Solving For X: A Step-by-Step Guide
Let's dive into the process of solving for x in the equation 3(2x - 4) = 5x + 6. This type of problem is a staple in algebra, and mastering it will build a solid foundation for more advanced math. We'll break down each step, making it easy to follow along and understand the logic behind it.
Understanding the Equation: A Foundation for Success
Before we jump into solving, it's crucial to understand what the equation is telling us. This algebraic equation presents a relationship between an unknown value (x) and known constants. Our goal is to isolate x on one side of the equation to determine its value. To do this effectively, we'll employ several key algebraic principles, including the distributive property and combining like terms. These are the foundational tools that will allow us to manipulate the equation while maintaining its balance and ultimately find the solution. Understanding these principles isn't just about solving this specific problem; it's about building a framework for tackling a wide range of algebraic challenges.
Think of an equation as a perfectly balanced scale. Whatever operation you perform on one side, you must also perform on the other to maintain that balance. This is the golden rule of equation solving, and it guides every step we take. We'll be using this principle to carefully unravel the equation, step by step, until we isolate x and reveal its value. So, let's begin our journey by understanding the first critical step: applying the distributive property.
Step 1: Applying the Distributive Property
The first step in solving this equation is to simplify both sides. We start by applying the distributive property to the left side of the equation. The distributive property states that a( b + c ) = ab + ac. In our case, we need to distribute the 3 across the terms inside the parentheses (2x and -4). This means we multiply 3 by both 2x and -4.
When we multiply 3 by 2x, we get 6x. Then, we multiply 3 by -4, which gives us -12. So, after applying the distributive property, the left side of the equation becomes 6x - 12. The equation now looks like this: 6x - 12 = 5x + 6. This step is crucial because it eliminates the parentheses, making the equation easier to work with. Remember, the distributive property is a powerful tool for simplifying expressions and equations. It allows us to break down complex expressions into simpler, more manageable terms. Now that we've simplified the left side, the next step involves gathering like terms to further isolate x. This will bring us closer to our ultimate goal of solving for x.
Step 2: Gathering Like Terms
Now that we've applied the distributive property, the next key step is to gather like terms. This means bringing all the terms containing x to one side of the equation and all the constant terms to the other side. This process helps to isolate x and simplify the equation further. Looking at our equation, 6x - 12 = 5x + 6, we have x terms on both sides (6x and 5x) and constant terms on both sides (-12 and 6). To gather the x terms, we can subtract 5x from both sides of the equation. This will eliminate the 5x term on the right side and move it to the left side. Subtracting 5x from both sides gives us: 6x - 5x - 12 = 5x - 5x + 6, which simplifies to x - 12 = 6.
Next, we need to gather the constant terms. To do this, we can add 12 to both sides of the equation. This will eliminate the -12 term on the left side and move it to the right side. Adding 12 to both sides gives us: x - 12 + 12 = 6 + 12, which simplifies to x = 18. By gathering like terms, we've successfully isolated x on one side of the equation. This is a significant milestone in our journey to solve for x. Now, let's move on to the final step: determining the value of x.
Step 3: Isolating x and Finding the Solution
After gathering like terms, we've arrived at a simplified equation: x = 18. This is a pivotal moment because we've successfully isolated x on one side of the equation. In this case, the equation directly tells us the value of x. There are no further steps needed to isolate x, as it is already by itself. The equation x = 18 clearly states that the value of x is 18. This means that if we substitute 18 for x in the original equation, 3(2x - 4) = 5x + 6, both sides of the equation will be equal. We can verify this by plugging 18 back into the original equation.
Let's substitute x with 18: 3(2(18) - 4) = 5(18) + 6. First, simplify inside the parentheses: 3(36 - 4) = 90 + 6. Then, continue simplifying: 3(32) = 96. Finally, we have 96 = 96, which confirms that our solution is correct. Therefore, the solution to the equation 3(2x - 4) = 5x + 6 is x = 18. This final step solidifies our understanding of the solution and reinforces the importance of verifying our answer. It's a good practice to always check your solution to ensure accuracy.
Conclusion: Mastering the Art of Solving for x
In conclusion, we've successfully navigated the process of solving for x in the equation 3(2x - 4) = 5x + 6. We started by understanding the equation and its components, then applied the distributive property to simplify the expression. Next, we gathered like terms to isolate x, and finally, we determined the value of x to be 18. This step-by-step approach is a valuable skill in algebra and beyond. The ability to solve for unknown variables is essential in various fields, including science, engineering, and finance. By mastering these fundamental algebraic techniques, you'll be well-equipped to tackle more complex mathematical problems.
Remember, the key to success in algebra is practice. The more equations you solve, the more comfortable you'll become with the process. Don't be afraid to make mistakes; they are learning opportunities. Each time you solve an equation, you're strengthening your problem-solving skills and building a deeper understanding of mathematical concepts. Keep practicing, and you'll become a confident and proficient equation solver. If you would like to continue learning more about Algebra check out great resources like Khan Academy Algebra.
