Solving 3(x+2) = -2(x-7) Find The Value Of X
Hey everyone! Today, we're diving deep into solving the algebraic equation 3(x+2) = -2(x-7). If you've ever felt a little lost when faced with equations like this, don't worry, you're in the right place. We're going to break it down step by step, so by the end of this article, you'll be solving these problems like a pro.
Understanding the Basics
Before we jump into the nitty-gritty, let’s quickly refresh some fundamental concepts. When we're solving an equation, our main goal is to isolate the variable – in this case, 'x' – on one side of the equation. This means we want to get 'x' by itself, so we can clearly see what its value is. To do this, we use various algebraic operations, making sure to keep the equation balanced. Whatever operation we perform on one side, we must also perform on the other side. This principle is crucial for maintaining the equality and finding the correct solution.
In our equation, 3(x+2) = -2(x-7), we have parentheses, which means we need to use the distributive property first. The distributive property states that a(b + c) = ab + ac. Applying this property will help us get rid of the parentheses and simplify the equation. Once we've distributed, we can combine like terms, move variables to one side, and constants to the other side, ultimately solving for 'x'. This process might sound like a lot of steps, but we'll take it one step at a time to ensure clarity and understanding. Remember, the key is to stay organized and focus on each step individually. With practice, these steps will become second nature, and you'll be able to solve similar equations with confidence.
Step-by-Step Solution
Let's break down the solution to 3(x+2) = -2(x-7) into clear, manageable steps. Grab your pen and paper, and let's work through this together!
Step 1: Distribute
The first thing we need to do is get rid of those parentheses. Remember the distributive property? We're going to apply it to both sides of the equation.
On the left side, we have 3(x+2). We multiply 3 by both 'x' and '2':
3 * x = 3x 3 * 2 = 6
So, 3(x+2) becomes 3x + 6.
Now, let's tackle the right side: -2(x-7). We multiply -2 by both 'x' and '-7':
-2 * x = -2x -2 * -7 = 14
So, -2(x-7) becomes -2x + 14.
Our equation now looks like this: 3x + 6 = -2x + 14. We've successfully distributed and simplified both sides, making it easier to work with.
Step 2: Combine Like Terms
Now that we've distributed, let's gather our 'x' terms on one side and our constants (the numbers) on the other. This is like sorting your socks – you want all the pairs together!
To get the 'x' terms together, we can add 2x to both sides of the equation. This will eliminate the -2x on the right side:
3x + 6 + 2x = -2x + 14 + 2x
This simplifies to:
5x + 6 = 14
Great! Now we have all our 'x' terms on the left side. Next, we need to move the constants to the right side. To do this, we subtract 6 from both sides:
5x + 6 - 6 = 14 - 6
This simplifies to:
5x = 8
We're getting closer! We've successfully combined like terms and have a simplified equation.
Step 3: Isolate the Variable
Our final step is to isolate 'x'. Right now, we have 5x = 8. To get 'x' by itself, we need to undo the multiplication. We do this by dividing both sides of the equation by 5:
5x / 5 = 8 / 5
This simplifies to:
x = 8/5
And there you have it! We've solved for 'x'. The value of x is 8/5. You can also express this as a decimal, which is 1.6.
Step 4: Verification (Optional but Recommended)
To be absolutely sure we've got the right answer, we can plug our solution back into the original equation. This is like checking your work to make sure you didn't make any mistakes.
Our original equation was 3(x+2) = -2(x-7). Let's substitute x = 8/5 into the equation:
3(8/5 + 2) = -2(8/5 - 7)
First, we need to deal with the parentheses. Let's find a common denominator to add and subtract the fractions:
3(8/5 + 10/5) = -2(8/5 - 35/5)
This simplifies to:
3(18/5) = -2(-27/5)
Now, multiply:
54/5 = 54/5
The left side equals the right side! This confirms that our solution, x = 8/5, is correct. High five!
Common Mistakes and How to Avoid Them
Solving algebraic equations can be tricky, and it's easy to make mistakes. Let's look at some common pitfalls and how to avoid them.
Sign Errors
One of the most common mistakes is messing up the signs, especially when distributing negative numbers. Remember, a negative times a negative is a positive, and a negative times a positive is a negative. Double-check your signs at each step to avoid these errors. When distributing -2 in our equation, it’s crucial to remember that -2 * -7 equals +14, not -14.
Incorrect Distribution
Another frequent mistake is not distributing properly. Make sure you multiply the number outside the parentheses by every term inside the parentheses. For example, in 3(x+2), you need to multiply 3 by both 'x' and '2'. Skipping one of the terms can lead to an incorrect solution. Always double-check that you've distributed correctly before moving on.
Combining Unlike Terms
It's tempting to try to simplify things too quickly, but you can only combine like terms. Like terms are those that have the same variable raised to the same power (e.g., 3x and 5x) or constants (e.g., 6 and 14). You can't combine 'x' terms with constants. Make sure you're only adding or subtracting terms that are alike.
Not Following Order of Operations
Remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)? Following the order of operations is crucial for solving equations correctly. Make sure you perform operations in the correct order to avoid errors. In our example, we dealt with the parentheses first by distributing, then we combined like terms, and finally, we isolated the variable.
Forgetting to Perform Operations on Both Sides
The golden rule of solving equations is that whatever you do to one side, you must do to the other. If you add a number to one side, you must add the same number to the other side. If you divide by a number on one side, you must divide by the same number on the other side. Forgetting this rule will throw off the balance of the equation and lead to an incorrect solution.
Skipping the Verification Step
We know it's tempting to skip this step, especially when you think you've nailed the solution. But verifying your answer is a fantastic way to catch any mistakes you might have made. Plug your solution back into the original equation and make sure it holds true. It's like having a built-in error detector!
Practice Problems
Alright, guys, let's put your newfound skills to the test! Here are a few practice problems for you to try. Remember, practice makes perfect!
- 4(x - 3) = -2(x + 5)
- 2(3x + 1) = 5(x - 2)
- -3(2x - 4) = 6(x + 1)
Work through these problems step by step, just like we did in the example. Don't forget to distribute, combine like terms, and isolate the variable. And, of course, verify your answers! The solutions are listed at the end of this article, but try to solve them on your own first. Working through these problems will help solidify your understanding and build your confidence.
Real-World Applications
You might be wondering, “When am I ever going to use this in real life?” Well, solving equations like 3(x+2) = -2(x-7) isn't just an abstract math exercise. It's a fundamental skill that's used in various real-world scenarios. Let’s explore a few examples:
Budgeting and Finance
Imagine you're planning a budget. You have a certain amount of money coming in, and you need to allocate it to different expenses. Equations can help you figure out how much you can spend on each category while staying within your budget. For example, you might need to calculate how much you can save each month to reach a financial goal, taking into account your income and expenses.
Cooking and Baking
Recipes often need to be scaled up or down. If a recipe calls for a certain amount of ingredients for a specific number of servings, you can use equations to adjust the quantities for a different number of servings. This is particularly useful when you're cooking for a large group or want to make a smaller batch of something.
Travel Planning
Planning a trip involves a lot of calculations. You might need to figure out how far you can travel on a certain amount of gas, how long it will take to reach your destination at a certain speed, or how much money you'll need for accommodations and activities. Equations can help you make these calculations accurately.
Engineering and Construction
Engineers and construction workers use equations all the time to design structures, calculate loads, and determine dimensions. Whether it's calculating the stress on a bridge or the amount of material needed for a building, equations are essential tools in these fields.
Problem-Solving in General
More broadly, the problem-solving skills you develop by solving equations are valuable in many areas of life. Breaking down a problem into smaller steps, identifying the key variables, and finding a solution are skills that can help you in your career, your personal life, and even in everyday decision-making.
So, the next time you're solving an equation, remember that you're not just doing math – you're developing skills that will help you in many different situations.
Conclusion
Great job, everyone! We've covered a lot in this article. We started with the basics, walked through a step-by-step solution of 3(x+2) = -2(x-7), discussed common mistakes, and even looked at some real-world applications. You've now got a solid foundation for solving algebraic equations. The key is practice, so keep working on those problems, and you'll become a math whiz in no time!
Remember, solving equations is like learning any new skill – it takes time and effort. Don't get discouraged if you don't get it right away. Keep practicing, and you'll see progress. And most importantly, have fun with it! Math can be challenging, but it can also be incredibly rewarding when you finally crack a tough problem.
If you have any questions or want to dive deeper into algebra, there are tons of resources available online and in libraries. Don't hesitate to explore and learn more. You've got this!
Practice Problems Solutions:
- x = 1/3
- x = -12
- x = 1/3
