Solving Linear Equations A Step By Step Guide For 2.8y + 6 + 0.2y = 5y - 14
Hey everyone! Today, we're diving deep into the fascinating world of linear equations. Specifically, we're going to tackle the equation 2.8y + 6 + 0.2y = 5y - 14. Linear equations might seem intimidating at first glance, but trust me, with a systematic approach, they're totally solvable. So, grab your pencils, and let's embark on this mathematical adventure together! We will break down each step, ensuring you understand the why behind the how. Understanding linear equations is crucial, as they form the bedrock of more advanced mathematical concepts and are widely applied in various fields, from physics and engineering to economics and computer science. Think of linear equations as the fundamental building blocks of mathematical modeling, allowing us to represent real-world relationships in a concise and manageable way. So, mastering these equations will open doors to a deeper understanding of the world around you.
Understanding Linear Equations
Before we jump into solving our specific equation, let's take a moment to grasp the essence of linear equations. At its heart, a linear equation is an algebraic equation where each term is either a constant or the product of a constant and a single variable. The variable is raised to the power of one, which is why it's called "linear" – because when graphed, these equations form a straight line. The general form of a linear equation in one variable is ax + b = c, where 'a', 'b', and 'c' are constants, and 'x' is the variable. Now, why are linear equations so important? Well, they pop up everywhere! From calculating distances and speeds to determining costs and profits, linear equations provide a framework for modeling countless real-world scenarios. They're the workhorses of mathematics, allowing us to quantify relationships and make predictions. In fact, linear equations are so fundamental that they are used in fields ranging from engineering and physics to economics and computer science. Imagine designing a bridge, calculating the trajectory of a rocket, or forecasting economic trends – all of these rely on the principles of linear equations. So, understanding linear equations is not just about solving problems in a textbook; it's about gaining a powerful tool for understanding and interacting with the world around us. Furthermore, linear equations serve as a stepping stone to more complex mathematical concepts. As you delve deeper into mathematics, you'll encounter systems of linear equations, linear inequalities, and linear programming, all of which build upon the foundational principles we're discussing here. So, mastering linear equations is like laying a solid foundation for your mathematical journey, paving the way for success in more advanced topics.
Step-by-Step Solution for 2.8y + 6 + 0.2y = 5y - 14
Alright, let's get our hands dirty and solve the equation 2.8y + 6 + 0.2y = 5y - 14. We'll break it down into manageable steps, making sure each step is crystal clear. The key to solving any linear equation is to isolate the variable (in this case, 'y') on one side of the equation. This involves a series of algebraic manipulations, each designed to simplify the equation while maintaining its balance. Think of it like a seesaw – whatever you do to one side, you must do to the other to keep it level.
Step 1: Combine Like Terms
The first thing we want to do is simplify each side of the equation by combining like terms. On the left side, we have two terms with 'y': 2.8y and 0.2y. Adding them together gives us 3y. So, the equation now looks like this: 3y + 6 = 5y - 14. Combining like terms is a fundamental step in solving any algebraic equation. It allows us to condense the equation, making it easier to work with. In this case, by combining 2.8y and 0.2y, we've reduced the number of terms on the left side, bringing us one step closer to isolating 'y'. This step is not just about simplification; it's about organization. By grouping similar terms together, we create a clearer picture of the equation's structure, making it easier to see the next steps required to solve it. Furthermore, combining like terms reduces the chances of making errors in subsequent steps. By consolidating terms, we minimize the number of individual operations we need to perform, thus lowering the risk of arithmetic mistakes. This is especially important when dealing with more complex equations, where the number of terms can be overwhelming. So, remember to always start by combining like terms – it's a simple yet powerful technique that can make a world of difference in your equation-solving journey.
Step 2: Move the Variable Terms to One Side
Next, we want to get all the 'y' terms on the same side of the equation. Let's subtract 3y from both sides: 3y + 6 - 3y = 5y - 14 - 3y. This simplifies to 6 = 2y - 14. The reason we subtract 3y from both sides is to eliminate the 'y' term from the left side of the equation. This is a crucial step in isolating the variable, as it brings us closer to having 'y' by itself on one side. By performing the same operation on both sides, we maintain the balance of the equation, ensuring that the equality remains true. This principle of maintaining balance is fundamental to all algebraic manipulations – it's the cornerstone of solving equations. Think of it like a seesaw – if you remove weight from one side, you must remove the same amount of weight from the other side to keep it level. In this case, subtracting 3y from both sides is like removing the same weight from both sides of the seesaw, ensuring that it remains balanced. This step not only isolates the variable but also simplifies the equation further. By eliminating the 'y' term from the left side, we've reduced the complexity of the equation, making it easier to see the next steps required to solve it. This is a common strategy in solving equations – to progressively simplify them until the variable is isolated.
Step 3: Move the Constant Terms to the Other Side
Now, let's isolate the 'y' term further by moving the constant terms to the other side. We'll add 14 to both sides: 6 + 14 = 2y - 14 + 14. This gives us 20 = 2y. Similar to the previous step, adding 14 to both sides is about maintaining the equation's balance while isolating the variable. By adding 14, we eliminate the constant term (-14) from the right side, bringing us closer to having 'y' by itself. This step is a crucial part of the overall strategy of isolating the variable. By systematically moving terms around, we're peeling away the layers surrounding 'y', one step at a time. Think of it like unwrapping a gift – each step reveals a little more of what's inside. In this case, each step brings us closer to uncovering the value of 'y'. This step also highlights the importance of inverse operations in solving equations. We added 14 because it's the inverse operation of subtraction, allowing us to cancel out the -14 on the right side. Understanding inverse operations is essential for mastering equation-solving, as it allows you to selectively eliminate terms and isolate the variable. Furthermore, this step demonstrates the power of algebraic manipulation. By strategically adding the same value to both sides, we've transformed the equation into a simpler form, making it easier to solve. This ability to manipulate equations is a fundamental skill in mathematics, allowing us to tackle a wide range of problems.
Step 4: Solve for y
Finally, to solve for 'y', we need to divide both sides by 2: 20 / 2 = 2y / 2. This results in y = 10. And there you have it! We've successfully solved the equation. Dividing both sides by 2 is the final step in isolating 'y'. Since 'y' is being multiplied by 2, we use the inverse operation of division to cancel out the 2 and leave 'y' by itself. This is the culmination of all the previous steps – a systematic process of simplification and manipulation that has led us to the solution. The result, y = 10, is the value that satisfies the original equation. If you substitute 10 for 'y' in the original equation, you'll find that both sides are equal. This is the ultimate test of a solution – it must make the equation true. This step also emphasizes the importance of understanding the relationship between multiplication and division. They are inverse operations, and this relationship is crucial for solving equations. By understanding how these operations work together, you can effectively isolate variables and find solutions. Furthermore, this step provides a sense of closure. After a series of algebraic manipulations, we've arrived at a clear and concise answer. This feeling of accomplishment is one of the joys of mathematics – the satisfaction of solving a problem and arriving at a solution.
Verification
To be absolutely sure, let's substitute y = 10 back into the original equation: 2.8(10) + 6 + 0.2(10) = 5(10) - 14. This simplifies to 28 + 6 + 2 = 50 - 14, which further simplifies to 36 = 36. The equation holds true, so our solution is correct! Verification is a critical step in the problem-solving process. It's a way to ensure that your answer is correct and that you haven't made any mistakes along the way. By substituting the solution back into the original equation, you're essentially testing whether it satisfies the equation's conditions. This process not only confirms the correctness of the solution but also reinforces your understanding of the equation itself. Think of it like a quality control check – you're making sure that the product (the solution) meets the required standards (the equation). If the solution doesn't satisfy the equation, it indicates that there's an error somewhere in your calculations, and you need to go back and review your steps. This is a valuable learning opportunity, as it allows you to identify and correct your mistakes, improving your problem-solving skills. Furthermore, verification builds confidence in your answer. Knowing that you've checked your solution and that it satisfies the equation gives you the assurance that you've solved the problem correctly. This confidence is essential for tackling more complex problems and for applying your mathematical skills in real-world situations.
Conclusion
So, there you have it! We've successfully navigated the linear equation 2.8y + 6 + 0.2y = 5y - 14 and found that y = 10. Remember, the key to mastering linear equations is to break them down into smaller, manageable steps. Combine like terms, isolate the variable, and always verify your solution. With practice, you'll become a linear equation-solving pro in no time! Linear equations are not just abstract mathematical concepts; they are powerful tools that can be used to model and solve real-world problems. From calculating distances and speeds to determining costs and profits, linear equations provide a framework for understanding and interacting with the world around us. So, the skills you've learned today are not just about solving equations in a textbook; they are about gaining a valuable tool for problem-solving in various contexts. Furthermore, mastering linear equations lays a solid foundation for more advanced mathematical concepts. As you delve deeper into mathematics, you'll encounter systems of linear equations, linear inequalities, and linear programming, all of which build upon the principles we've discussed here. So, the effort you put into understanding linear equations will pay off in the long run, opening doors to a wider range of mathematical knowledge and skills. Finally, remember that practice makes perfect. The more you work with linear equations, the more comfortable and confident you'll become in solving them. So, don't be afraid to tackle new problems, and keep practicing! With persistence and a systematic approach, you'll master the art of solving linear equations and unlock their power to solve real-world problems.