Solving For N: 11(n-1)+35=3n - Find The Value!

Let's dive into solving the equation 11(n-1)+35=3n for the variable n. This is a classic algebra problem that involves distributing, combining like terms, and isolating the variable. We'll break down each step to make it super clear and easy to follow. Understanding how to solve for n in such equations is a fundamental skill in mathematics, crucial for more advanced topics. So, let's get started and unlock the value of n!

Understanding the Equation

The given equation is 11(n-1)+35=3n. Before we jump into solving, let's take a moment to understand what this equation represents. It's a linear equation, meaning that the highest power of our variable n is 1. These equations typically have one solution, which is the value of n that makes the equation true. Our goal is to isolate n on one side of the equation to find that value. This involves using algebraic manipulations, always ensuring that we maintain the balance of the equation – whatever operation we perform on one side, we must also perform on the other. This principle of balance is the cornerstone of equation solving.

Breaking Down the Components

• 11(n-1): This part involves the distributive property. We need to multiply 11 by both n and -1 inside the parentheses.
• +35: This is a constant term added to the result of the distribution.
• =3n: This is the other side of the equation, with 3n representing a term with the variable n.

Why is this important?

Knowing how to solve for n in this type of equation is crucial because it's a building block for more complex mathematical problems. Linear equations appear in various fields, including physics, engineering, economics, and computer science. Whether you're calculating the trajectory of a projectile, balancing a budget, or designing an algorithm, the ability to manipulate and solve equations is indispensable. Furthermore, mastering this skill enhances your problem-solving abilities, teaching you to approach challenges logically and systematically.

Step-by-Step Solution

Now, let's go through the step-by-step solution of the equation 11(n-1)+35=3n. We'll break it down into manageable parts, explaining each operation and its purpose. This process will not only give you the answer but also deepen your understanding of algebraic techniques.

Step 1: Distribute the 11

Our first step is to apply the distributive property to the term 11(n-1). This means we multiply 11 by both n and -1:

11 * n = 11n

11 * -1 = -11

So, 11(n-1) becomes 11n - 11. Now, let’s rewrite the entire equation with this simplification:

11n - 11 + 35 = 3n

Why do we distribute? Distribution allows us to remove the parentheses and combine like terms, which is a crucial step in simplifying and solving equations. It's like unwrapping a package – we need to get inside to see what we're working with!

Step 2: Combine Like Terms

Next, we need to combine the like terms on the left side of the equation. Like terms are those that have the same variable raised to the same power, or constants. In this case, we have two constants: -11 and +35. Let's combine them:

-11 + 35 = 24

Now our equation looks like this:

11n + 24 = 3n

Why combine like terms? Combining like terms simplifies the equation, making it easier to isolate the variable. It's like organizing your toolbox – grouping similar tools together helps you find what you need quickly.

Step 3: Move the Variable Terms to One Side

Our goal is to isolate n, so we need to get all the terms with n on one side of the equation. Let's subtract 3n from both sides:

11n + 24 - 3n = 3n - 3n

This simplifies to:

8n + 24 = 0

Why move variable terms? Moving variable terms to one side and constant terms to the other is a fundamental technique in solving equations. It sets the stage for isolating the variable and finding its value. Think of it as sorting your laundry – separating the whites from the colors before washing.

Step 4: Move the Constant Term to the Other Side

Now, let's move the constant term (+24) to the other side of the equation by subtracting 24 from both sides:

8n + 24 - 24 = 0 - 24

This simplifies to:

8n = -24

Why move the constant term? Moving the constant term isolates the variable term, bringing us closer to our goal of solving for n. It's like clearing your desk – removing the clutter so you can focus on the important papers.

Step 5: Isolate n

Finally, to isolate n, we need to divide both sides of the equation by the coefficient of n, which is 8:

8n / 8 = -24 / 8

This gives us:

n = -3

Why divide by the coefficient? Dividing by the coefficient of the variable is the final step in isolating the variable. It's like unlocking the last lock on a treasure chest – once you do it, the treasure (the value of n) is yours!

Checking the Solution

It's always a good idea to check your solution to make sure it's correct. To do this, we substitute our value of n (-3) back into the original equation:

11(n-1) + 35 = 3n

11(-3-1) + 35 = 3(-3)

11(-4) + 35 = -9

-44 + 35 = -9

-9 = -9

The left side of the equation equals the right side, so our solution n = -3 is correct!

Why check the solution? Checking your solution ensures accuracy and helps prevent errors. It's like proofreading your work – catching mistakes before they cause problems. This step is especially important in mathematics, where a small error can lead to a wrong answer.

Conclusion

In conclusion, we have successfully solved the equation 11(n-1)+35=3n and found that n = -3. This process involved several key algebraic techniques, including distribution, combining like terms, and isolating the variable. Understanding and mastering these techniques is essential for success in mathematics and related fields. Remember, practice makes perfect, so keep solving equations and building your skills!

For further learning and practice on algebraic equations, you can visit resources like Khan Academy's Algebra 1 section, which offers a wealth of lessons, exercises, and videos.