Unlock Algebra: Solve 4(5n + 8722) = 3(5n + 8722) - 14

by Jhon Lennon 55 views

Hey algebra adventurers! Today, we're diving deep into a super interesting equation that might look a little intimidating at first glance: 4(5n + 8722) = 3(5n + 8722) - 14. Don't let those big numbers or the parentheses fool you, guys. We're going to break this down step-by-step, and by the end, you'll be a master at solving it. Algebra is all about patterns and logical steps, and this problem is a perfect example of that. We'll explore the distributive property, how to combine like terms, and isolate the variable 'n' to find its true value. Get ready to boost your math skills and impress your friends with your newfound algebraic prowess. This isn't just about finding 'n'; it's about understanding the process and building confidence in your problem-solving abilities. So, grab your favorite thinking cap, maybe a snack, and let's get started on this mathematical journey. We'll cover everything from the basics of simplifying expressions to the final triumphant reveal of 'n'. It's going to be fun, I promise!

Understanding the Equation: What's Really Going On?

Alright team, let's first get a solid grip on the equation we're tackling: 4(5n + 8722) = 3(5n + 8722) - 14. At its heart, this is a linear equation, meaning the highest power of our variable 'n' is just one. The goal, as always in algebra, is to find the value of 'n' that makes both sides of the equation equal. See those parentheses? They're like little traps designed to make you think twice, but they're actually your best friends when you know how to handle them. The key here is the distributive property. This magical property tells us that when you have a number multiplying a set of terms inside parentheses, you need to multiply that number by each term inside. So, on the left side, we have '4' outside '(5n + 8722)', and on the right side, we have '3' outside '(5n + 8722)'. Notice anything special? Yep, the expression (5n + 8722) appears on both sides of the equation! This is a crucial observation that can actually simplify our approach significantly. We're not just blindly applying the distributive property; we're also looking for shortcuts and patterns. Think of it like this: you have four identical gift bags, each containing 5 candies and 8722 marbles. On the other side, you have three of those same gift bags, but then you also have 14 loose marbles. You want to figure out how many candies ('n') are in each bag so that the total number of items on both sides is equal. This real-world analogy helps visualize the algebraic concepts. So, before we jump into calculations, take a moment to really see the structure of the equation. Recognize that 5n + 8722 is a common block. This understanding is the first major step towards a smooth and efficient solution. We're building a strong foundation before we start constructing the solution!

Step 1: Applying the Distributive Property

Now, let's get our hands dirty with some actual math, starting with the distributive property. Remember what we said? Multiply the number outside the parentheses by each term inside. Let's start with the left side of the equation: 4(5n + 8722).

  • Multiply 4 by 5n: 4 * 5n = 20n
  • Multiply 4 by 8722: 4 * 8722 = 34888

So, the left side simplifies to 20n + 34888.

Now, let's tackle the right side: 3(5n + 8722) - 14.

  • First, distribute the 3 to the terms inside the parentheses:

    • Multiply 3 by 5n: 3 * 5n = 15n
    • Multiply 3 by 8722: 3 * 8722 = 26166

  • So, the part with the parentheses becomes 15n + 26166.

  • Now, we still have that '- 14' hanging around on the right side. Let's combine the constant terms: 26166 - 14 = 26152.

Putting the right side back together, it simplifies to 15n + 26152.

After applying the distributive property and simplifying, our original equation 4(5n + 8722) = 3(5n + 8722) - 14 now looks like this: 20n + 34888 = 15n + 26152. See? It's already looking much more manageable! We've successfully removed the parentheses and combined some terms. This is a huge win, guys! It shows that even complex-looking equations can be systematically simplified. Take a moment to appreciate this transformation. It's the power of following the rules of algebra. We haven't even solved for 'n' yet, but we've already made significant progress by making the equation cleaner and easier to work with. This is a common strategy in problem-solving: simplify first, then solve. Keep that momentum going!

Step 2: Isolating the Variable 'n' (The Fun Part!)

Alright, we've simplified our equation to 20n + 34888 = 15n + 26152. Our next mission, should we choose to accept it (and we totally should!), is to get all the terms with 'n' on one side of the equation and all the constant numbers on the other. This is what we call isolating the variable. Think of it as a tug-of-war, where we want all the 'n's on one team and all the numbers on the other. To do this, we use the golden rule of algebra: whatever you do to one side of the equation, you must do to the other side to keep it balanced.

Let's start by moving the 'n' terms. We have 20n on the left and 15n on the right. It's usually easier to work with positive numbers, so let's subtract 15n from both sides:

  • 20n + 34888 - 15n = 15n + 26152 - 15n

  • On the left side, 20n - 15n gives us 5n. So, the left side becomes 5n + 34888.

  • On the right side, 15n - 15n cancels out, leaving us with just 26152.

Our equation now looks like this: 5n + 34888 = 26152.

Great job! We've successfully moved all the 'n' terms to the left. Now, let's move the constant terms. We have '+ 34888' on the left and we want to get rid of it. To do that, we subtract 34888 from both sides:

  • 5n + 34888 - 34888 = 26152 - 34888

  • On the left side, 34888 - 34888 cancels out, leaving us with just 5n.

  • On the right side, we calculate 26152 - 34888. Since we're subtracting a larger number from a smaller one, the result will be negative. 26152 - 34888 = -8736.

So, our equation is now: 5n = -8736.

We are so close to the finish line, guys! We've isolated the term with 'n'. The next (and final) step is to get 'n' all by itself.

Step 3: Solving for 'n' and Final Check

We've arrived at the pivotal point: 5n = -8736. This equation tells us that 5 times the value of 'n' equals -8736. To find out what 'n' is, we need to perform the inverse operation of multiplication, which is division. We'll divide both sides of the equation by 5:

  • 5n / 5 = -8736 / 5

  • On the left side, 5n / 5 simplifies to just n.

  • On the right side, we need to calculate -8736 / 5.

Let's do the division: 8736 ÷ 5.

  • 8 divided by 5 is 1 with a remainder of 3.
  • Bring down the 7, making it 37. 37 divided by 5 is 7 with a remainder of 2.
  • Bring down the 3, making it 23. 23 divided by 5 is 4 with a remainder of 3.
  • Bring down the 6, making it 36. 36 divided by 5 is 7 with a remainder of 1.

So, 8736 / 5 is 1747 with a remainder of 1, or 1747.2. Since our original number was negative, our result will also be negative.

Therefore, n = -1747.2.

The Final Check: Is Our Answer Correct?

It's always, always, always a good idea to check our answer. This is where we plug our calculated value of 'n' back into the original equation to see if both sides are indeed equal. This step confirms our work and builds confidence. Let's substitute n = -1747.2 into 4(5n + 8722) = 3(5n + 8722) - 14.

First, let's calculate the value of (5n + 8722):

  • 5 * (-1747.2) = -8736
  • Now add 8722: -8736 + 8722 = -14

So, the expression (5n + 8722) equals -14.

Now, let's plug this value back into the original equation:

  • Left side: 4 * (-14) = -56
  • Right side: 3 * (-14) - 14
    • 3 * (-14) = -42
    • -42 - 14 = -56

Look at that! The left side (-56) equals the right side (-56). Our solution is correct! Isn't that satisfying? You've successfully navigated a complex algebraic equation, applied the distributive property, isolated the variable, and verified your answer. High fives all around, you algebra superstars!

Conclusion: You've Conquered the Equation!

And there you have it, folks! We've successfully solved the equation 4(5n + 8722) = 3(5n + 8722) - 14, finding that n = -1747.2. We walked through each step methodically: understanding the structure, applying the distributive property, simplifying, isolating the variable 'n', and finally, checking our work. Remember, the key takeaways are to stay calm, break down the problem into smaller steps, and always remember the golden rule of keeping both sides of the equation balanced. The distributive property is your best friend when dealing with parentheses, and isolating the variable is all about strategic addition, subtraction, multiplication, and division. Don't be discouraged by large numbers or seemingly complex expressions; they often simplify beautifully. Keep practicing these types of problems, and you'll find your confidence and speed increasing with every equation you solve. Math is a skill that improves with practice, just like learning an instrument or a new sport. So keep at it, celebrate your successes, and remember that you've got this! You're not just solving equations; you're building valuable problem-solving skills that will serve you well in all areas of life. Awesome job, everyone!