Solving -7[-2x - 5 + 3(x + 1)] = 3x - 6 A Step-by-Step Guide

In the realm of algebra, solving equations is a fundamental skill. Equations are mathematical statements that assert the equality of two expressions. The goal of solving an equation is to find the value(s) of the variable(s) that make the equation true. Linear equations, characterized by variables raised to the power of one, are the building blocks of more complex algebraic concepts. In this comprehensive guide, we will delve into the step-by-step process of solving the linear equation $-7[-2x - 5 + 3(x + 1)] = 3x - 6$. This equation involves multiple operations, including the distributive property, combining like terms, and isolating the variable. By mastering these techniques, you will be well-equipped to tackle a wide range of algebraic problems. This guide aims to provide a clear, concise, and accessible explanation of each step, ensuring that you develop a strong understanding of the underlying principles. Whether you are a student learning algebra for the first time or someone looking to refresh your skills, this guide will serve as a valuable resource. The beauty of mathematics lies in its logical structure and the systematic approach to problem-solving. By breaking down complex equations into simpler steps, we can unlock their solutions and gain a deeper appreciation for the elegance of algebra. So, let's embark on this journey together and unravel the mysteries of linear equations.

Step 1: Simplify the Expression Inside the Brackets

The initial step in solving the equation $-7[-2x - 5 + 3(x + 1)] = 3x - 6$ involves simplifying the expression within the innermost brackets. This requires applying the distributive property, which states that for any real numbers a, b, and c, a(b + c) = ab + ac. In our case, we need to distribute the 3 across the terms inside the parentheses (x + 1). By carefully applying the distributive property, we can eliminate the parentheses and pave the way for further simplification. This step is crucial because it reduces the complexity of the equation, making it easier to manage and solve. The distributive property is a cornerstone of algebraic manipulation, and its correct application is essential for accurate results. It allows us to transform expressions by multiplying a factor across multiple terms, effectively expanding the expression. In the given equation, the distributive property helps us to get rid of parentheses, which is a necessary step to simplify the equation. After applying the distributive property, we can then combine like terms, further simplifying the equation. Simplifying the expression inside the brackets sets the stage for the subsequent steps, such as combining like terms and isolating the variable. By systematically simplifying the equation, we can avoid errors and arrive at the correct solution. This initial simplification is like laying the foundation for a building; a strong foundation ensures the stability and integrity of the entire structure. In algebra, a well-simplified equation is much easier to solve, leading to a smoother and more efficient problem-solving process. Therefore, mastering the technique of simplifying expressions within brackets is a key skill for anyone seeking proficiency in algebra.

-7[-2x - 5 + 3(x + 1)] = 3x - 6
-7[-2x - 5 + 3x + 3] = 3x - 6

Step 2: Combine Like Terms Inside the Brackets

After applying the distributive property, the next crucial step in solving the equation $-7[-2x - 5 + 3x + 3] = 3x - 6$ is to combine like terms within the brackets. Like terms are terms that have the same variable raised to the same power. In this case, we have -2x and 3x as like terms, as well as -5 and +3 as like terms. Combining like terms involves adding or subtracting their coefficients. This process simplifies the expression by reducing the number of terms, making the equation more manageable. Combining like terms is a fundamental algebraic technique that helps to consolidate similar elements within an expression. It is based on the principle that terms with the same variable part can be combined by adding or subtracting their numerical coefficients. This step is crucial for simplifying complex expressions and equations, making them easier to work with. In the context of solving equations, combining like terms helps to isolate the variable and ultimately find its value. By reducing the number of terms, we reduce the complexity of the equation, making it easier to solve. The process of combining like terms is similar to organizing a collection of objects; we group together similar items to make the collection more organized and easier to understand. In algebra, this organization helps us to see the structure of the equation more clearly and to manipulate it more effectively. For example, in our equation, combining the 'x' terms (-2x and 3x) and the constant terms (-5 and +3) simplifies the equation, making it easier to apply further algebraic operations. This step is not only about simplifying the equation but also about developing a clear and logical approach to problem-solving in algebra. Mastering the skill of combining like terms is essential for success in algebra and beyond, as it forms the basis for more advanced algebraic techniques.

-7[-2x - 5 + 3x + 3] = 3x - 6
-7[x - 2] = 3x - 6

Step 3: Distribute the -7

With the expression inside the brackets simplified, the next step in solving the equation $-7[x - 2] = 3x - 6$ is to distribute the -7 across the terms inside the brackets. This again involves applying the distributive property, but this time with a negative factor. It's crucial to pay close attention to the signs when distributing negative numbers, as incorrect signs can lead to significant errors in the solution. Distributing the -7 means multiplying both the 'x' term and the constant term (-2) by -7. This step further expands the equation, removing the brackets and setting the stage for isolating the variable. The distributive property is a cornerstone of algebraic manipulation, and its correct application is essential for accurate results. It allows us to transform expressions by multiplying a factor across multiple terms, effectively expanding the expression. In this case, distributing the -7 is a critical step in simplifying the equation and moving towards the solution. The careful application of the distributive property, especially with negative numbers, demonstrates a solid understanding of algebraic principles. It's a process that requires attention to detail, ensuring that each term is correctly multiplied and that the signs are handled appropriately. This step is not just about removing the brackets; it's about transforming the equation into a form that is easier to solve. By distributing the -7, we create individual terms that can then be combined with like terms on the other side of the equation. This process is akin to preparing ingredients for a recipe; each ingredient needs to be properly prepared before it can be combined to create the final dish. In algebra, each step in solving an equation is a preparation for the next, leading us closer to the solution. Therefore, mastering the distributive property is a fundamental skill for anyone seeking proficiency in algebra.

-7[x - 2] = 3x - 6
-7x + 14 = 3x - 6

Step 4: Isolate the Variable Terms

After distributing the -7, the equation becomes -7x + 14 = 3x - 6. The next critical step in solving for 'x' is to isolate the variable terms. This means gathering all the terms containing 'x' on one side of the equation and all the constant terms on the other side. To achieve this, we can add 7x to both sides of the equation. This eliminates the -7x term on the left side and moves the 'x' term to the right side. Isolating the variable terms is a fundamental strategy in solving algebraic equations. It allows us to separate the terms containing the unknown variable from the constant terms, paving the way for the final step of solving for the variable. This process is based on the principle of maintaining equality; whatever operation we perform on one side of the equation, we must also perform on the other side to keep the equation balanced. In this case, adding 7x to both sides is a strategic move that helps us to consolidate the 'x' terms on one side. This step is crucial for simplifying the equation and making it easier to solve. By isolating the variable terms, we are essentially creating a situation where the variable is the only unknown on one side of the equation, making it possible to determine its value. This process is similar to sorting a mixed bag of items; we separate the different types of items into distinct groups to make them easier to count and manage. In algebra, isolating the variable terms allows us to focus on the variable and its relationship to the constants in the equation. This clarity is essential for solving the equation accurately and efficiently. Therefore, mastering the technique of isolating variable terms is a key skill for anyone seeking proficiency in algebra.

-7x + 14 = 3x - 6
-7x + 14 + 7x = 3x - 6 + 7x
14 = 10x - 6

Step 5: Isolate the Constant Terms

Following the isolation of variable terms, the equation is now 14 = 10x - 6. The next step is to isolate the constant terms. This involves moving all the constant terms to the side of the equation opposite the variable term. To do this, we can add 6 to both sides of the equation. This eliminates the -6 on the right side and moves the constant term to the left side. Isolating the constant terms is a crucial step in solving for the variable. It complements the isolation of variable terms by bringing together all the numerical values on one side of the equation. This process is based on the same principle of maintaining equality; whatever operation we perform on one side of the equation, we must also perform on the other side to keep the equation balanced. In this case, adding 6 to both sides is a strategic move that helps us to consolidate the constant terms on one side. This step is essential for simplifying the equation and making it easier to solve. By isolating the constant terms, we are essentially creating a situation where we have a numerical value on one side of the equation and a single term containing the variable on the other side. This setup makes it straightforward to determine the value of the variable by performing a final operation. This process is similar to organizing a room; we gather similar items together in specific areas to make the room more organized and functional. In algebra, isolating the constant terms allows us to focus on the numerical relationships in the equation, making it easier to solve for the unknown variable. Therefore, mastering the technique of isolating constant terms is a key skill for anyone seeking proficiency in algebra.

14 = 10x - 6
14 + 6 = 10x - 6 + 6
20 = 10x

Step 6: Solve for x

With the constant terms isolated, the equation is now 20 = 10x. The final step in solving for 'x' is to divide both sides of the equation by the coefficient of x, which is 10. This isolates 'x' on one side of the equation, giving us the solution. Dividing both sides by 10 is a direct application of the principle of maintaining equality; by performing the same operation on both sides, we ensure that the equation remains balanced. This step is the culmination of all the previous steps, leading us to the value of the unknown variable. Solving for 'x' is the ultimate goal of solving the equation, and it represents the answer to the problem. This process is similar to the final step in a recipe; after preparing all the ingredients and following the instructions, we arrive at the finished dish. In algebra, solving for 'x' is the final step in a logical sequence of operations, leading us to the solution. This step requires a clear understanding of algebraic principles and the ability to apply them correctly. The result of this step is the value of 'x' that satisfies the original equation. This value can then be used to check the solution by substituting it back into the original equation and verifying that both sides of the equation are equal. Therefore, mastering the technique of solving for 'x' is the ultimate goal of learning algebra, as it allows us to find the solutions to a wide range of mathematical problems.

20 = 10x
20 / 10 = 10x / 10
x = 2

Therefore, the solution to the equation $-7[-2x - 5 + 3(x + 1)] = 3x - 6$ is x = 2.

In conclusion, solving the equation $-7[-2x - 5 + 3(x + 1)] = 3x - 6$ involves a series of systematic steps, each building upon the previous one. These steps include simplifying expressions within brackets, combining like terms, applying the distributive property, and isolating the variable. By mastering these techniques, you can confidently solve a wide range of linear equations. The process of solving algebraic equations is not just about finding the answer; it's about developing a logical and systematic approach to problem-solving. Each step in the process is a deliberate action, guided by algebraic principles and the goal of isolating the variable. This step-by-step approach is applicable not only to mathematics but also to other areas of life where problem-solving is essential. The skills learned in algebra, such as logical reasoning, attention to detail, and the ability to break down complex problems into simpler steps, are valuable assets in any field. As you continue to practice and refine your algebraic skills, you will become more confident in your ability to tackle challenging problems. The journey of learning algebra is a journey of intellectual growth and empowerment. By mastering the fundamental concepts and techniques, you unlock the door to a deeper understanding of mathematics and its applications in the world around us. So, embrace the challenge, persevere through the difficulties, and celebrate the satisfaction of finding solutions. The world of algebra is vast and fascinating, and the skills you acquire will serve you well in your academic and professional pursuits. Remember, practice makes perfect, and with each equation you solve, you are building a stronger foundation for future success.