Solving For 'w': A Step-by-Step Guide

Oct 19, 2025 by ADMIN 38 views

Hey guys! Let's dive into the world of algebra and learn how to solve for 'w'. The equation we're tackling today is: $-\frac{3}{2} w-3=\frac{1}{2} w+2-\frac{3}{4} w+\frac{5}{4}$ . Don't worry if it looks a little intimidating at first. We'll break it down step-by-step to make it super easy to understand. Solving for a variable, in this case 'w', means we want to isolate 'w' on one side of the equation and find its numerical value. This is a fundamental skill in mathematics, used in everything from basic arithmetic to advanced calculus. Understanding how to manipulate equations to isolate a variable is key to solving a wide range of problems in various fields, including science, engineering, and even economics. We'll use a combination of algebraic principles to simplify the equation and get 'w' all by itself. This will involve combining like terms, and using inverse operations (addition/subtraction, multiplication/division) to cancel out terms on one side of the equation and move them to the other. Remember, the goal is always to keep the equation balanced, so whatever you do to one side, you must do to the other. Let's get started and have some fun!

Step-by-Step Solution

First things first, our main focus will be simplifying the equation and grouping like terms. It can be easy to make errors, so we'll be careful in each step. Here’s how we'll solve for 'w':

Combine 'w' terms: On the right side of the equation, we have $\frac{1}{2} w$ and $-\frac{3}{4} w$ . To combine these, we need a common denominator, which is 4. So, we rewrite $\frac{1}{2} w$ as $\frac{2}{4} w$ . Now, we can combine the terms: $\frac{2}{4} w - \frac{3}{4} w = -\frac{1}{4} w$ . Our equation now looks like this: $-\frac{3}{2} w - 3 = -\frac{1}{4} w + 2 + \frac{5}{4}$ . This is a crucial step. It simplifies the right-hand side, making the next steps easier. Always remember that combining like terms is about making the equation cleaner and more manageable.
Combine constant terms: On the right side of the equation, we have $2$ and $\frac{5}{4}$ . To combine these, we can rewrite $2$ as $\frac{8}{4}$ . Now, combine them: $\frac{8}{4} + \frac{5}{4} = \frac{13}{4}$ . The equation now becomes: $-\frac{3}{2} w - 3 = -\frac{1}{4} w + \frac{13}{4}$ . Again, this simplifies our equation further and makes the following steps more direct. Constant terms are the numbers without any variables; combining them consolidates the numbers to make it simpler to isolate the variable.
Move 'w' terms to one side: Let's get all the 'w' terms on the left side. To do this, we'll add $\frac{1}{4} w$ to both sides of the equation. This gives us: $-\frac{3}{2} w + \frac{1}{4} w - 3 = \frac{13}{4}$ . Rewriting $-\frac{3}{2} w$ with a common denominator, we get $-\frac{6}{4} w + \frac{1}{4} w - 3 = \frac{13}{4}$ . Combining the 'w' terms, we get $-\frac{5}{4} w - 3 = \frac{13}{4}$ . Remember, always perform the same operation on both sides of the equation to maintain balance.
Move constant terms to the other side: We want to isolate the 'w' term, so let's get rid of the -3 on the left side. Add 3 to both sides: $-\frac{5}{4} w = \frac{13}{4} + 3$ . Rewriting 3 with a common denominator, we get $\frac{12}{4}$ . Thus $-\frac{5}{4} w = \frac{13}{4} + \frac{12}{4} = \frac{25}{4}$ . Now we're getting close to isolating 'w'.
Isolate 'w': Finally, we need to get 'w' by itself. We do this by dividing both sides of the equation by $-\frac{5}{4}$ . This gives us $w = \frac{25}{4} / -\frac{5}{4}$ . Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of $-\frac{5}{4}$ is $-\frac{4}{5}$ . So, we have $w = \frac{25}{4} * -\frac{4}{5} = -5$ . And there you have it! The solution to the equation. Isn't that cool? It's all about methodically working through the problem, step by step, being careful with your operations. That's solving equations in a nutshell.

The Answer

So, after all the steps, the value of 'w' is -5. Congrats, you solved the equation! It may seem like a lot of steps, but with practice, it becomes easier and faster. Remember to always double-check your work, and don't be afraid to ask for help if you need it. The core concept is about understanding how to use the basic rules of algebra.

Why is Solving for 'w' Important?

Understanding how to solve for 'w' and other variables is a fundamental skill in mathematics and has applications in countless areas. In science, you'll use it to calculate unknown values in formulas, like finding the speed of an object or the amount of a chemical reaction. In engineering, you'll use it to design and analyze structures, circuits, and systems. Even in everyday life, you might use it to calculate budgets, solve financial problems, or understand the cost of a project. More importantly, it helps you develop logical thinking and problem-solving skills which will be valuable in any career.

Applications in Real Life

Let's consider some specific examples where solving for 'w' (or other variables) comes in handy:

Physics: In physics, you may use it to find an unknown quantity, like the time it takes an object to fall, given the distance and acceleration due to gravity. The formulas often require you to rearrange and solve for a specific variable.
Finance: In finance, you might use it to calculate the interest rate on a loan, or to determine how long it will take for an investment to grow to a certain value. Financial models frequently involve solving algebraic equations.
Computer Science: In computer science, it is used in programming to solve equations, optimize algorithms, and build models. Equations are everywhere in the software development process.

These are just a few of the many real-world applications where solving for 'w', or any variable, is essential. From complex engineering problems to simple calculations, it's a vital tool.

Tips for Success

Here are some tips for success to help you become a master of solving equations:

Practice regularly: The more you practice, the more comfortable you'll become with the steps and the different types of equations. Practice with various types of equations, not just the ones with fractions.
Understand the concepts: Don't just memorize the steps. Make sure you understand why you're doing each operation. This will help you solve different types of problems and will strengthen your understanding of algebraic principles.
Take your time: Don't rush through the steps. Slow down and carefully consider each operation to avoid mistakes. Many errors arise from trying to solve the equation too quickly.
Check your work: Always check your answer by plugging it back into the original equation. If the equation is balanced, your solution is correct. This is the best way to verify your work and learn from mistakes.
Use visual aids: Draw diagrams or use visual aids to help understand the concepts. Sometimes, a visual representation can make a complex problem much easier to solve. Things like number lines can be useful.
Don't be afraid to ask for help: If you're struggling, ask your teacher, a friend, or an online resource for help. There are plenty of resources available to help you learn and succeed. Remember there are no silly questions.

Conclusion

So, there you have it! You've learned how to solve for 'w' in a complex equation. Remember the key steps: combine like terms, isolate the variable, and always check your work. These principles will help you tackle any algebraic equation. Keep practicing, and you'll become a pro in no time! Keep in mind that understanding and practicing this type of problem helps build your analytical and problem-solving skills, which are transferable to virtually every other field of study and employment. Also, don't forget to have some fun while learning!