Solving For N: 6n - 7(n+1) = 6n - Step-by-Step Guide

Oct 24, 2025 by ADMIN 53 views

Solving for n in the Equation 6n - 7(n+1) = 6n: A Comprehensive Guide

Hey guys! Today, we're going to dive into solving a pretty neat algebraic equation. We've got 6n - 7(n+1) = 6n, and our mission, should we choose to accept it (and we do!), is to find the value of n that makes this equation true. Don't worry, it's not as daunting as it looks. We'll break it down step-by-step, so even if algebra isn't your best friend, you'll be able to follow along. So, grab your pencils, and let's get started!

Understanding the Basics of Algebraic Equations

Before we jump straight into solving, let's quickly recap what an algebraic equation actually is. Think of it like a balanced scale. On one side, you've got an expression (like our 6n - 7(n+1)), and on the other, you've got another expression (in our case, 6n). The equals sign (=) is the fulcrum, the point that keeps everything balanced. Our goal is to figure out what value of n will keep the scale perfectly level. In simpler terms, we want to find the n that makes both sides of the equation equal.

Key concepts in algebra that we'll be using include the distributive property (which helps us get rid of those parentheses) and combining like terms (which simplifies our equation). Remember, the golden rule is whatever you do to one side of the equation, you must do to the other to maintain that balance.

The Importance of Order of Operations

You've probably heard of PEMDAS or BODMAS – these are acronyms that help us remember the order of operations: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). We'll be using this throughout our solution to make sure we're tackling things in the correct order. Messing up the order can lead to the wrong answer, and we definitely don't want that!

Step-by-Step Solution to 6n - 7(n+1) = 6n

Okay, let's get our hands dirty and solve this equation. We'll go through each step meticulously, so you can see exactly what's happening and why.

Step 1: Distribute the -7

The first thing we need to do is tackle those parentheses. We've got -7(n+1), which means we need to distribute the -7 to both the n and the +1 inside the parentheses. This means multiplying -7 by n and then multiplying -7 by 1.

So, -7 multiplied by n is -7n, and -7 multiplied by 1 is -7. This transforms our equation to:

6n - 7n - 7 = 6n

Step 2: Combine Like Terms on the Left Side

Now, let's simplify the left side of the equation. We've got two terms with n: 6n and -7n. These are like terms, which means we can combine them.

6n minus 7n is -1n, which we can simply write as -n. So, our equation now looks like this:

-n - 7 = 6n

Step 3: Isolate the Variable Terms

Our next goal is to get all the terms with n on one side of the equation. Currently, we've got -n on the left and 6n on the right. Let's move the -n to the right side. To do this, we add n to both sides of the equation. Remember, whatever we do to one side, we have to do to the other to keep things balanced!

So, we add n to both sides:

-n - 7 + n = 6n + n

This simplifies to:

-7 = 7n

Step 4: Solve for n

We're almost there! We've got -7 = 7n. Now, we need to isolate n completely. It's currently being multiplied by 7, so to undo that multiplication, we need to divide both sides of the equation by 7.

Dividing both sides by 7, we get:

-7 / 7 = 7n / 7

This simplifies to:

-1 = n

Step 5: The Solution

We've done it! We've found the value of n that satisfies the equation. Our solution is:

n = -1

Verifying the Solution

It's always a good idea to check our answer to make sure we haven't made any mistakes along the way. To do this, we'll substitute our solution, n = -1, back into the original equation and see if both sides are equal.

Our original equation was:

6n - 7(n+1) = 6n

Substitute n = -1:

6(-1) - 7(-1+1) = 6(-1)

Now, let's simplify:

-6 - 7(0) = -6

-6 - 0 = -6

-6 = -6

Hey, look at that! Both sides are equal, which means our solution, n = -1, is correct. Awesome!

Common Mistakes to Avoid

Solving algebraic equations can be tricky, and it's easy to make mistakes if you're not careful. Here are a few common pitfalls to watch out for:

Forgetting to Distribute: When you have a number or variable multiplied by something in parentheses, make sure you distribute it to every term inside the parentheses. This is a classic mistake, so double-check your work!
Incorrectly Combining Like Terms: Only combine terms that are truly alike. You can't combine a term with n with a constant term (a term without a variable). Make sure you're adding or subtracting the coefficients (the numbers in front of the variables) correctly.
Not Following the Order of Operations: PEMDAS/BODMAS is your friend! Make sure you're tackling operations in the correct order. Parentheses/Brackets first, then Exponents/Orders, then Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right).
Forgetting to Apply Operations to Both Sides: Remember the golden rule! Whatever you do to one side of the equation, you MUST do to the other side to maintain balance. If you add something to the left side, you need to add the exact same thing to the right side.
Sign Errors: Keep a close eye on your signs (positive and negative). A small sign error can throw off your entire solution. It's easy to make a mistake, so take your time and double-check.

Tips and Tricks for Mastering Algebraic Equations

Want to become a whiz at solving algebraic equations? Here are a few tips and tricks to help you on your way:

Practice, Practice, Practice: The more you practice, the more comfortable you'll become with solving equations. Work through lots of examples, and don't be afraid to make mistakes – they're a great learning opportunity!
Show Your Work: It might seem tedious, but writing out each step of your solution can help you catch errors and understand the process better. It's also easier to go back and see where you went wrong if you make a mistake.
Check Your Answers: Always verify your solution by plugging it back into the original equation. This is the best way to make sure you haven't made any errors.
Use Visual Aids: If you're a visual learner, try drawing diagrams or using manipulatives to help you understand the concepts. Algebra tiles can be especially helpful for visualizing equations.
Break It Down: If you're faced with a complex equation, break it down into smaller, more manageable steps. This can make the problem seem less daunting and easier to solve.
Don't Be Afraid to Ask for Help: If you're stuck, don't hesitate to ask a teacher, tutor, or friend for help. Sometimes a fresh perspective can make all the difference.

Conclusion: You've Cracked the Code!

There you have it, guys! We've successfully solved the equation 6n - 7(n+1) = 6n and found that n = -1. We've also covered some important algebraic concepts, common mistakes to avoid, and tips for mastering equations. Remember, practice makes perfect, so keep working at it, and you'll become an algebra pro in no time.

Solving algebraic equations is a fundamental skill in mathematics, and it opens the door to more advanced topics. So, keep honing your skills, and you'll be well-equipped to tackle any mathematical challenge that comes your way. Keep up the great work!