Solving $\sqrt{6y} = \sqrt{36 + 2y}$ A Step-by-Step Guide

Jul 10, 2025 by ADMIN 60 views

$Solving Radical Equations A Step-by-Step Guide to $\sqrt{6y} = \sqrt{36 + 2y}$$

Introduction

In the realm of mathematics, solving equations is a fundamental skill. Among the various types of equations, radical equations, which involve square roots or other radicals, can sometimes present a challenge. This article provides a detailed walkthrough of how to solve the radical equation $\sqrt{6y} = \sqrt{36 + 2y}$, breaking down each step with clear justifications. By understanding the underlying principles, you'll be well-equipped to tackle similar problems with confidence. Whether you're a student learning algebra or simply brushing up on your math skills, this guide will help you master the art of solving radical equations.

Step-by-Step Solution of $\sqrt{6y} = \sqrt{36 + 2y}$

1. The Original Equation: $\sqrt{6y} = \sqrt{36 + 2y}$

We begin with the given equation, which sets the stage for our solution. The original equation $\sqrt{6y} = \sqrt{36 + 2y}$ is a radical equation because it involves square roots. To solve this type of equation, our primary goal is to eliminate the radicals and isolate the variable y. This involves performing operations on both sides of the equation to maintain balance and ensure we arrive at the correct solution. Understanding the initial setup is crucial, as it dictates the subsequent steps we will take. This first step lays the groundwork for a systematic approach to solving the equation, ensuring that we follow the correct algebraic procedures to arrive at the correct value for y. Our first step in solving this equation involves understanding the structure of the equation itself, identifying the radicals, and recognizing the variable we need to isolate. By clearly understanding the initial problem, we set ourselves up for a successful solution.

2. Squaring Both Sides: $(\sqrt{6y})^2 = (\sqrt{36 + 2y})^2$ which simplifies to $6y = 36 + 2y$

To eliminate the square roots, we square both sides of the equation. This is a crucial step in solving radical equations because it allows us to remove the radical signs and work with a simpler, algebraic equation. Squaring both sides of the equation $\sqrt{6y} = \sqrt{36 + 2y}$ means we apply the exponent of 2 to both the left-hand side and the right-hand side. When we square a square root, the radical is effectively canceled out. On the left side, $(\sqrt{6y})^2$ simplifies to $6y$. On the right side, $(\sqrt{36 + 2y})^2$ simplifies to $36 + 2y$. By performing this operation, we transform the equation from one involving square roots to a linear equation, which is much easier to solve. The result, $6y = 36 + 2y$, is a linear equation in terms of y, and it sets us up for the next steps in isolating the variable. This step is justified by the property that if a = b, then a² = b², which is a fundamental principle in algebra. This simplification is key to moving forward and finding the solution for y.

3. Subtracting 2y from Both Sides: $6y - 2y = 36 + 2y - 2y$ which simplifies to $4y = 36$

Now, we aim to isolate the y term on one side of the equation. To do this, we subtract 2y from both sides. Subtracting 2y from both sides of the equation $6y = 36 + 2y$ is a strategic move to consolidate the terms involving y on one side. This operation maintains the balance of the equation, as we are performing the same action on both sides. When we subtract 2y from the left side, $6y - 2y$ simplifies to $4y$. On the right side, subtracting 2y cancels out the 2y term, leaving us with just 36. This simplification leads to the equation $4y = 36$, which is a much simpler form and brings us closer to isolating y. This step is justified by the subtraction property of equality, which states that if a = b, then a - c = b - c. This property is a cornerstone of algebraic manipulation, allowing us to rearrange equations while preserving their validity. By isolating the y term, we are setting up the final step to solve for y. This step is critical in solving the equation and requires a solid understanding of algebraic principles.

4. Dividing Both Sides by 4: $\frac{4y}{4} = \frac{36}{4}$ which simplifies to $y = 9$

To finally isolate y, we divide both sides of the equation by 4. Dividing both sides of the equation $4y = 36$ by 4 is the final step in isolating the variable y. This operation maintains the balance of the equation, as we are performing the same action on both sides. When we divide the left side, $4y$, by 4, we are left with just y. On the right side, dividing 36 by 4 gives us 9. Therefore, the equation simplifies to $y = 9$, which is the solution to the original radical equation. This step is justified by the division property of equality, which states that if a = b, then a/c = b/c, provided that c is not zero. In this case, c is 4, and it is perfectly valid to divide both sides by 4. This final step directly reveals the value of y, providing the solution to the equation. It’s a clear and decisive action that concludes the algebraic manipulation process.

5. Verification: Substitute $y = 9$ into the original equation $\sqrt{6y} = \sqrt{36 + 2y}$

To ensure our solution is correct, we substitute y = 9 back into the original equation. Verification is a critical step in solving any equation, especially radical equations, because it helps us confirm that our solution is correct and doesn't introduce any extraneous roots. An extraneous root is a solution that emerges from the algebraic process but does not satisfy the original equation. By substituting $y = 9$ back into the original equation $\sqrt{6y} = \sqrt{36 + 2y}$, we are checking whether the left-hand side (LHS) and the right-hand side (RHS) are equal. Substituting gives us $\sqrt{6(9)} = \sqrt{36 + 2(9)}$ which simplifies to $\sqrt{54} = \sqrt{36 + 18}$. Further simplification yields $\sqrt{54} = \sqrt{54}$, which is a true statement. Since both sides of the equation are equal when y = 9, we can confidently say that our solution is valid. This step is not just a formality; it's a safeguard against potential errors and a way to ensure the accuracy of our result. By confirming that our solution satisfies the original equation, we complete the problem-solving process with confidence.

Conclusion

Solving radical equations involves a series of algebraic steps, each with its own justification. By systematically squaring both sides, isolating the variable, and verifying the solution, we can confidently solve equations like $\sqrt{6y} = \sqrt{36 + 2y}$. This step-by-step guide provides a clear understanding of the process, empowering you to tackle similar problems with ease. Remember, the key is to understand the underlying principles and apply them methodically. With practice, you'll become proficient in solving radical equations and other algebraic challenges. Understanding each step and its justification is crucial for mastering this skill, ensuring accuracy and confidence in your mathematical abilities.

Keywords

Solving radical equations, step-by-step solution, algebraic manipulation, square roots, isolating variables, verification, extraneous roots, subtraction property of equality, division property of equality, balancing equations, linear equations, mathematical problem-solving, equation solving methods.