Solving For X In 6x - 12y = 18 A Step By Step Guide

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In this article, we will delve into the process of solving the linear equation 6x - 12y = 18 for the variable x. Linear equations are fundamental in mathematics and appear in various real-world applications, from physics and engineering to economics and computer science. Mastering the techniques to solve these equations is crucial for anyone seeking a solid foundation in algebra and beyond. We will break down the steps involved in isolating x and expressing it in terms of y, providing a clear and concise explanation for each step. This detailed guide aims to equip you with the necessary skills to confidently tackle similar problems and build a deeper understanding of algebraic manipulations. By the end of this article, you'll be able to approach linear equations with ease and precision. Our main objective is to transform the given equation into a form where x is isolated on one side, expressing its value in terms of y. This involves applying algebraic operations to both sides of the equation while maintaining the equality. The solution will reveal the relationship between x and y, allowing us to determine the value of x for any given value of y. Understanding these concepts is not just about solving equations; it's about developing critical thinking and problem-solving skills that are valuable in many aspects of life. So, let's embark on this journey of algebraic exploration and unlock the secrets of linear equations together!

Step-by-Step Solution

To solve the equation 6x - 12y = 18 for x, we will follow a systematic approach. Our goal is to isolate x on one side of the equation. This involves performing algebraic operations on both sides to maintain the equality. Let's break down the process into manageable steps:

1. Isolate the Term with x

Our first step is to isolate the term containing x, which is 6x. To do this, we need to eliminate the -12y term on the left side of the equation. We can achieve this by adding 12y to both sides of the equation. Remember, whatever operation we perform on one side, we must perform on the other side to maintain balance. This is a fundamental principle of algebra, ensuring that the equation remains valid throughout the solution process. Adding the same value to both sides doesn't change the equality, but it helps us to simplify the equation and move closer to isolating x. This principle is applicable to all types of equations, making it a cornerstone of algebraic manipulation.

Original equation: 6x - 12y = 18

Adding 12y to both sides:

6x - 12y + 12y = 18 + 12y

Simplifying the left side, -12y + 12y cancels out:

6x = 18 + 12y

Now we have successfully isolated the term with x on the left side. The equation is now in a simpler form, making it easier to proceed with the next step. This process of isolating terms is a crucial technique in solving equations and is used extensively in various mathematical contexts. It allows us to focus on the variable we want to solve for and eliminate the clutter of other terms.

2. Solve for x

Now that we have the equation 6x = 18 + 12y, we need to isolate x completely. Currently, x is being multiplied by 6. To undo this multiplication, we will perform the inverse operation, which is division. We will divide both sides of the equation by 6. Just like adding or subtracting, dividing both sides by the same non-zero number maintains the equality. This is another fundamental principle in solving equations. Dividing both sides ensures that the relationship between the two sides remains consistent, leading us to the correct solution. It's important to remember that we cannot divide by zero, as this operation is undefined in mathematics. In this case, dividing by 6 is perfectly valid and will help us isolate x.

Dividing both sides by 6:

(6x) / 6 = (18 + 12y) / 6

On the left side, 6 cancels out, leaving x:

x = (18 + 12y) / 6

Now, we can simplify the right side by dividing each term in the numerator by 6:

x = 18/6 + 12y/6

Performing the divisions:

x = 3 + 2y

We can rewrite this as:

x = 2y + 3

Therefore, we have successfully solved the equation for x. The solution x = 2y + 3 expresses x in terms of y. This means that for any given value of y, we can substitute it into this equation to find the corresponding value of x. This solution represents a linear relationship between x and y, which can be visualized as a straight line on a graph.

Final Answer

The solution to the equation 6x - 12y = 18 for x is:

x = 2y + 3

This corresponds to option A in the given choices. We have shown, step-by-step, how to isolate x and arrive at this solution. Understanding the principles of algebraic manipulation is key to solving linear equations and other mathematical problems. This solution provides a clear and concise relationship between x and y, allowing for easy calculation of x given any value of y. The ability to solve for variables in equations is a fundamental skill in mathematics and has wide-ranging applications in various fields.

Why Other Options are Incorrect

It's important to understand why the other options provided are incorrect. This helps solidify your understanding of the solution process and prevent similar errors in the future. Let's examine each incorrect option:

  • B. x = 2y + 18: This option is incorrect because it fails to correctly divide the constant term 18 by 6 during the simplification process. While the 12y term is correctly divided, the constant term remains unchanged, leading to an incorrect solution. The error highlights the importance of applying the same operation to all terms when dividing both sides of an equation.

  • C. x = 12y + 18: This option is significantly incorrect. It seems to misunderstand the initial steps of isolating x. The 12y term was not properly moved to the right side of the equation, and the division by 6 was completely missed. This error demonstrates a lack of understanding of the basic principles of equation solving.

  • D. x = 12y + 3: This option makes a similar mistake to option B by not correctly handling the 12y term during the division step. The 12y term should have been divided by 6, resulting in 2y, not 12y. This error underscores the need for careful attention to detail when performing algebraic manipulations.

By analyzing these incorrect options, we can gain a deeper appreciation for the correct solution and the importance of each step in the process. It's not enough to simply arrive at the answer; understanding why the other options are wrong is crucial for building a strong foundation in algebra.

Practice Problems

To further solidify your understanding of solving linear equations for a specific variable, here are a few practice problems. Try to solve them using the steps outlined in this guide. Remember to show your work and double-check your answers.

  1. Solve for y: 3x + 5y = 15
  2. Solve for a: 7a - 2b = 21
  3. Solve for p: 4p + 8q = 32

Working through these practice problems will help you develop confidence and proficiency in solving linear equations. The key is to practice consistently and pay attention to the details. If you encounter any difficulties, revisit the steps outlined in this guide or seek additional resources. Remember, mastering these skills will benefit you in various areas of mathematics and beyond.

Conclusion

In conclusion, solving the equation 6x - 12y = 18 for x involves a series of algebraic manipulations aimed at isolating x. The correct solution is x = 2y + 3. This process highlights the importance of understanding and applying the fundamental principles of algebra, such as maintaining equality by performing the same operations on both sides of the equation. By following a systematic approach and paying attention to detail, you can confidently solve similar linear equations. The ability to solve for variables is a crucial skill in mathematics and has wide-ranging applications in various fields, making it a valuable asset for anyone seeking to excel in STEM disciplines and beyond. We hope this comprehensive guide has provided you with a clear understanding of the solution process and empowered you to tackle algebraic challenges with confidence. Remember, practice makes perfect, so continue to work on similar problems to further enhance your skills and deepen your understanding of these concepts.