Solving For N In The Equation N + 1 = 4(n - 8)

In the realm of mathematics, equations serve as puzzles waiting to be solved. One such equation is n + 1 = 4(n - 8), a seemingly simple expression that holds a hidden value for the variable 'n'. Our mission is to unravel this mystery and determine the numerical value that satisfies this equation. In this article, we will embark on a journey through the steps involved in solving for 'n', providing a clear and concise explanation along the way. So, let's dive into the world of algebra and uncover the solution to this intriguing equation.

1. Understanding the Equation

Before we delve into the mechanics of solving the equation, it's crucial to grasp the fundamental concepts it embodies. The equation n + 1 = 4(n - 8) represents a relationship between two expressions. On the left-hand side, we have 'n + 1', which signifies the sum of the variable 'n' and the number 1. On the right-hand side, we encounter '4(n - 8)', indicating the product of 4 and the difference between 'n' and 8. The equals sign (=) acts as a bridge, asserting that the values of these two expressions are identical. Our goal is to isolate the variable 'n' on one side of the equation, thereby revealing its numerical value.

2. Distributing the 4

The first step in our algebraic quest involves simplifying the right-hand side of the equation. To achieve this, we employ the distributive property, a fundamental principle in mathematics. The distributive property dictates that multiplying a number by a sum or difference is equivalent to multiplying the number by each term within the parentheses individually. In our case, we need to distribute the 4 across the expression '(n - 8)'. This entails multiplying 4 by 'n' and 4 by '-8'.

Performing this distribution, we obtain: 4 * n = 4n and 4 * (-8) = -32. Consequently, the right-hand side of the equation transforms from '4(n - 8)' to '4n - 32'. Now, our equation appears as follows: n + 1 = 4n - 32. We have successfully eliminated the parentheses, bringing us closer to isolating 'n'.

3. Gathering 'n' Terms

Our next objective is to consolidate all the terms containing 'n' onto one side of the equation. To accomplish this, we need to strategically move terms across the equals sign. The key principle here is that performing the same operation on both sides of an equation maintains its balance. In this instance, we choose to subtract 'n' from both sides of the equation. This action will eliminate 'n' from the left-hand side and transfer it to the right-hand side.

Subtracting 'n' from both sides, we get: (n + 1) - n = (4n - 32) - n. Simplifying this, we find that 'n' on the left-hand side cancels out, leaving us with '1'. On the right-hand side, '4n - n' simplifies to '3n'. Thus, our equation now stands as: 1 = 3n - 32. We have successfully gathered the 'n' terms on one side, bringing us closer to our solution.

4. Isolating the 'n' Term

Now that we have all the 'n' terms on one side, our focus shifts to isolating the term containing 'n'. In our equation, '3n' is accompanied by '-32'. To isolate '3n', we need to eliminate this constant term. We can achieve this by adding 32 to both sides of the equation. This operation will effectively cancel out the '-32' on the right-hand side, leaving us with just '3n'.

Adding 32 to both sides, we obtain: 1 + 32 = (3n - 32) + 32. Simplifying this, we find that '1 + 32' equals '33', and '-32 + 32' cancels out, leaving us with just '3n'. Therefore, our equation now reads: 33 = 3n. We have successfully isolated the term containing 'n', bringing us one step closer to the final answer.

5. Solving for 'n'

We have reached the final stage of our algebraic journey: solving for 'n'. Our equation currently states 33 = 3n. To determine the value of 'n', we need to undo the multiplication by 3. The inverse operation of multiplication is division. Therefore, we will divide both sides of the equation by 3. This action will isolate 'n' on the right-hand side, revealing its numerical value.

Dividing both sides by 3, we get: 33 / 3 = (3n) / 3. Simplifying this, we find that '33 / 3' equals '11', and '(3n) / 3' simplifies to 'n'. Consequently, our equation transforms into: 11 = n. We have successfully solved for 'n'! The value of 'n' that satisfies the equation n + 1 = 4(n - 8) is 11.

6. Verification

To ensure the accuracy of our solution, it's always prudent to verify our answer. We can do this by substituting the value we found for 'n' back into the original equation. If both sides of the equation yield the same result, our solution is correct. In our case, we found that n = 11. Let's substitute this value into the original equation: n + 1 = 4(n - 8).

Substituting n = 11, we get: 11 + 1 = 4(11 - 8). Simplifying the left-hand side, we find that 11 + 1 = 12. Simplifying the right-hand side, we first evaluate the expression within the parentheses: 11 - 8 = 3. Then, we multiply 4 by 3, which gives us 4 * 3 = 12. Thus, the right-hand side also equals 12. Since both sides of the equation equal 12 when we substitute n = 11, our solution is verified. We have confidently confirmed that the value of 'n' that satisfies the equation is indeed 11.

Conclusion

In this article, we embarked on a step-by-step journey to solve the equation n + 1 = 4(n - 8). We began by understanding the equation and its components. Then, we employed the distributive property to simplify the right-hand side. Next, we gathered the 'n' terms on one side and isolated the term containing 'n'. Finally, we solved for 'n' by dividing both sides of the equation by 3. Our solution revealed that the value of 'n' that satisfies the equation is 11. To ensure the accuracy of our solution, we verified our answer by substituting it back into the original equation. This process affirmed that our solution was correct. Through this exercise, we have not only solved an algebraic equation but also reinforced our understanding of fundamental mathematical principles. The ability to solve equations is a crucial skill in mathematics and beyond, empowering us to tackle a wide range of problems in various fields. So, let us continue to explore the fascinating world of mathematics and unlock its endless possibilities.