Solving -16 + 5n = -7(-6 + 8n) + 3 A Step-by-Step Guide
Introduction
In this detailed guide, we will thoroughly explore the process of solving the linear equation -16 + 5n = -7(-6 + 8n) + 3. Linear equations, foundational in mathematics, appear across numerous disciplines, making proficiency in solving them crucial. This article aims to provide a step-by-step solution, offering clear explanations to enhance understanding and problem-solving skills. We will delve into each stage, ensuring clarity and accuracy in our solution. Our approach will not only focus on the mechanics of solving the equation but also on understanding the underlying principles of algebraic manipulation. This comprehensive methodology will empower you to tackle similar problems confidently and effectively. Whether you're a student looking to improve your algebra skills or someone seeking a refresher, this guide offers valuable insights and practical techniques for solving linear equations.
Step 1: Distribute the -7 on the Right Side of the Equation
The first key step in simplifying the equation -16 + 5n = -7(-6 + 8n) + 3 is to distribute the -7 across the terms within the parentheses on the right side. This process involves multiplying -7 by both -6 and 8n. When multiplying -7 by -6, we remember that the product of two negative numbers is positive, so -7 multiplied by -6 equals 42. Next, we multiply -7 by 8n. Since we are multiplying a negative number by a positive number, the result will be negative. Thus, -7 times 8n equals -56n. Applying this distribution, the equation transforms from its original state to -16 + 5n = 42 - 56n + 3. This step is crucial because it eliminates the parentheses, making it easier to combine like terms in the subsequent steps. The distributive property is a fundamental concept in algebra, allowing us to simplify complex expressions and equations. Mastering this step is essential for solving a wide range of algebraic problems, including linear equations like this one. By carefully applying the distributive property, we lay the groundwork for solving the equation efficiently and accurately. This transformation sets the stage for further simplification and isolation of the variable 'n'.
Step 2: Combine Like Terms on the Right Side
Following the distribution, our equation now reads -16 + 5n = 42 - 56n + 3. The next crucial step in simplifying this equation involves combining the like terms on the right side. Like terms are those that contain the same variable raised to the same power, or in this case, the constant terms. On the right side, we have two constant terms: 42 and 3. To combine these, we simply add them together. Adding 42 and 3 gives us a total of 45. Therefore, the right side of the equation simplifies to 45 - 56n. After this combination, our equation is further simplified to -16 + 5n = 45 - 56n. This step is vital for making the equation more manageable and paving the way for isolating the variable 'n'. By combining like terms, we reduce the complexity of the equation, making it easier to manipulate and solve. This process helps to streamline the equation, bringing us closer to finding the value of 'n'. Understanding how to identify and combine like terms is a fundamental skill in algebra, essential for solving equations and simplifying expressions efficiently. The ability to combine like terms accurately is key to progressing through the solution process smoothly and correctly.
Step 3: Add 56n to Both Sides of the Equation
Now that our equation is simplified to -16 + 5n = 45 - 56n, the subsequent step involves moving all terms containing the variable 'n' to one side of the equation. To achieve this, we focus on the term -56n on the right side. The most effective way to eliminate this term from the right side and move it to the left is by adding its additive inverse, which is +56n, to both sides of the equation. This is based on the fundamental algebraic principle that adding the same quantity to both sides of an equation maintains the equality. By adding 56n to both sides, we transform the equation. On the left side, adding 56n to 5n results in 61n, so the left side becomes -16 + 61n. On the right side, adding 56n to -56n cancels each other out, leaving only 45. Therefore, the equation now reads -16 + 61n = 45. This step is pivotal in isolating the variable 'n'. By moving all 'n' terms to one side, we bring the equation closer to a form where 'n' can be easily solved. This manipulation is a key technique in solving algebraic equations, allowing us to streamline the process and focus on isolating the variable of interest.
Step 4: Add 16 to Both Sides of the Equation
With the equation now in the form -16 + 61n = 45, the next crucial step is to isolate the term containing 'n' on one side. To achieve this, we need to eliminate the constant term -16 from the left side of the equation. The most effective way to do this is by adding its additive inverse, which is +16, to both sides of the equation. This action is based on the fundamental principle that adding the same value to both sides of an equation maintains the balance and equality. When we add 16 to the left side of the equation, it cancels out the -16, leaving us with just 61n. On the right side, adding 16 to 45 results in 61. Consequently, the equation transforms to 61n = 61. This step is essential because it isolates the term with 'n', bringing us closer to solving for the value of 'n'. Isolating the variable is a fundamental technique in solving algebraic equations, and this step demonstrates that principle clearly. By strategically adding 16 to both sides, we simplify the equation and set the stage for the final step of solving for 'n'. This maneuver is a key component in the process of solving linear equations, allowing us to efficiently determine the value of the unknown variable.
Step 5: Divide Both Sides by 61
Having simplified the equation to 61n = 61, the final step in solving for 'n' involves isolating 'n' completely. Currently, 'n' is being multiplied by 61. To undo this multiplication and solve for 'n', we need to perform the inverse operation, which is division. We divide both sides of the equation by 61. This action is based on the principle that dividing both sides of an equation by the same non-zero number maintains the equality. When we divide 61n by 61, the 61s cancel each other out, leaving 'n' by itself on the left side of the equation. On the right side, dividing 61 by 61 results in 1. Therefore, the equation simplifies to n = 1. This final step definitively solves for the value of 'n', which is 1. This process of division is a fundamental technique in algebra, used to isolate variables and solve equations. By dividing both sides by the coefficient of 'n', we successfully determined the value of the unknown variable. This step highlights the importance of using inverse operations to solve for variables in algebraic equations. The solution n = 1 is the value that satisfies the original equation, making the equation a true statement.
Final Answer
After meticulously following the steps of simplification and isolation, we have arrived at the solution for the linear equation -16 + 5n = -7(-6 + 8n) + 3. Through a series of algebraic manipulations, including distribution, combining like terms, and applying inverse operations, we have successfully isolated the variable 'n'. Our step-by-step process led us to the final solution, which is n = 1. This result signifies that when 'n' is replaced with 1 in the original equation, both sides of the equation are equal, thus satisfying the equation. The journey to this solution has highlighted the importance of algebraic principles, such as maintaining equality by performing the same operations on both sides, and the strategic use of inverse operations to isolate variables. Understanding and applying these principles are crucial for solving a wide array of mathematical problems. Therefore, the final answer to the given linear equation is definitively n = 1. This conclusion not only solves the specific problem at hand but also reinforces the understanding of fundamental algebraic techniques.
