Solving Algebraic Equations A Step-by-Step Guide For 4(3x - 6) + 2(3x - 1) = 2

Introduction to Solving Algebraic Equations

In the realm of mathematics, solving equations is a fundamental skill. Algebraic equations, in particular, form the backbone of many mathematical concepts and real-world applications. This article aims to provide a comprehensive, step-by-step guide on how to solve a specific algebraic equation: 4(3x - 6) + 2(3x - 1) = 2. We will break down each step, ensuring clarity and understanding, making it easier for students and anyone interested in brushing up on their algebra skills. Mastering these skills is crucial for success in higher mathematics and various fields that rely on quantitative analysis.

This detailed exploration will not only show you the mechanics of solving this particular equation but also highlight the underlying principles of algebraic manipulation. By understanding these principles, you'll be better equipped to tackle a wide range of algebraic problems. We'll cover everything from the distributive property to combining like terms and isolating the variable. So, let's dive in and unravel the intricacies of this equation!

Understanding the Equation: 4(3x - 6) + 2(3x - 1) = 2

Before we jump into solving the equation, it's crucial to understand what the equation 4(3x - 6) + 2(3x - 1) = 2 represents. This is a linear equation in one variable, 'x'. Linear equations are characterized by having the variable raised to the power of 1, and they graph as a straight line. The goal of solving this equation is to find the value of 'x' that makes the equation true. In other words, we want to find the value of 'x' that, when substituted into the equation, makes the left side equal to the right side.

The equation itself is composed of several parts. We have two terms involving 'x': 4(3x - 6) and 2(3x - 1). These terms involve the distributive property, which we'll discuss in detail in the next section. The equation also includes constant terms (-6 and -1) and the constant 2 on the right side of the equation. Understanding the structure of the equation is the first step towards solving it. It allows us to plan our approach and identify the necessary steps to isolate 'x'.

Furthermore, recognizing the type of equation we're dealing with—in this case, a linear equation—helps us anticipate the nature of the solution. Linear equations typically have one unique solution, which means there's only one value of 'x' that satisfies the equation. This knowledge gives us a target and helps us verify our solution later on. By breaking down the equation into its components and understanding its structure, we set the stage for a successful solution.

Step 1: Applying the Distributive Property

The first key step in simplifying the equation 4(3x - 6) + 2(3x - 1) = 2 is to apply the distributive property. This property states that a(b + c) = ab + ac. In simpler terms, it means we need to multiply the number outside the parentheses by each term inside the parentheses. Let's apply this to our equation. First, we'll distribute the 4 in the term 4(3x - 6). This gives us 4 * 3x - 4 * 6, which simplifies to 12x - 24. Next, we'll distribute the 2 in the term 2(3x - 1). This gives us 2 * 3x - 2 * 1, which simplifies to 6x - 2.

Now, we can rewrite the original equation with these simplified terms: 12x - 24 + 6x - 2 = 2. By applying the distributive property, we've eliminated the parentheses, which makes the equation easier to work with. This step is crucial because it allows us to combine like terms in the next step. Without distributing, we wouldn't be able to simplify the equation effectively. The distributive property is a fundamental concept in algebra, and mastering it is essential for solving equations. It's a versatile tool that appears in many different types of algebraic problems, so understanding it thoroughly will greatly enhance your problem-solving abilities.

By carefully applying the distributive property, we've transformed the equation into a more manageable form. This sets the stage for the subsequent steps, where we'll combine like terms and isolate the variable 'x'. This initial simplification is often the key to unlocking the solution to an algebraic equation, and it demonstrates the power of applying mathematical properties to make complex problems more accessible.

Step 2: Combining Like Terms

After applying the distributive property, our equation looks like this: 12x - 24 + 6x - 2 = 2. The next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this equation, 12x and 6x are like terms, and -24 and -2 are also like terms. To combine like terms, we simply add or subtract their coefficients (the numbers in front of the variables) and add or subtract the constants.

Let's start by combining the 'x' terms: 12x + 6x = 18x. Now, let's combine the constant terms: -24 - 2 = -26. So, after combining like terms, our equation becomes 18x - 26 = 2. This step significantly simplifies the equation, making it much easier to solve. Combining like terms reduces the number of terms in the equation, which streamlines the process of isolating the variable. It's a fundamental technique in algebra and is used extensively in solving various types of equations.

The importance of this step cannot be overstated. By grouping similar terms together, we reduce the complexity of the equation and bring it closer to a form where we can easily isolate 'x'. This process not only simplifies the equation but also helps to clarify the relationship between the variable and the constants. Combining like terms is a crucial step in maintaining the balance of the equation while moving towards the solution. It’s a skill that’s applicable across a wide range of algebraic problems, making it a valuable tool in your mathematical arsenal. Mastering this technique will significantly improve your ability to solve equations efficiently and accurately.

Step 3: Isolating the Variable

Now that we've combined like terms, our equation is 18x - 26 = 2. The next critical step is to isolate the variable, 'x'. This means we want to get 'x' by itself on one side of the equation. To do this, we need to undo the operations that are being performed on 'x'. In this case, 'x' is being multiplied by 18 and then 26 is being subtracted. We'll reverse these operations in the opposite order.

First, we'll undo the subtraction of 26 by adding 26 to both sides of the equation. Remember, it's crucial to perform the same operation on both sides to maintain the equality. Adding 26 to both sides gives us: 18x - 26 + 26 = 2 + 26, which simplifies to 18x = 28. Now, 'x' is only being multiplied by 18. To undo this multiplication, we'll divide both sides of the equation by 18. This gives us: 18x / 18 = 28 / 18, which simplifies to x = 28/18. We can further simplify the fraction 28/18 by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us x = 14/9.

Isolating the variable is the heart of solving algebraic equations. It's the process of systematically undoing operations to reveal the value of the unknown. This step requires a clear understanding of inverse operations and the importance of maintaining balance in the equation. By carefully applying these principles, we've successfully isolated 'x' and found its value. This process is not only applicable to linear equations but also to more complex equations. The ability to isolate variables is a cornerstone of algebraic problem-solving and is essential for tackling a wide variety of mathematical challenges.

Step 4: Simplifying the Solution

After isolating the variable in the equation 4(3x - 6) + 2(3x - 1) = 2, we arrived at the solution x = 28/18. The final, yet crucial, step is to simplify the solution. Simplifying fractions means reducing them to their lowest terms, which makes the solution clearer and easier to understand. In this case, both the numerator (28) and the denominator (18) are divisible by 2. Dividing both by 2 gives us x = 14/9. This fraction cannot be simplified further because 14 and 9 have no common factors other than 1.

Therefore, the simplified solution to the equation is x = 14/9. This is the most concise and accurate way to express the answer. Simplifying solutions is not just about aesthetics; it's about presenting the answer in its most understandable form. In many contexts, a simplified fraction is preferred over an unsimplified one because it clearly communicates the value of the variable. Moreover, simplifying fractions can sometimes reveal further insights or connections within the problem.

In addition to simplifying fractions, it's also important to check whether the solution makes sense in the context of the original problem. In this case, since we were solving a linear equation, we expect a single numerical solution, which we have found. Simplifying the solution is a final polish that ensures the answer is clear, accurate, and ready to be used in further calculations or applications. It's a skill that reinforces the importance of precision and attention to detail in mathematics.

Step 5: Verifying the Solution

Once we've found a solution to an algebraic equation, such as x = 14/9 for the equation 4(3x - 6) + 2(3x - 1) = 2, it's crucial to verify the solution. Verification involves substituting the solution back into the original equation to check if it makes the equation true. This step ensures that we haven't made any errors in our calculations and that our solution is correct. To verify our solution, we'll substitute x = 14/9 into the original equation:

4(3(14/9) - 6) + 2(3(14/9) - 1) = 2

First, let's simplify the terms inside the parentheses: 3(14/9) = 42/9, which simplifies to 14/3. Now, we have:

4(14/3 - 6) + 2(14/3 - 1) = 2

Next, we need to subtract the constants inside the parentheses. To do this, we'll convert the constants to fractions with a denominator of 3: 6 = 18/3 and 1 = 3/3. So, we have:

4(14/3 - 18/3) + 2(14/3 - 3/3) = 2

Now, we can perform the subtractions: 14/3 - 18/3 = -4/3 and 14/3 - 3/3 = 11/3. Our equation becomes:

4(-4/3) + 2(11/3) = 2

Now, we'll multiply: 4(-4/3) = -16/3 and 2(11/3) = 22/3. So, we have:

-16/3 + 22/3 = 2

Finally, we add the fractions: -16/3 + 22/3 = 6/3, which simplifies to 2. So, we have:

2 = 2

Since the left side of the equation equals the right side, our solution x = 14/9 is correct. Verifying the solution is a critical step in the problem-solving process. It provides confidence in the answer and helps catch any mistakes that may have been made along the way. This practice reinforces the importance of accuracy and attention to detail in mathematics.

Conclusion

In conclusion, solving the algebraic equation 4(3x - 6) + 2(3x - 1) = 2 involves a series of steps, each building upon the previous one. We began by understanding the equation and recognizing it as a linear equation in one variable. Then, we applied the distributive property to eliminate parentheses, followed by combining like terms to simplify the equation. The next crucial step was isolating the variable 'x' by using inverse operations, which led us to the solution x = 28/18. We then simplified this fraction to its lowest terms, resulting in x = 14/9. Finally, we verified our solution by substituting it back into the original equation, confirming its correctness.

This step-by-step approach not only provides the solution to this specific equation but also illustrates the fundamental principles of algebraic manipulation. These principles, such as the distributive property, combining like terms, and isolating the variable, are applicable to a wide range of algebraic problems. Mastering these skills is essential for success in mathematics and related fields. By understanding the logic behind each step, you can approach similar problems with confidence and accuracy.

Algebraic equations are a cornerstone of mathematics, and the ability to solve them is a valuable skill. This guide has aimed to provide a clear and comprehensive explanation of the process, making it accessible to learners of all levels. Remember, practice is key to mastering these skills. By working through various problems and applying these techniques, you'll develop a strong foundation in algebra and enhance your problem-solving abilities. The journey through algebra is one of discovery and empowerment, and with each equation solved, you'll gain a deeper appreciation for the power and beauty of mathematics.