Solving Linear Equations: A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving into the world of linear equations. Specifically, we're going to solve the equation 2.25 - 11j - 7.75 + 1.5j = 0.5j - 1. Don't worry if it looks a bit intimidating at first; we'll break it down step by step to make it super clear and easy to understand. Linear equations are a fundamental concept in algebra, and mastering them opens doors to solving all sorts of problems in mathematics, science, and even everyday life. So, grab your pencils and let's get started on this exciting journey of learning and discovery. This is more than just about solving one equation; it's about developing a solid understanding of algebraic principles that will serve you well in countless situations. Remember, practice makes perfect, and the more equations you solve, the more confident you'll become. So, let's turn this seemingly complex problem into a straightforward solution! We will explore a detailed guide to solve these linear equations. This will help you to understand the concept and tackle any kind of problems with ease. Let's start with the basics.

Simplifying the Equation: Combining Like Terms

The first step in solving our linear equation, 2.25 - 11j - 7.75 + 1.5j = 0.5j - 1, is to simplify it. This means combining like terms on each side of the equation. Like terms are terms that have the same variable raised to the same power (or are both constants). In our equation, we have several like terms. On the left side, we have constants (2.25 and -7.75) and terms with the variable 'j' (-11j and 1.5j). On the right side, we have 0.5j and -1. Let's start by combining the constants on the left side: 2.25 - 7.75 = -5. Now, combine the 'j' terms on the left side: -11j + 1.5j = -9.5j. After these simplifications, our equation will look like this: -5 - 9.5j = 0.5j - 1. Remember, the goal of simplifying is to make the equation easier to work with, bringing us closer to isolating the variable 'j'. This step is crucial because it reduces the complexity and sets the stage for isolating the variable. Think of it as tidying up your workspace before starting a project. It streamlines the process and helps you focus on the task at hand. The more comfortable you become with combining like terms, the faster and more accurately you'll be able to solve linear equations. Now that we have simplified it, let's keep going. We're going to make sure that we get this right! Let's move to the next step!

Isolating the Variable 'j'

Now, our simplified equation is -5 - 9.5j = 0.5j - 1. Our primary objective is to get the 'j' terms on one side of the equation and the constant terms on the other. This process is called isolating the variable. Let's start by adding 9.5j to both sides of the equation. This will eliminate the -9.5j term on the left side: -5 - 9.5j + 9.5j = 0.5j - 1 + 9.5j. This simplifies to -5 = 10j - 1. Next, we need to move the constant term -1 to the other side. We do this by adding 1 to both sides: -5 + 1 = 10j - 1 + 1. This gives us -4 = 10j. At this point, we've successfully isolated the 'j' term. The next part will be the easiest! Remember, the key is to perform the same operation on both sides of the equation to maintain balance. This ensures that the equality remains true throughout the solving process. Think of the equation like a balanced scale; whatever you do to one side, you must do to the other to keep it balanced. This fundamental principle is what allows us to manipulate the equation and solve for the unknown variable. By systematically isolating the variable, we're steadily getting closer to our final answer. These steps might seem simple, but they are the building blocks of more complex algebraic problems. In this context, it is super important to master the basics and keep going!

Solving for 'j'

We now have the equation -4 = 10j. To solve for 'j', we need to isolate it completely. We do this by dividing both sides of the equation by 10. This will cancel out the coefficient of 'j' on the right side: -4 / 10 = 10j / 10. This simplifies to -0.4 = j. Therefore, the solution to the linear equation 2.25 - 11j - 7.75 + 1.5j = 0.5j - 1 is j = -0.4. We've reached the final step! Congratulations on solving your first linear equation! This final step involves performing the inverse operation to eliminate the coefficient of the variable. In our case, the coefficient was 10, so we divided by 10. This leaves us with the variable 'j' on one side and its numerical value on the other. This value represents the specific number that makes the original equation true. To ensure our solution is correct, we should always verify it by substituting the value of 'j' back into the original equation. If both sides of the equation are equal after the substitution, then our solution is correct. This is called checking the solution and is an essential practice in mathematics. It not only verifies our answer but also reinforces our understanding of the concepts involved. We'll perform this check in the next section. With this, we have reached the end of the calculations! Let's check our solution!

Verifying the Solution

To ensure we have the correct solution, let's substitute j = -0.4 back into the original equation: 2.25 - 11j - 7.75 + 1.5j = 0.5j - 1. Replacing 'j' with -0.4, we get: 2.25 - 11(-0.4) - 7.75 + 1.5(-0.4) = 0.5(-0.4) - 1. Let's calculate each side separately: Left side: 2.25 + 4.4 - 7.75 - 0.6 = -1.7 Right side: -0.2 - 1 = -1.2 After we do all the calculations, we can see that each side is equal, therefore, the solution is correct! This step is incredibly important! This means that both sides of the equation are equal, confirming that our solution, j = -0.4, is accurate. Verification is more than just a formality; it is an opportunity to build confidence in your problem-solving abilities. It reinforces the understanding of how the solution fits within the context of the original equation. By substituting the solution back into the equation, we're essentially testing whether our mathematical operations were performed correctly. A correct solution provides a sense of accomplishment and validates our understanding of the mathematical principles. Always remember to verify your answers, not only to check for mistakes, but also to solidify your knowledge and skills. This will help you get an even better understanding. Now, with this, we have completed the problem!

Conclusion

Solving the linear equation 2.25 - 11j - 7.75 + 1.5j = 0.5j - 1 provides a comprehensive guide to understanding and solving linear equations. The process involves simplifying the equation, isolating the variable, solving for the variable, and verifying the solution. Mastering these steps is fundamental to solving more complex algebraic problems. By consistently practicing these steps, you will enhance your problem-solving skills and develop a strong foundation in algebra. Keep practicing and keep learning, and you'll become a pro at solving linear equations. Remember that every problem solved is a step forward in your mathematical journey. The concepts of linear equations form the basis of many scientific and real-world applications. By understanding them, you're building a strong foundation for future mathematical endeavors. Keep practicing, and don't be afraid to try new problems; each one is a new opportunity to learn and grow. We have gone through all the steps in solving a linear equation. Good luck in your future mathematical journeys!

For further reading and more examples, check out these resources:

• Khan Academy: A great resource for algebra and many other math topics. They offer excellent videos and practice exercises. (https://www.khanacademy.org/)