Solving Equations Step-by-Step -4(x-4) + 5x = 5 - 2
In the realm of mathematics, solving linear equations is a fundamental skill. Linear equations, characterized by variables raised to the first power, are ubiquitous in various fields, from everyday problem-solving to advanced scientific applications. Mastering the art of simplifying and solving these equations is crucial for anyone seeking a strong foundation in mathematics. This article delves into the step-by-step process of solving the equation -4(x-4) + 5x = 5 - 2, providing a detailed explanation of each operation and the underlying mathematical principles. We will explore the concepts of the distributive property, combining like terms, and isolating the variable, all essential tools in the arsenal of any aspiring mathematician. By the end of this guide, you will not only understand how to solve this specific equation but also gain a deeper appreciation for the broader techniques applicable to a wide range of linear equations. This foundational knowledge will empower you to tackle more complex mathematical challenges with confidence and proficiency. Remember, practice is key to mastery, so be sure to work through various examples and exercises to solidify your understanding.
Understanding the Equation: -4(x-4) + 5x = 5 - 2
Before diving into the solution, let's break down the equation -4(x-4) + 5x = 5 - 2 and understand its components. This linear equation involves a single variable, 'x', and our goal is to find the value of 'x' that satisfies the equation. The left side of the equation, -4(x-4) + 5x, contains a term with parentheses, which indicates the need for the distributive property. The distributive property states that a(b + c) = ab + ac, meaning we need to multiply -4 by both 'x' and '-4' inside the parentheses. Additionally, we have the term 5x, which represents 5 multiplied by the variable 'x'. On the right side of the equation, we have 5 - 2, a simple arithmetic operation that needs to be performed. To effectively solve this equation, we will follow a systematic approach. First, we will apply the distributive property to eliminate the parentheses. Then, we will combine like terms on each side of the equation to simplify it. Finally, we will isolate the variable 'x' by performing inverse operations until we arrive at the solution. This step-by-step process ensures accuracy and clarity in our solution. Understanding the structure of the equation and the properties that govern its manipulation is crucial for success in solving not just this equation, but any linear equation. With a solid grasp of these fundamentals, you'll be well-equipped to tackle more complex mathematical problems.
Step 1: Applying the Distributive Property
The first crucial step in simplifying the equation -4(x-4) + 5x = 5 - 2 is applying the distributive property. As mentioned earlier, the distributive property states that a(b + c) = ab + ac. In our equation, we have -4(x-4), which means we need to multiply -4 by both 'x' and '-4'. Multiplying -4 by 'x' gives us -4x. Multiplying -4 by -4 gives us +16 (remember, a negative times a negative equals a positive). Therefore, -4(x-4) becomes -4x + 16. Now, we can rewrite the equation as -4x + 16 + 5x = 5 - 2. The distributive property has successfully eliminated the parentheses, allowing us to proceed with further simplification. This property is a cornerstone of algebraic manipulation, enabling us to handle expressions involving parentheses and other grouping symbols. Without the distributive property, solving equations like this would be significantly more challenging. It's important to remember the signs when applying the distributive property, especially when dealing with negative numbers. A common mistake is to forget to distribute the negative sign, leading to an incorrect simplification. Mastering the distributive property is essential for success in algebra and beyond. It's a tool you'll use frequently in more advanced mathematical concepts, so a solid understanding now will pay dividends in the future.
Step 2: Combining Like Terms
After applying the distributive property, our equation now reads -4x + 16 + 5x = 5 - 2. The next step in simplifying the equation is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this equation, we have two terms with the variable 'x': -4x and +5x. We can combine these terms by adding their coefficients. The coefficient of -4x is -4, and the coefficient of 5x is 5. Adding these coefficients, we get -4 + 5 = 1. Therefore, -4x + 5x simplifies to 1x, which is usually written simply as x. Now, our equation becomes x + 16 = 5 - 2. Combining like terms is a fundamental step in simplifying algebraic expressions and equations. It allows us to reduce the number of terms and make the equation easier to solve. This process relies on the commutative and associative properties of addition, which allow us to rearrange and regroup terms without changing the value of the expression. Identifying like terms is crucial for successful simplification. Look for terms with the same variable and the same exponent. Constant terms (terms without a variable) can also be combined. Mastering the art of combining like terms will significantly improve your ability to solve a wide range of algebraic problems. It's a skill that will serve you well throughout your mathematical journey.
Step 3: Simplifying the Right Side of the Equation
Having simplified the left side of the equation, let's now focus on the right side. Our equation currently reads x + 16 = 5 - 2. The right side, 5 - 2, is a simple arithmetic operation. Subtracting 2 from 5 gives us 3. Therefore, the right side of the equation simplifies to 3. Now, our equation becomes x + 16 = 3. Simplifying both sides of the equation is crucial for isolating the variable and solving for its value. In this case, the right side required a straightforward subtraction, but in other equations, you might encounter more complex arithmetic operations, such as multiplication, division, or exponents. The key is to perform these operations correctly and efficiently to arrive at a simplified expression. Remember the order of operations (PEMDAS/BODMAS) when simplifying expressions with multiple operations. This ensures that you perform the operations in the correct sequence, leading to an accurate result. A simplified equation is easier to work with and reduces the chances of making errors in subsequent steps. So, always strive to simplify both sides of the equation as much as possible before proceeding to isolate the variable.
Step 4: Isolating the Variable
Our equation is now simplified to x + 16 = 3. The final step in solving for x is to isolate the variable. This means getting 'x' by itself on one side of the equation. To do this, we need to eliminate the +16 on the left side. We can achieve this by performing the inverse operation, which is subtraction. We subtract 16 from both sides of the equation. Subtracting 16 from the left side, x + 16 - 16, leaves us with just x. Subtracting 16 from the right side, 3 - 16, gives us -13. Therefore, our equation becomes x = -13. We have successfully isolated the variable 'x' and found its value. Isolating the variable is a fundamental technique in solving equations. It involves performing inverse operations to undo the operations that are being applied to the variable. For example, if a number is being added to the variable, we subtract that number from both sides. If a number is multiplying the variable, we divide both sides by that number. It's crucial to perform the same operation on both sides of the equation to maintain equality. This ensures that the solution remains valid. Mastering the art of isolating the variable is essential for solving a wide range of equations, from simple linear equations to more complex algebraic problems. With practice, you'll become proficient at identifying the necessary inverse operations and applying them effectively.
Solution: x = -13
After meticulously following each step, we have arrived at the solution to the equation -4(x-4) + 5x = 5 - 2. By applying the distributive property, combining like terms, simplifying both sides of the equation, and isolating the variable, we have determined that x = -13. This solution represents the value of 'x' that makes the equation true. To verify our solution, we can substitute -13 back into the original equation and check if both sides are equal. Substituting -13 for 'x' in the original equation, we get: -4(-13-4) + 5(-13) = 5 - 2. Simplifying the left side, we have: -4(-17) - 65 = 68 - 65 = 3. The right side is 5 - 2 = 3. Since both sides of the equation are equal when x = -13, our solution is verified. This process of verification is a crucial step in problem-solving, as it ensures the accuracy of our answer. It's always a good practice to check your solutions, especially in mathematical problems. A correct solution demonstrates not only your understanding of the concepts but also your ability to apply them accurately. With x = -13, we have successfully solved the equation.
Conclusion
In conclusion, we have successfully navigated the process of solving the linear equation -4(x-4) + 5x = 5 - 2, arriving at the solution x = -13. This journey has highlighted the importance of several key mathematical concepts and techniques. We began by understanding the structure of the equation and identifying the operations involved. Then, we applied the distributive property to eliminate parentheses, a crucial step in simplifying algebraic expressions. Next, we combined like terms, further reducing the complexity of the equation. We simplified both sides of the equation to make it easier to work with. Finally, we isolated the variable 'x' by performing inverse operations, ultimately revealing its value. Throughout this process, we emphasized the importance of accuracy, precision, and a systematic approach to problem-solving. Each step was carefully explained, with attention to the underlying mathematical principles. This comprehensive guide aims to not only solve the specific equation but also to equip you with the skills and knowledge necessary to tackle a wide range of linear equations. Remember, mathematics is a journey of continuous learning and practice. By mastering the fundamentals, you build a strong foundation for more advanced concepts. So, continue to explore, practice, and challenge yourself, and you'll find yourself becoming increasingly confident and proficient in the world of mathematics.
