Solving (2/3)(x-6) = 14 A Step-by-Step Guide

Hey there, math enthusiasts! Let's dive into the fascinating world of equations and tackle the problem (2/3)(x-6) = 14. This equation might seem a bit daunting at first, but don't worry, we'll break it down step-by-step and conquer it together. Understanding how to solve equations like this is a fundamental skill in mathematics, and it opens doors to more advanced concepts. We will not only solve this particular equation, but also discuss the general strategies you can use to solve any linear equation. So, buckle up, grab your pencils, and let's embark on this mathematical adventure!

Understanding the Basics of Equations

Before we jump into the specifics of our equation, let's make sure we're all on the same page with the basics. An equation, in its simplest form, is a mathematical statement that asserts the equality of two expressions. Think of it like a balanced scale – what's on one side must be equal to what's on the other. The goal of solving an equation is to isolate the variable (in our case, 'x') on one side of the equation, revealing its value. To achieve this, we use a variety of algebraic techniques that maintain the balance of the equation. These techniques often involve performing the same operation on both sides, like adding, subtracting, multiplying, or dividing. Mastering these techniques is crucial for success in algebra and beyond. Equations are not just abstract concepts; they are the language we use to describe relationships and solve problems in the real world. From calculating the trajectory of a rocket to predicting the growth of a population, equations are the unsung heroes behind countless innovations and discoveries.

Common Strategies for Solving Equations

There are several key strategies that are commonly used when solving equations. These strategies form the foundation of algebraic manipulation and are essential for tackling a wide range of problems. First, we have the distributive property, which allows us to simplify expressions by multiplying a term across a sum or difference within parentheses. This is particularly useful when dealing with equations like ours, where we have a fraction multiplied by an expression in parentheses. Next, we have the concept of inverse operations. Every mathematical operation has an inverse that undoes it – addition and subtraction are inverses, and multiplication and division are inverses. By applying inverse operations, we can isolate the variable and solve for its value. Another important strategy is combining like terms. If we have multiple terms with the same variable or constant terms on the same side of the equation, we can simplify the equation by combining them. Finally, we must always remember to maintain balance. Any operation we perform on one side of the equation must also be performed on the other side to preserve the equality. With these strategies in mind, let's get back to our equation and see how we can apply them.

Step-by-Step Solution of (2/3)(x-6) = 14

Alright, let's get our hands dirty and solve this equation! We'll break it down into manageable steps, making sure each step is clear and easy to follow. Remember, the key is to isolate 'x' on one side of the equation. Our equation is (2/3)(x-6) = 14. The first step we'll take is to get rid of the fraction (2/3) that's hanging out in front of the parentheses. We can do this by multiplying both sides of the equation by the reciprocal of 2/3, which is 3/2. This is a classic move in equation-solving, and it helps to simplify things considerably. By multiplying both sides by 3/2, we maintain the balance of the equation while eliminating the fraction. This leaves us with a much simpler equation to work with.

Step 1: Multiply Both Sides by 3/2

So, let's multiply both sides of the equation by 3/2. We have:

(3/2) * (2/3)(x-6) = 14 * (3/2)

On the left side, the (3/2) and (2/3) cancel each other out, leaving us with just (x-6). On the right side, we have 14 * (3/2), which simplifies to 21. So, our equation now looks like this:

x - 6 = 21

See how much simpler it is already? We've successfully eliminated the fraction and made the equation easier to handle. This step highlights the power of using inverse operations to simplify equations. By multiplying by the reciprocal, we effectively undid the multiplication by 2/3, bringing us closer to isolating 'x'. Now that we have a more streamlined equation, we can move on to the next step in our solving process.

Step 2: Isolate x by Adding 6 to Both Sides

Now that we have x - 6 = 21, our next goal is to isolate 'x' completely. To do this, we need to get rid of the '- 6' that's attached to it. Remember our discussion about inverse operations? The inverse of subtraction is addition, so we'll add 6 to both sides of the equation. This will effectively cancel out the '- 6' on the left side, leaving 'x' by itself. This step is another crucial application of the principle of maintaining balance in an equation. By adding 6 to both sides, we ensure that the equality remains true. Adding 6 to both sides gives us:

x - 6 + 6 = 21 + 6

On the left side, the '- 6' and '+ 6' cancel each other out, leaving us with just 'x'. On the right side, 21 + 6 equals 27. So, our equation now looks like this:

x = 27

And there we have it! We've successfully isolated 'x' and found its value. This step demonstrates how adding the same value to both sides of an equation can help us to isolate the variable we're trying to solve for.

The Solution and Its Significance

So, after all that algebraic maneuvering, we've arrived at our solution: x = 27. That's the value of 'x' that makes the equation (2/3)(x-6) = 14 true. But what does this solution really mean? Well, it means that if we substitute 27 for 'x' in the original equation, the left side will equal the right side. Let's verify this to make sure our solution is correct. This process of verification is a critical step in problem-solving, as it helps us to catch any errors we might have made along the way. It also reinforces our understanding of the equation and its solution. Verifying the solution provides a sense of closure and confidence in our answer.

Verification: Plugging x = 27 Back into the Original Equation

Let's plug x = 27 back into the original equation and see if it holds true:

(2/3)(x-6) = 14

Substitute x = 27:

(2/3)(27-6) = 14

Simplify inside the parentheses:

(2/3)(21) = 14

Multiply:

14 = 14

It checks out! The left side equals the right side, confirming that x = 27 is indeed the correct solution. This verification step is a powerful tool for building confidence in your problem-solving abilities. It's like a final seal of approval on your work. Furthermore, it highlights the importance of accuracy in each step of the solution process. A small error early on can lead to an incorrect solution, which is why carefulness and verification are essential.

General Strategies for Solving Linear Equations

Now that we've successfully solved this particular equation, let's zoom out and talk about some general strategies that you can use to tackle any linear equation. Linear equations are equations where the variable appears only to the first power (no x², x³, etc.), and they are the bread and butter of algebra. Mastering the techniques for solving them will set you up for success in more advanced math courses. These strategies are not just about memorizing steps; they are about developing a flexible and adaptable approach to problem-solving. They encourage you to think critically about the structure of an equation and to choose the most efficient path to a solution. By internalizing these strategies, you'll be able to confidently approach a wide variety of linear equations.

Key Techniques for Tackling Linear Equations

There are a few key techniques that are particularly useful when solving linear equations. First, always aim to simplify both sides of the equation as much as possible before you start isolating the variable. This might involve distributing, combining like terms, or clearing fractions. Simplifying upfront can make the equation much easier to work with in the long run. Second, remember the power of inverse operations. Use them strategically to undo operations that are preventing you from isolating the variable. Third, maintain balance at all costs! Any operation you perform on one side of the equation must also be performed on the other side to preserve the equality. Finally, verify your solution by plugging it back into the original equation. This will catch any errors and give you confidence in your answer. These techniques are not just about solving equations; they are about developing a systematic and rigorous approach to mathematical problem-solving.

Conclusion: Mastering Equations for Mathematical Success

Congratulations! You've successfully solved the equation (2/3)(x-6) = 14 and gained valuable insights into the world of equation-solving. We've walked through the step-by-step solution, verified our answer, and discussed general strategies for tackling linear equations. Remember, mastering equations is a fundamental skill in mathematics, and it opens doors to more advanced concepts and applications. The ability to solve equations is not just a mathematical skill; it's a life skill. It teaches you how to break down complex problems into smaller, manageable steps, and it empowers you to find solutions. Keep practicing, keep exploring, and keep challenging yourself – the world of mathematics is vast and fascinating, and there's always something new to discover!

Practice Makes Perfect

The best way to truly master equation-solving is through practice. The more you practice, the more comfortable you'll become with the techniques, and the more confident you'll be in your abilities. So, don't be afraid to tackle challenging problems and to make mistakes along the way – mistakes are learning opportunities! Seek out additional practice problems in textbooks, online resources, or from your teacher. And remember, there's no substitute for putting in the time and effort to develop your skills. With consistent practice, you'll be well on your way to becoming an equation-solving master!