Solving For X: A Step-by-Step Guide To The Equation

Hey math enthusiasts! Let's dive into a common algebra problem: solving for x in an equation. We're going to break down the equation 1.2(3x + 5) = 3.6x + 6 step by step. Don't worry if equations seem intimidating at first; with a clear approach, they become much more manageable. This guide is designed to make the process as easy as possible, providing explanations and tips along the way. Whether you're a student tackling homework or just brushing up on your math skills, this guide will provide you with the necessary knowledge to handle similar problems confidently. Understanding how to solve for a variable is a fundamental skill in algebra, applicable in various fields, from science and engineering to economics and even everyday problem-solving. By the end of this guide, you'll not only solve the given equation but also build a solid foundation for tackling more complex algebraic challenges. The key to mastering algebra, or any mathematical discipline, lies in practice. Each step involves specific operations that need to be carried out methodically. In this context, the goal is to isolate the variable 'x' on one side of the equation, meaning we want to get to something like 'x =' followed by a number that represents the value of 'x' that satisfies the original equation. Let's start with the first step which is to distribute the 1.2 across the terms within the parentheses, we get 3.6x + 6 = 3.6x + 6, and proceed from there.

Step 1: Distributing and Simplifying

Alright, guys, let's kick things off with the first step: distributing the 1.2. This is a super important step in solving many algebraic equations. We need to multiply the 1.2 by each term inside the parentheses. In the original equation, we have 1.2(3x + 5) = 3.6x + 6. Now, let's break it down: 1.2 times 3x equals 3.6x, and 1.2 times 5 equals 6. So, after distributing, our equation becomes 3.6x + 6 = 3.6x + 6. Notice how the left side has been expanded. Distributing is essentially applying the distributive property of multiplication over addition, which states that for any numbers a, b, and c, a(b + c) = ab + ac. This property is fundamental to algebra and enables us to manipulate and simplify equations. Remember, the goal here is always to make the equation easier to work with by removing parentheses and combining like terms where possible. Without this step, we'd struggle to solve the equation effectively. This process sets the stage for isolating x. Make sure you're careful with the calculations – even a small mistake can throw off the entire solution. Double-checking your work after each step is always a good idea. Another crucial point is understanding that you're not changing the equation's value when distributing; you're simply rewriting it in a different, more usable form. The terms on both sides still hold the same mathematical relationship, but the expanded format gives us more opportunities to isolate x. In fact, in this instance, we see that the left side of the equation is identical to the right side of the equation. This indicates there are infinite solutions, where any value of x would make this true.

Step 2: Combining Like Terms

Okay, in this particular equation, combining like terms isn't as straightforward because we've got the same terms on both sides of the equation after distribution. When we simplify an equation, the general strategy is to get all the x terms on one side and the constant terms (the numbers without x) on the other side. So, what do we do when we've got 3.6x on both sides? In this case, we have to recognize that since the equation simplifies to 3.6x + 6 = 3.6x + 6, it means that no matter what value we assign to x, the equation will always hold true. When you get to a step where both sides of the equation are identical, it indicates either an infinite number of solutions or that the original equation is not correctly set up. In equations where there is a unique solution for x, after this step you'd typically have an equation like ax = b, where 'a' and 'b' are constants. Then, you'd divide both sides by 'a' to isolate x. But, since this isn't the case here, we take a different approach. Since this equation is essentially asking us to determine what value we can assign x, it's clear there are an infinite number of solutions.

Step 3: Isolating the Variable

Now, let's talk about the key to solving for x: isolating the variable. The goal is to get x all by itself on one side of the equation. To do this, we need to perform operations on both sides of the equation to eliminate the other terms. In many cases, you'll be adding or subtracting terms from both sides. For instance, if you have 3x + 5 = 11, you would subtract 5 from both sides to get 3x = 6. Then, you would divide both sides by 3 to find that x = 2. Remember, whatever you do to one side of the equation, you must do to the other to keep the equation balanced. This is a fundamental rule in algebra. In this case, since we had 3.6x + 6 = 3.6x + 6, it means that any value of x would be a solution to the equation. So, we're not able to isolate the variable in the way we usually would, because the equation is already simplified in a way that shows us that any number can be substituted in place of 'x'.

Step 4: Solving for x (or Recognizing the Outcome)

In our particular equation 1.2(3x + 5) = 3.6x + 6, the outcome reveals a special case. When we distributed and simplified, we ended up with 3.6x + 6 = 3.6x + 6. This means that both sides of the equation are identical. Because of this, any value you plug in for x will make the equation true, which gives us an infinite number of solutions. It's important to recognize these special cases. It's not always going to be the standard scenario where you get a single value for x. Sometimes, you might find that x cancels out completely, and you're left with a true statement like 6 = 6 (as we have here), which tells us there are infinite solutions. On the flip side, you could end up with a false statement like 5 = 7, which would mean there's no solution. So, in this instance, we can say that any real number can be a solution to this equation, and the solution set is represented as all real numbers.

Conclusion: Understanding the Solution

So, there you have it! In the equation 1.2(3x + 5) = 3.6x + 6, we found that there are infinite solutions. This outcome is a result of the equation simplifying in a way that leaves us with identical expressions on both sides. The key takeaways from this problem are:

• Distribution: Always distribute any number outside parentheses to each term inside. This is essential for simplifying the equation.
• Combining Like Terms: Grouping terms with x together and constants together helps isolate x.
• Isolating the Variable: The goal is to get x alone on one side by performing inverse operations on both sides of the equation.
• Special Cases: Recognize that some equations may have no solutions or infinite solutions. If both sides of the equation are identical after simplification, you have an infinite number of solutions. If the variables cancel out, and you end up with a false statement, there's no solution.

By following these steps and understanding the principles behind them, you'll be well-equipped to solve similar algebraic equations. Keep practicing, and you'll become more confident in your math skills! Remember that even though this equation had infinite solutions, the process is still valuable for understanding how to approach different types of equations in the future. Keep practicing, and you'll find that solving equations becomes easier and more intuitive over time! Don’t hesitate to practice more problems, and remember, with each problem you solve, you're building a stronger foundation in algebra. Keep it up, and you'll do great!