Solving For W In The Equation (5w + 4)/3 = (3w)/2

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In this article, we'll walk through the process of finding the value of 'w' that satisfies the equation (5w + 4)/3 = (3w)/2. This is a common type of algebraic problem that involves solving for a variable in an equation with fractions. We will use step-by-step instructions to ensure clarity and understanding. This article will be very useful for students and anyone wanting to refresh their algebraic skills. Understanding how to solve such equations is essential for more advanced mathematical concepts. Our goal is to make the solution process clear, concise, and easy to follow.

Understanding the Problem

Before we dive into solving the equation, let's first ensure we understand the problem at hand. The equation we're dealing with is:

(5w + 4) / 3 = (3w) / 2

Here, 'w' is the variable we need to find. The equation states that the expression (5w + 4) divided by 3 is equal to the expression (3w) divided by 2. To solve for 'w', we need to manipulate the equation in a way that isolates 'w' on one side.

This involves several algebraic steps, including eliminating fractions, distributing terms, combining like terms, and finally, isolating the variable. Each step is crucial to arriving at the correct solution. We will break down each of these steps in detail to ensure clarity.

Step-by-Step Solution

Now, let's go through the step-by-step solution to find the value of 'w'.

Step 1: Eliminate the Fractions

To eliminate the fractions, we need to find the least common multiple (LCM) of the denominators, which are 3 and 2. The LCM of 3 and 2 is 6. We multiply both sides of the equation by 6 to get rid of the fractions:

6 * [(5w + 4) / 3] = 6 * [(3w) / 2]

This simplifies to:

2 * (5w + 4) = 3 * (3w)

Step 2: Distribute the Terms

Next, we distribute the numbers on both sides of the equation:

2 * 5w + 2 * 4 = 3 * 3w

Which simplifies to:

10w + 8 = 9w

Step 3: Combine Like Terms

Now, we want to get all the terms with 'w' on one side of the equation. Subtract 9w from both sides:

10w - 9w + 8 = 9w - 9w

This simplifies to:

w + 8 = 0

Step 4: Isolate the Variable

Finally, to isolate 'w', subtract 8 from both sides of the equation:

w + 8 - 8 = 0 - 8

This gives us:

w = -8

Therefore, the value of w that makes the equation true is -8.

Conclusion

In summary, we found that the value of w that satisfies the equation (5w + 4) / 3 = (3w) / 2 is -8. We achieved this by systematically eliminating fractions, distributing terms, combining like terms, and isolating the variable. This step-by-step approach is essential for solving algebraic equations effectively. Understanding these steps will help you tackle similar problems with confidence.

Choosing the Correct Answer

Now that we have solved the equation and found that w = -8, let's look at the multiple-choice options provided:

A. -8 B. -2 C. -1 D. 8

The correct answer is A. -8, which matches our solution. It's always a good practice to double-check your answer by substituting it back into the original equation to ensure it holds true.

Importance of Understanding Algebraic Equations

Algebraic equations form the foundation of many mathematical and scientific disciplines. Solving these equations is a crucial skill that is applied in various fields, including engineering, physics, computer science, and economics. Understanding the principles of algebra not only helps in academic pursuits but also in practical problem-solving in everyday life.

Common Mistakes to Avoid

When solving algebraic equations, there are several common mistakes that students often make. Being aware of these can help you avoid them and improve your accuracy.

1. Incorrectly Distributing Terms

One common mistake is not properly distributing terms. For example, in the equation 2 * (5w + 4), it's important to multiply both 5w and 4 by 2. A mistake would be to only multiply one of the terms.

2. Not Eliminating Fractions Properly

When dealing with fractions, it's essential to eliminate them by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. Forgetting to do this, or calculating the LCM incorrectly, can lead to incorrect solutions.

3. Combining Unlike Terms

Another common mistake is to combine terms that are not like terms. For example, you cannot add 10w and 8 directly because they are not like terms. You can only combine terms that have the same variable raised to the same power.

4. Sign Errors

Sign errors are also frequent. Be careful when adding or subtracting negative numbers, and always double-check your work to ensure you haven't made any sign mistakes.

5. Not Isolating the Variable Completely

The goal of solving for a variable is to isolate it completely on one side of the equation. Make sure you perform all the necessary steps to get the variable by itself.

By being mindful of these common mistakes and practicing regularly, you can improve your problem-solving skills and avoid errors.

Tips for Solving Algebraic Equations

To become proficient in solving algebraic equations, here are some helpful tips:

  1. Understand the Basics: Ensure you have a solid understanding of the fundamental principles of algebra, including the order of operations, combining like terms, and the properties of equality.
  2. Practice Regularly: Practice is key to mastering any mathematical skill. The more you practice, the more comfortable you will become with the process of solving equations.
  3. Show Your Work: Always show each step of your solution. This makes it easier to track your progress and identify any mistakes.
  4. Check Your Answers: After solving an equation, substitute your solution back into the original equation to verify that it is correct. This will help you catch any errors and build confidence in your solutions.
  5. Break Down Complex Problems: If you encounter a complex equation, break it down into smaller, more manageable steps. This will make the problem less daunting and easier to solve.
  6. Use Visual Aids: Visual aids, such as diagrams or graphs, can sometimes help you understand and solve equations more easily.
  7. Seek Help When Needed: Don't hesitate to ask for help if you are struggling with a particular type of equation. Your teacher, classmates, or online resources can provide valuable assistance.

By following these tips and staying persistent, you can enhance your algebraic skills and become a more confident problem solver.

Practice Problems

To reinforce your understanding, here are some practice problems similar to the one we solved. Try solving them on your own, and then check your answers:

  1. Solve for x: (3x + 5) / 2 = (2x) / 3
  2. Solve for y: (4y - 2) / 5 = (y) / 4
  3. Solve for z: (6z + 1) / 3 = (4z) / 2

Working through these problems will give you more practice and solidify your understanding of how to solve equations with fractions.

Real-World Applications

Understanding how to solve algebraic equations is not just an academic exercise; it has many real-world applications. For example, engineers use algebraic equations to design structures and systems, physicists use them to describe the laws of nature, and economists use them to model economic behavior. In everyday life, you might use algebraic equations to calculate budgets, plan trips, or make financial decisions.

Final Thoughts

Solving algebraic equations is a fundamental skill that is valuable in many areas of life. By understanding the basic principles and practicing regularly, you can become proficient at solving equations and apply this skill to a variety of problems. Remember to break down complex problems into smaller steps, show your work, and check your answers. With persistence and practice, you can master algebra and use it to solve problems in many different contexts.

By working through this article and the practice problems, you should now have a solid understanding of how to solve equations of this type. Keep practicing, and you'll continue to improve your skills in algebra!