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

Solving algebraic equations is a fundamental skill in mathematics. It's not just about finding the value of an unknown variable; it's about understanding the underlying principles of equality and how to manipulate expressions to isolate the variable. In this article, we will walk through the step-by-step process of solving a given equation and simplifying the answer. By mastering these techniques, you'll be well-equipped to tackle more complex mathematical problems.

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

To solve equations effectively, understanding the given equation is the first crucial step. Our main goal is to find the value of x that satisfies the equation -2(4 + 3x) = 3(x - 2). This equation involves the variable x, constants, and parentheses, indicating that we need to apply the distributive property and combine like terms to isolate x. Equations are mathematical statements that assert the equality of two expressions. Solving an equation means finding the value(s) of the variable(s) that make the equation true. The given equation, -2(4 + 3x) = 3(x - 2), is a linear equation, which means that the highest power of the variable x is 1. Linear equations are generally easier to solve than equations with higher powers of the variable. When you encounter an equation, take a moment to analyze its structure. Identify the variables, constants, and operations involved. This initial assessment will help you choose the most appropriate strategy for solving the equation. In this case, we see that we have parentheses on both sides of the equation, which suggests that we should start by applying the distributive property. The distributive property states that a(b + c) = ab + ac. Applying this property will allow us to remove the parentheses and simplify the equation. Understanding the properties of operations and how they apply to different types of expressions is essential for solving equations. Another important aspect of understanding an equation is recognizing the order of operations. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which operations should be performed. By following the order of operations, we can ensure that we simplify expressions correctly. This equation is a classic example of a linear equation that requires careful application of algebraic principles to solve accurately. Make sure to check your solution by substituting it back into the original equation to verify that it makes the equation true. This step helps prevent errors and ensures that the solution is correct. In summary, understanding the equation is a critical first step in the problem-solving process. It involves identifying the variables, constants, and operations, recognizing the structure of the equation, and determining the most appropriate strategy for solving it. With a solid understanding of the equation, you can proceed with confidence to the next steps in the solution.

Applying the Distributive Property

The distributive property is a key concept when simplifying algebraic expressions. In our equation, -2(4 + 3x) = 3(x - 2), the parentheses indicate that we need to apply the distributive property to both sides. This property states that a(b + c) = ab + ac. On the left side of the equation, we distribute -2 to both 4 and 3x: -2 * 4 = -8 and -2 * 3x = -6x. So, -2(4 + 3x) becomes -8 - 6x. On the right side of the equation, we distribute 3 to both x and -2: 3 * x = 3x and 3 * -2 = -6. So, 3(x - 2) becomes 3x - 6. Now, our equation looks like this: -8 - 6x = 3x - 6. The distributive property allows us to eliminate the parentheses, making the equation easier to manipulate and solve. This step is crucial because it transforms the equation into a form where we can start combining like terms. The distributive property is not only used in equations but also in various algebraic manipulations, making it a fundamental skill to master. When applying the distributive property, pay close attention to the signs of the terms involved. A negative sign outside the parentheses will change the signs of all the terms inside the parentheses. This is a common area for errors, so double-checking your work is essential. Remember, the distributive property is a cornerstone of algebraic simplification. By mastering its application, you can confidently tackle a wide range of equations and expressions. This step sets the stage for the subsequent steps in solving the equation, such as combining like terms and isolating the variable. In summary, applying the distributive property is a critical step in simplifying algebraic expressions and equations. It allows us to remove parentheses and transform the equation into a more manageable form. By carefully distributing and paying attention to signs, we can accurately simplify the equation and proceed towards finding the solution. After applying the distributive property, the equation is now in a form that is easier to solve, and we can move on to the next step, which is combining like terms.

Combining Like Terms and Isolating x

After applying the distributive property, our equation is now -8 - 6x = 3x - 6. The next step in solving for x is to combine like terms and isolate x. To do this, we need to gather all the terms with x on one side of the equation and all the constant terms on the other side. Let’s start by adding 6x to both sides of the equation to eliminate the -6x term on the left side: -8 - 6x + 6x = 3x - 6 + 6x. This simplifies to -8 = 9x - 6. Now, we add 6 to both sides of the equation to isolate the term with x: -8 + 6 = 9x - 6 + 6. This simplifies to -2 = 9x. Finally, to isolate x, we divide both sides of the equation by 9: -2 / 9 = 9x / 9. This gives us x = -2/9. Combining like terms involves identifying terms that have the same variable raised to the same power and then adding or subtracting their coefficients. In this case, -6x and 3x are like terms. Isolating the variable is a fundamental technique in algebra. It involves performing operations on both sides of the equation to get the variable by itself on one side. This process often involves using inverse operations, such as addition and subtraction, or multiplication and division. The goal is to undo the operations that are being performed on the variable. By isolating x, we have found the value of x that satisfies the equation. It’s always a good idea to check your solution by substituting it back into the original equation to ensure that it is correct. This step helps catch any errors that may have occurred during the solution process. In summary, combining like terms and isolating x are essential steps in solving equations. By carefully manipulating the equation and using inverse operations, we can isolate the variable and find its value. This process requires a solid understanding of algebraic principles and attention to detail. With practice, these techniques become second nature, allowing you to solve a wide range of equations efficiently. Now that we have isolated x, we have found the solution to the equation.

The Solution and Simplification

After combining like terms and isolating x, we found that x = -2/9. This is the solution to the equation -2(4 + 3x) = 3(x - 2). The solution is already in its simplest form, as -2 and 9 have no common factors other than 1. Therefore, no further simplification is needed. The solution x = -2/9 is a rational number, which means it can be expressed as a fraction of two integers. It is also a negative number, which is important to note as it provides additional context to the solution. When solving equations, it is crucial to check your solution to ensure that it is correct. To do this, substitute x = -2/9 back into the original equation: -2(4 + 3(-2/9)) = 3((-2/9) - 2). First, simplify the expressions inside the parentheses: 4 + 3(-2/9) = 4 - 6/9 = 4 - 2/3 = 12/3 - 2/3 = 10/3 and (-2/9) - 2 = -2/9 - 18/9 = -20/9. Now, substitute these values back into the equation: -2(10/3) = 3(-20/9). This simplifies to -20/3 = -60/9. To compare these fractions, we can simplify -60/9 by dividing both the numerator and denominator by 3: -60/9 = -20/3. Since -20/3 = -20/3, the solution x = -2/9 is correct. The solution to an equation is the value(s) of the variable(s) that make the equation true. In this case, x = -2/9 is the only value that satisfies the equation -2(4 + 3x) = 3(x - 2). When the solution is a fraction, it is essential to ensure that it is in its simplest form. This means that the numerator and denominator have no common factors other than 1. In this case, -2/9 is already in its simplest form. In summary, the solution x = -2/9 is the final answer to the equation -2(4 + 3x) = 3(x - 2). It is a simplified rational number that satisfies the equation. Checking the solution by substituting it back into the original equation confirms its correctness. Now that we have found the solution and verified it, we can confidently conclude that x = -2/9 is the correct answer.

Conclusion

In conclusion, solving the equation -2(4 + 3x) = 3(x - 2) involved several key steps: applying the distributive property, combining like terms, and isolating x. By carefully following these steps, we arrived at the solution x = -2/9. This process highlights the importance of understanding fundamental algebraic principles and techniques. Solving equations is a cornerstone of mathematics and is essential for various applications in science, engineering, and everyday life. The ability to manipulate equations and isolate variables is a valuable skill that can be applied to a wide range of problems. The steps we followed in this article provide a systematic approach to solving linear equations. This approach can be adapted and applied to more complex equations as well. Remember, practice is key to mastering these skills. The more equations you solve, the more comfortable and confident you will become in your ability to solve them. The solution x = -2/9 is a specific example of a rational number solution. Not all equations have rational solutions; some may have integer solutions, while others may have irrational or complex solutions. Understanding the different types of solutions is an important aspect of equation solving. In this article, we focused on a linear equation, which is an equation where the highest power of the variable is 1. Linear equations have a unique solution, as we found in this case. More complex equations, such as quadratic equations, may have multiple solutions. The process of solving equations is not just about finding the answer; it is also about developing problem-solving skills and logical reasoning. By breaking down a problem into smaller steps and applying the appropriate techniques, you can effectively solve even challenging equations. In summary, solving equations is a fundamental skill in mathematics that requires a solid understanding of algebraic principles and techniques. By following a systematic approach and practicing regularly, you can develop the ability to solve a wide range of equations with confidence. The solution x = -2/9 is a testament to the power of these techniques and the importance of mastering them. We hope this article has provided you with a clear and comprehensive understanding of how to solve this equation and simplify your answer.