Solving For X: A Step-by-Step Guide To Equation 8(x+5)

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Hey guys! Today, we're diving deep into the world of algebra to tackle an equation that might seem a bit intimidating at first glance. But don't worry, we're going to break it down step by step, so by the end of this article, you'll be solving equations like a pro. Our mission? To find the value of 'x' in the equation 8(x + 5) + 3 = -3(x - 7) - 2. So, grab your pencils, notebooks, and let's get started!

Understanding the Equation

Before we jump into the solution, let's take a moment to understand what this equation is all about. We have a linear equation, which means that the highest power of our variable 'x' is 1. These types of equations represent a straight line when graphed, hence the name 'linear.' Our goal is to isolate 'x' on one side of the equation so that we can determine its value. To do this, we'll need to use a combination of algebraic operations, such as distribution, combining like terms, and inverse operations.

The beauty of algebra lies in its ability to represent relationships between numbers and variables. In this particular equation, we see 'x' interacting with several constants and coefficients. Our job is to carefully unravel these interactions to reveal the hidden value of 'x.' Think of it like a puzzle – each step we take brings us closer to the final solution. Remember, the key is to maintain balance. Whatever operation we perform on one side of the equation, we must also perform on the other side to keep the equation true. This principle is the cornerstone of solving algebraic equations, ensuring that we're not changing the fundamental relationship, but rather transforming it into a more revealing form.

Step 1: Distribute

The first order of business is to get rid of those parentheses. We do this by using the distributive property, which means we multiply the number outside the parentheses by each term inside. Let's start with the left side of the equation: 8(x + 5). We multiply 8 by x and 8 by 5, which gives us 8x + 40. Don't forget the +3 that's already there, so the left side now looks like 8x + 40 + 3.

Now, let's tackle the right side: -3(x - 7). Here, we multiply -3 by x and -3 by -7. Remember that multiplying two negative numbers results in a positive number, so -3 times -7 is +21. This gives us -3x + 21. And we still have the -2 hanging around, making the right side -3x + 21 - 2. Distribution is like unpacking a gift – we're revealing the individual terms that were hidden inside the parentheses. It's a crucial step because it allows us to combine like terms and simplify the equation. By distributing carefully, we ensure that we're accurately representing the original equation in a more manageable form. This step is all about expanding the equation, setting the stage for the subsequent steps where we'll start to condense and isolate our variable.

Step 2: Combine Like Terms

Now that we've distributed, it's time to simplify each side of the equation by combining like terms. Like terms are those that have the same variable raised to the same power (or no variable at all, in which case they are constants). On the left side, we have 8x + 40 + 3. The like terms here are 40 and 3, which we can add together to get 43. So, the left side simplifies to 8x + 43.

On the right side, we have -3x + 21 - 2. The like terms here are 21 and -2. Subtracting 2 from 21 gives us 19, so the right side simplifies to -3x + 19. Combining like terms is like tidying up a room – we're grouping similar items together to make things more organized and easier to work with. In our equation, this step reduces the number of terms, making the equation less cluttered and more approachable. It's a fundamental step in solving equations because it streamlines the process, allowing us to focus on the essential components. By combining like terms, we're essentially simplifying the expression on each side of the equation, bringing us closer to isolating 'x' and finding its value. This step transforms the equation into a more concise form, highlighting the relationship between the variable and the constants.

Step 3: Move the x Terms to One Side

Our next goal is to get all the terms with 'x' on one side of the equation. It doesn't matter which side we choose, but let's aim for the side that will give us a positive coefficient for 'x' to make things a bit easier. We have 8x + 43 = -3x + 19. To move the -3x from the right side to the left side, we add 3x to both sides of the equation. This gives us 8x + 3x + 43 = -3x + 3x + 19, which simplifies to 11x + 43 = 19.

Moving the 'x' terms is like gathering all the members of a team together – we're consolidating the variable terms to one side so we can focus on isolating 'x'. The principle behind this step is the addition property of equality, which states that adding the same quantity to both sides of an equation maintains the equality. By strategically adding 3x to both sides, we eliminate the 'x' term from the right side and bring it over to the left side. This is a crucial step in solving for 'x' because it groups the variable terms together, paving the way for the next steps where we'll isolate 'x' completely. The decision to move the 'x' terms to the left side in this case was driven by the desire to keep the coefficient of 'x' positive, which often simplifies the subsequent calculations.

Step 4: Isolate the x Term

Now we want to isolate the 'x' term, which means getting rid of the constant term on the same side. We have 11x + 43 = 19. To get rid of the +43, we subtract 43 from both sides of the equation. This gives us 11x + 43 - 43 = 19 - 43, which simplifies to 11x = -24.

Isolating the 'x' term is like clearing a path – we're removing any obstacles that are preventing 'x' from being completely alone on one side of the equation. This step relies on the subtraction property of equality, which states that subtracting the same quantity from both sides of an equation maintains the equality. By subtracting 43 from both sides, we effectively cancel out the +43 on the left side, leaving only the 'x' term and its coefficient. This is a pivotal step in solving for 'x' because it brings us one step closer to the final solution. With the 'x' term now isolated, we can focus on the last operation needed to determine the value of 'x'. This step is all about precision and attention to detail, ensuring that we're accurately manipulating the equation to reveal the value of our variable.

Step 5: Solve for x

Finally, we're in the home stretch! We have 11x = -24. To solve for x, we need to divide both sides of the equation by the coefficient of x, which is 11. This gives us 11x / 11 = -24 / 11, which simplifies to x = -24/11.

Solving for 'x' is like reaching the summit of a mountain – we've navigated all the challenges and arrived at our destination. This final step utilizes the division property of equality, which states that dividing both sides of an equation by the same non-zero quantity maintains the equality. By dividing both sides by 11, we isolate 'x' and determine its value. The result, x = -24/11, is the solution to our equation. This means that if we substitute -24/11 for 'x' in the original equation, both sides will be equal. This step is the culmination of all our efforts, bringing together the principles of algebra to reveal the answer. With this solution in hand, we can confidently say that we've successfully solved the equation and found the value of 'x'.

The Solution

So, the solution to the equation 8(x + 5) + 3 = -3(x - 7) - 2 is x = -24/11. We did it!

Checking Our Work

It's always a good idea to check our work to make sure we haven't made any mistakes along the way. To do this, we substitute our solution, x = -24/11, back into the original equation and see if both sides are equal.

Original equation: 8(x + 5) + 3 = -3(x - 7) - 2

Substitute x = -24/11: 8(-24/11 + 5) + 3 = -3(-24/11 - 7) - 2

Let's simplify each side separately.

Left side: 8(-24/11 + 5) + 3 = 8(-24/11 + 55/11) + 3 = 8(31/11) + 3 = 248/11 + 3 = 248/11 + 33/11 = 281/11

Right side: -3(-24/11 - 7) - 2 = -3(-24/11 - 77/11) - 2 = -3(-101/11) - 2 = 303/11 - 2 = 303/11 - 22/11 = 281/11

Since both sides simplify to 281/11, our solution x = -24/11 is correct!

Tips for Solving Equations

  • Stay Organized: Keep your work neat and organized. Write each step clearly and align the equal signs.
  • Double-Check: Always double-check your work, especially when dealing with negative signs.
  • Simplify: Simplify both sides of the equation as much as possible before moving terms around.
  • Practice: The more you practice, the better you'll become at solving equations. Try different types of equations to challenge yourself.
  • Check Your Answer: Always substitute your solution back into the original equation to verify that it's correct.

Conclusion

Solving equations might seem challenging at first, but with a step-by-step approach and a little practice, you can conquer any equation that comes your way. Remember the key steps: distribute, combine like terms, move the x terms, isolate the x term, and solve for x. And don't forget to check your work! You've got this! Happy solving, guys!