Solving For N In The Equation 2(n+1)-3=5n-(n-4) A Step-by-Step Guide

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Hey there, math enthusiasts! Today, we're diving into a fascinating algebraic equation that might seem a bit daunting at first glance, but trust me, it's totally solvable. We're going to break down the equation 2(n+1)-3=5n-(n-4), step by step, to find the elusive value of 'n'. So, grab your thinking caps, and let's get started!

Understanding the Equation

Before we jump into solving, let's make sure we all understand what the equation is telling us. In essence, we have two expressions on either side of an equals sign. The left side is 2(n+1)-3, and the right side is 5n-(n-4). Our mission, should we choose to accept it (and we do!), is to find the value of 'n' that makes both sides of the equation equal. Think of it like a balancing scale; we need to figure out what 'n' needs to be so that the scale is perfectly balanced.

Embarking on the Algebraic Adventure

Now, let's roll up our sleeves and dive into the nitty-gritty of solving this equation. The key to success here is following the order of operations and applying some basic algebraic principles. Don't worry, we'll take it one step at a time!

Step 1: Distribute the Love (and the Numbers)

The first thing we need to do is simplify both sides of the equation by distributing any numbers that are hanging out in front of parentheses. On the left side, we have 2(n+1). This means we need to multiply the 2 by both the 'n' and the 1 inside the parentheses. So, 2 * n is 2n, and 2 * 1 is 2. This transforms the left side of our equation into 2n + 2 - 3.

On the right side, we have 5n-(n-4). The negative sign in front of the parentheses means we're essentially distributing a -1. So, -1 * n is -n, and -1 * -4 is +4 (remember, a negative times a negative is a positive!). This turns the right side of the equation into 5n - n + 4.

Now, our equation looks like this: 2n + 2 - 3 = 5n - n + 4. We've successfully distributed the numbers and eliminated the parentheses. High five!

Step 2: Combine Like Terms – It's All About Grouping!

The next step is to combine any like terms on each side of the equation. Like terms are those that have the same variable (in this case, 'n') or are just plain old numbers (constants). On the left side, we have 2n, which is a term with 'n', and then we have 2 - 3, which are both constants. Combining the constants, 2 - 3 equals -1. So, the left side simplifies to 2n - 1.

On the right side, we have 5n and -n, which are both terms with 'n', and then we have +4, which is a constant. Combining the 'n' terms, 5n - n is the same as 5n - 1n, which equals 4n. So, the right side simplifies to 4n + 4.

Our equation is now looking much cleaner: 2n - 1 = 4n + 4. We're making progress, guys!

Step 3: Isolate the Variable – Get 'n' on Its Own!

Our goal is to get 'n' all by itself on one side of the equation. To do this, we need to move all the terms with 'n' to one side and all the constants to the other side. Let's start by moving the 2n term from the left side to the right side. To do this, we subtract 2n from both sides of the equation. Remember, whatever we do to one side, we have to do to the other to keep the equation balanced.

So, subtracting 2n from both sides, we get: 2n - 1 - 2n = 4n + 4 - 2n. This simplifies to -1 = 2n + 4 (because 2n - 2n cancels out, and 4n - 2n equals 2n).

Now, let's move the constant +4 from the right side to the left side. To do this, we subtract 4 from both sides: -1 - 4 = 2n + 4 - 4. This simplifies to -5 = 2n (because 4 - 4 cancels out).

We're almost there! We've isolated the 'n' term on the right side.

Step 4: Solve for 'n' – The Grand Finale!

The final step is to solve for 'n' by getting rid of the coefficient (the number in front of 'n'). In this case, the coefficient is 2. To get rid of it, we divide both sides of the equation by 2: -5 / 2 = 2n / 2. This simplifies to -5/2 = n (because 2n / 2 equals n).

So, we've found it! The value of 'n' that satisfies the equation 2(n+1)-3=5n-(n-4) is -5/2 or -2.5. Woohoo! We did it!

Verification: Let's Check Our Answer!

It's always a good idea to check our answer to make sure we didn't make any mistakes along the way. To do this, we substitute -5/2 for 'n' in the original equation and see if both sides are equal.

Original equation: 2(n+1)-3=5n-(n-4)

Substituting n = -5/2: 2(-5/2 + 1) - 3 = 5(-5/2) - (-5/2 - 4)

Let's simplify each side:

Left side: 2(-5/2 + 2/2) - 3 = 2(-3/2) - 3 = -3 - 3 = -6

Right side: 5(-5/2) - (-5/2 - 8/2) = -25/2 - (-13/2) = -25/2 + 13/2 = -12/2 = -6

Both sides equal -6, so our answer is correct! We've successfully verified that n = -5/2 is the solution to the equation.

Key Takeaways and Pro Tips

Solving algebraic equations might seem tricky at first, but with practice, it becomes second nature. Here are some key takeaways and pro tips to keep in mind:

  • Order of Operations: Always follow the order of operations (PEMDAS/BODMAS) – Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
  • Distribution: When you have a number in front of parentheses, distribute it to each term inside the parentheses.
  • Combine Like Terms: Simplify each side of the equation by combining like terms.
  • Isolate the Variable: Get the variable term on one side and the constants on the other side.
  • Solve for the Variable: Divide or multiply to get the variable by itself.
  • Verification: Always check your answer by substituting it back into the original equation.
  • Stay Organized: Keep your work neat and organized to avoid mistakes.

Conclusion: You've Conquered the Equation!

Congratulations, mathletes! You've successfully navigated the world of algebra and found the value of 'n' in the equation 2(n+1)-3=5n-(n-4). You've mastered the art of distribution, combining like terms, isolating variables, and solving for 'n'. Pat yourselves on the back – you deserve it!

Remember, math is like a puzzle, and every equation is a new challenge waiting to be solved. Keep practicing, keep exploring, and keep those mathematical gears turning. Until next time, happy solving!

Further Exploration and Practice

If you're feeling adventurous and want to hone your algebraic skills even further, here are some suggestions:

  • Practice Problems: Seek out more equations similar to this one and try solving them on your own. There are tons of resources online and in textbooks.
  • Online Resources: Websites like Khan Academy and Mathway offer excellent lessons and practice exercises on algebra.
  • Tutoring: If you're struggling with certain concepts, consider seeking help from a tutor or teacher.
  • Real-World Applications: Look for real-world examples of how algebra is used. You might be surprised to see how it pops up in everyday life!

Remember, the key to mastering math is consistent practice and a willingness to embrace the challenge. Keep up the great work, and you'll be solving even the most complex equations in no time!

So go forth, my mathematical friends, and conquer the world of equations! You've got this!