: Verify the Solution

To ensure the accuracy of our solution, it is always a good practice to verify it. Verifying the solution involves substituting the value we found for y back into the original equation and checking if the equation holds true. This step is not merely a formality; it is a crucial safeguard against errors that may have crept in during the solution process. It provides a concrete confirmation that our answer is correct and that we have successfully solved the equation. Substitution is the key operation here, but the underlying principle is that a true solution must satisfy the original equation. By plugging the value of y back into the equation, we are essentially testing whether it fits the equation's requirements. If the equation balances, then we can be confident that our solution is correct. If not, then we know that an error has been made somewhere along the way, and we need to retrace our steps to identify and correct it. This verification process is an integral part of mathematical problem-solving, fostering accuracy and building confidence in our results. It reinforces the importance of not just finding an answer, but also ensuring that it is the right answer.

Substituting y = -2 back into the original equation -5y + 6 = 16:

-5(-2) + 6 = 16

10 + 6 = 16

16 = 16

Since the equation holds true, our solution y = -2 is correct.

Conclusion

In summary, we have successfully solved the equation -5y + 6 = 16 by following a clear, step-by-step process. We first isolated the term containing the variable by subtracting 6 from both sides of the equation. Then, we isolated the variable itself by dividing both sides by -5. Finally, we verified our solution by substituting it back into the original equation. This methodical approach, grounded in the principles of equality and inverse operations, is the key to solving linear equations effectively. Understanding these steps is not just about solving this particular equation; it's about developing a robust problem-solving strategy that can be applied to a wide range of algebraic problems. Solving such equations is a fundamental skill in mathematics, and mastering it opens doors to more advanced concepts and applications. The process involves not just manipulation of numbers and symbols, but also a deep understanding of the underlying principles that govern these operations. Each step is a logical progression, guided by the goal of isolating the variable and revealing its value. The ability to solve linear equations is a valuable asset, empowering individuals to tackle mathematical challenges with confidence and precision. It is a skill that transcends the classroom, finding relevance in various real-world scenarios where mathematical thinking is essential.

The solution to the equation -5y + 6 = 16 is y = -2. This process demonstrates the fundamental principles of solving linear equations, a crucial skill in algebra and beyond. By mastering these techniques, you'll be well-equipped to tackle more complex mathematical problems.