Solving The Equation: $5(n+2)+6(-1-4n)=4-7n+n$

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Hey guys! Today, we're diving into a fun little math problem. We need to solve the equation 5(n+2)+6(−1−4n)=4−7n+n5(n+2)+6(-1-4n)=4-7n+n. Don't worry, it looks more complicated than it actually is. We'll break it down step by step, so it's super easy to follow. Our main goal here is to find the value of 'n' that makes this equation true. We'll use some basic algebraic principles, like the distributive property and combining like terms. So, grab your pencils, and let's get started! Remember, math can be enjoyable if you approach it methodically and take your time to understand each step. Let’s jump right into it and see how we can crack this equation together! We're going to make math less intimidating and more accessible, so everyone feels confident tackling these kinds of problems. Let’s do this!

Step-by-Step Solution

1. Distribute the numbers outside the parentheses

The first thing we need to do is get rid of those parentheses. We do this by distributing the numbers outside the parentheses to the terms inside. So, we'll multiply 5 by both 'n' and 2, and we'll multiply 6 by both -1 and -4n.

  • 5∗(n+2)=5n+105 * (n + 2) = 5n + 10
  • 6∗(−1−4n)=−6−24n6 * (-1 - 4n) = -6 - 24n

So, our equation now looks like this:

5n+10−6−24n=4−7n+n5n + 10 - 6 - 24n = 4 - 7n + n

This step is crucial because it simplifies the equation and allows us to combine like terms later on. Remember, the distributive property is a fundamental concept in algebra, and it's something you'll use a lot. We're essentially expanding the expression to make it easier to work with. Taking the time to ensure this step is done correctly will set the stage for a smoother solution process. Now that we've successfully distributed, we can move on to the next step: combining those like terms. This is where we group together terms with the same variable and constant terms to further simplify our equation. So, let's keep going and see what we can do!

2. Combine like terms on both sides of the equation

Now, let's combine the like terms on each side of the equation. On the left side, we have '5n' and '-24n', which are like terms because they both contain the variable 'n'. We also have '+10' and '-6', which are constants (numbers without a variable). On the right side, we have '-7n' and '+n', which are also like terms.

  • Combining '5n' and '-24n' gives us: 5n−24n=−19n5n - 24n = -19n
  • Combining '+10' and '-6' gives us: 10−6=410 - 6 = 4
  • Combining '-7n' and '+n' gives us: −7n+n=−6n-7n + n = -6n

So, our equation now looks like this:

−19n+4=4−6n-19n + 4 = 4 - 6n

Combining like terms is a vital step in simplifying algebraic equations. It helps us to consolidate the expression, making it easier to isolate the variable we're solving for. By grouping similar terms together, we reduce the complexity of the equation, which is crucial for accurately determining the value of 'n'. This step ensures that we're working with a more manageable form of the equation, making the subsequent steps clearer and more straightforward. Make sure to double-check that you've combined all like terms correctly before moving on, as any errors here will impact the final result. Let’s continue on our journey to solve for 'n'!

3. Move the variable terms to one side of the equation

Next, we want to get all the terms with 'n' on one side of the equation. It doesn't matter which side we choose, but let's move them to the left side in this case. To do this, we'll add '6n' to both sides of the equation.

−19n+4+6n=4−6n+6n-19n + 4 + 6n = 4 - 6n + 6n

This simplifies to:

−13n+4=4-13n + 4 = 4

The goal here is to isolate the variable term. We want all the 'n' terms on one side and all the constant terms on the other side. This is a fundamental technique in solving equations. By adding the same term to both sides, we maintain the balance of the equation while moving the terms where we want them. It's like a mathematical balancing act! This step is a key part of the solving process, as it sets us up for the final isolation of the variable. Always remember to perform the same operation on both sides to keep the equation balanced. Now that we've moved the variable terms, let's move on to isolating 'n' completely. We're getting closer to our solution!

4. Move the constant terms to the other side of the equation

Now, let's move the constant term (+4) from the left side to the right side. To do this, we'll subtract 4 from both sides of the equation.

−13n+4−4=4−4-13n + 4 - 4 = 4 - 4

This simplifies to:

−13n=0-13n = 0

Moving the constant terms helps us further isolate the variable. This step follows the same principle as moving variable terms: we perform the same operation on both sides to maintain the equation's balance. Isolating the variable is like peeling away the layers to reveal the core value we're seeking. By subtracting 4 from both sides, we've successfully cleared the left side of any constants, leaving us with just the term involving 'n'. We're almost there! The final step will involve dealing with the coefficient of 'n' to fully uncover its value. So, let’s keep pushing forward and finish this problem!

5. Solve for n

Finally, to solve for 'n', we need to get rid of the coefficient -13. We do this by dividing both sides of the equation by -13.

−13n−13=0−13\frac{-13n}{-13} = \frac{0}{-13}

This gives us:

n=0n = 0

And there you have it! We've solved the equation. Dividing both sides by the coefficient is the final act of isolating the variable. This step gives us the direct value of 'n' that satisfies the original equation. When we divide 0 by any non-zero number, the result is always 0. So, in this case, n equals 0. This completes our solution process. We've taken a potentially daunting equation and broken it down into manageable steps, applying fundamental algebraic principles along the way. Congratulations on making it to the end! Solving for a variable is a cornerstone of algebra, and mastering these techniques will serve you well in more advanced math problems. Let's celebrate our success!

Final Answer

Therefore, the solution to the equation 5(n+2)+6(−1−4n)=4−7n+n5(n+2)+6(-1-4n)=4-7n+n is:

n=0n = 0

So, guys, we've cracked it! The value of 'n' that satisfies the equation is 0. I hope you found this step-by-step solution helpful and easy to understand. Remember, math problems can seem tough at first, but breaking them down into smaller steps makes them much more manageable. Keep practicing, and you'll become a pro at solving these in no time! If you have any questions or want to try another problem, just let me know. Happy solving!