Solve For W: A Step-by-Step Guide To 10/3 = 5w
Hey everyone! Today, we're diving into a fundamental algebraic problem: solving for a variable. In this case, we're tackling the equation 10/3 = 5w. Don't worry, it's not as daunting as it might look! We'll break it down step-by-step, so you'll be a pro at solving these types of equations in no time. Whether you're a student brushing up on your algebra skills or just someone who enjoys a good mathematical challenge, this guide is for you. So, let's put on our thinking caps and get started!
Understanding the Equation
Before we jump into solving, let's make sure we understand what the equation 10/3 = 5w is telling us. In simple terms, this equation states that the fraction 10/3 is equal to 5 times the value of the variable 'w'. Our goal is to isolate 'w' on one side of the equation so we can determine its value. Think of it like a puzzle – we need to manipulate the equation while keeping both sides balanced until we reveal the hidden value of 'w'. This involves using inverse operations, which are operations that undo each other. For example, multiplication and division are inverse operations, as are addition and subtraction. By applying these operations strategically, we can gradually simplify the equation and bring 'w' into the spotlight. Remember, the key is to perform the same operation on both sides of the equation to maintain the equality. If we add, subtract, multiply, or divide one side, we must do the exact same thing to the other side. This ensures that the equation remains balanced and that we arrive at the correct solution. So, with this understanding in mind, let's move on to the first step in solving for 'w'. We'll start by identifying the operation that's currently affecting 'w' and then applying its inverse to both sides of the equation. This process will gradually peel away the layers until we have 'w' all by itself, revealing its true value. Stay with me, and you'll see how straightforward it can be!
Step 1: Isolate 'w'
In our equation, 10/3 = 5w, 'w' is being multiplied by 5. To isolate 'w', we need to perform the inverse operation of multiplication, which is division. We'll divide both sides of the equation by 5. This is a crucial step because it maintains the balance of the equation. Remember, whatever we do to one side, we must do to the other. So, let's divide both sides by 5: (10/3) / 5 = (5w) / 5
Now, let's simplify this. Dividing a fraction by a whole number can sometimes seem tricky, but it's quite manageable. We can rewrite the division by 5 as multiplication by its reciprocal, which is 1/5. So, the left side of the equation becomes (10/3) * (1/5). On the right side, dividing 5w by 5 simply leaves us with 'w', which is exactly what we want! So, our equation now looks like this: (10/3) * (1/5) = w. This is a significant step forward. We've successfully isolated 'w' on one side of the equation. Now, all that's left to do is simplify the left side to find the value of 'w'. This involves multiplying the two fractions together. Remember, when multiplying fractions, we multiply the numerators (the top numbers) and the denominators (the bottom numbers). This will give us a single fraction that represents the value of 'w'. So, let's move on to the next step and perform this multiplication. We're getting closer and closer to our final answer!
Step 2: Simplify the Equation
Now we need to simplify the left side of the equation: (10/3) * (1/5) = w. To multiply fractions, we multiply the numerators and the denominators. So, 10 multiplied by 1 is 10, and 3 multiplied by 5 is 15. This gives us the fraction 10/15. Our equation now looks like this: 10/15 = w. But we're not quite done yet! The fraction 10/15 can be simplified further. Both 10 and 15 are divisible by 5. So, we can divide both the numerator and the denominator by 5 to reduce the fraction to its simplest form. Dividing 10 by 5 gives us 2, and dividing 15 by 5 gives us 3. Therefore, the simplified fraction is 2/3. This means that the value of 'w' is 2/3. We've successfully simplified the equation and found our solution! It's always a good idea to double-check your work, especially when dealing with fractions. So, let's quickly review the steps we've taken to ensure we haven't made any mistakes. We started by dividing both sides of the original equation by 5 to isolate 'w'. Then, we simplified the resulting fraction by multiplying and reducing it to its simplest form. This process has led us to the solution: w = 2/3. Now, let's write out our final answer clearly and concisely.
Final Answer
Therefore, the solution to the equation 10/3 = 5w is:
w = 2/3
We've successfully solved for 'w'! This is a fantastic example of how we can use algebraic principles to isolate variables and find their values. Remember, the key is to perform the same operations on both sides of the equation to maintain balance and to use inverse operations to gradually simplify the equation. Solving for variables is a fundamental skill in algebra and mathematics in general. It's a skill that will be used time and time again in more complex problems and applications. So, mastering this basic concept is crucial for building a strong foundation in mathematics. I encourage you to practice solving similar equations to solidify your understanding. The more you practice, the more comfortable and confident you'll become in your ability to solve for variables. And remember, mathematics is not just about memorizing formulas and procedures; it's about understanding the underlying concepts and principles. Once you grasp these concepts, you'll find that mathematics becomes much more accessible and even enjoyable. So, keep practicing, keep exploring, and keep challenging yourself. You've got this! And if you ever get stuck, don't hesitate to review the steps we've covered in this guide. We've broken down the process into clear and manageable steps, so you can easily follow along and find the solution. Congratulations on solving for 'w'! You've taken another step forward in your mathematical journey.
Practice Problems
To really solidify your understanding of solving for variables, let's try a few practice problems. These problems are similar to the one we just solved, but they'll give you a chance to apply the same principles in different contexts. Remember, the key is to isolate the variable by performing inverse operations on both sides of the equation. So, grab a pencil and paper, and let's get started!
- Solve for x: 2x = 8/5
- Solve for y: 1/4 = 3y
- Solve for z: 7z = 14/3
These problems will give you a good opportunity to practice dividing fractions and simplifying your answers. Remember to always reduce your fractions to their simplest form. And don't be afraid to make mistakes! Mistakes are a natural part of the learning process. When you make a mistake, take the time to understand why you made it and how you can avoid making it again in the future. This is how you grow and improve your skills. If you're feeling stuck, try reviewing the steps we took in the original problem. You can also break the problem down into smaller steps. First, identify the operation that's currently affecting the variable. Then, determine the inverse operation you need to perform. Finally, apply that operation to both sides of the equation and simplify. With practice, you'll become more and more confident in your ability to solve these types of equations. And remember, solving for variables is a fundamental skill that will be invaluable in your future mathematical studies. So, keep practicing and keep learning! You're doing great!
Solutions to Practice Problems:
- x = 4/5
- y = 1/12
- z = 2/3
