Solving For X: (1/2)x + 9 = (-2/3)x Explained
Hey guys! Let's dive into solving this equation for x: (1/2)x + 9 = (-2/3)x. It might look a little intimidating at first, but don't worry, we'll break it down step by step. Understanding how to solve linear equations like this is a fundamental skill in mathematics, and it's super useful in many real-world applications, from calculating finances to understanding physics. So, grab your thinking caps, and let's get started!
Understanding the Basics of Linear Equations
Before we jump into the specifics of this equation, let's quickly recap what a linear equation is. A linear equation is an equation in which the highest power of the variable (in this case, 'x') is 1. These equations, when graphed, form a straight line – hence the name 'linear'. Our goal in solving these equations is to isolate the variable on one side of the equation, so we know what value of 'x' makes the equation true. To do this, we use inverse operations, which are operations that "undo" each other (like addition and subtraction, or multiplication and division). Remember, whatever we do to one side of the equation, we must do to the other side to keep the equation balanced. This principle of maintaining balance is the golden rule of equation solving!
When tackling linear equations, there are a few key strategies we can employ. The most important is combining like terms. This involves grouping together terms that contain the same variable (like 'x' terms) and constant terms (numbers without variables). We can also use the distributive property to eliminate parentheses if they're present. Furthermore, eliminating fractions early on often simplifies the equation and makes it easier to work with. In our equation, (1/2)x + 9 = (-2/3)x, we have 'x' terms and a constant, and we also have fractions, so we'll need to use these strategies to solve it effectively. Now that we have the basics down, let's jump into the nitty-gritty of solving our specific equation!
Step-by-Step Solution
Okay, let's get our hands dirty and solve the equation (1/2)x + 9 = (-2/3)x. Here’s how we’ll do it step-by-step:
1. Eliminate Fractions
Fractions can sometimes make equations look messier than they are. To get rid of them, we need to find the least common multiple (LCM) of the denominators. In our equation, the denominators are 2 and 3. The LCM of 2 and 3 is 6. So, we'll multiply every term in the equation by 6. This is crucial – we need to multiply all terms to maintain the balance of the equation. This step is like giving the same dose of medicine to everyone who's sick; it ensures fairness and correctness in our mathematical treatment.
6 * (1/2)x + 6 * 9 = 6 * (-2/3)x
This simplifies to:
3x + 54 = -4x
See how much cleaner the equation looks now? We've successfully eliminated the fractions, making it easier to work with. This step is a game-changer because it transforms the equation into a friendlier form, paving the way for the next steps.
2. Combine 'x' Terms
Our next goal is to get all the 'x' terms on one side of the equation. It doesn't matter which side we choose, but it's often easier to choose the side that will result in a positive coefficient for 'x'. In this case, adding 4x to both sides will do the trick. Remember, we're using inverse operations here – we're adding 4x to “undo” the -4x on the right side. This is like moving pieces in a puzzle to bring similar colors or shapes together, making the overall picture clearer.
3x + 4x + 54 = -4x + 4x
This simplifies to:
7x + 54 = 0
Now, we have all our 'x' terms on the left side, which is exactly what we wanted. We're one step closer to isolating 'x' and finding its value.
3. Isolate 'x'
We're almost there! Now we need to isolate 'x' completely. To do this, we first need to get rid of the constant term (54) on the left side. We can do this by subtracting 54 from both sides. Again, we're using the inverse operation to “undo” the addition. This is like peeling away the outer layers of an onion to get to its core – in our case, 'x'.
7x + 54 - 54 = 0 - 54
This simplifies to:
7x = -54
Now, we have 7x on one side and a constant on the other. The final step to isolate 'x' is to divide both sides by 7. This will give us the value of 'x'. This step is like the final brushstroke on a painting, bringing the entire piece into completion.
7x / 7 = -54 / 7
This gives us:
x = -54/7
4. The Solution
And there we have it! The solution to the equation (1/2)x + 9 = (-2/3)x is x = -54/7. We’ve successfully isolated 'x' and found its value. This solution is like the key that unlocks a door, revealing the answer we've been searching for.
Verification
It's always a good idea to check our solution to make sure we didn't make any mistakes along the way. To do this, we substitute our solution (x = -54/7) back into the original equation and see if both sides of the equation are equal. This step is like proofreading an essay to catch any errors – it ensures the accuracy and validity of our work.
Original equation:
(1/2)x + 9 = (-2/3)x
Substitute x = -54/7:
(1/2)(-54/7) + 9 = (-2/3)(-54/7)
Simplify:
-27/7 + 9 = 36/7
To add -27/7 and 9, we need to convert 9 to a fraction with a denominator of 7:
9 = 63/7
Now we can add:
-27/7 + 63/7 = 36/7
So, the left side of the equation is:
36/7
The right side of the equation is already 36/7, so we have:
36/7 = 36/7
Since both sides of the equation are equal, our solution x = -54/7 is correct! Verification is like the final seal of approval, confirming that our solution is accurate and reliable.
Common Mistakes to Avoid
When solving equations like this, there are a few common pitfalls that students often stumble into. Let's highlight some of these so you can avoid them:
- Forgetting to Multiply All Terms: When eliminating fractions, it's crucial to multiply every term in the equation by the LCM, not just the fractional terms. Missing a term can throw off the entire solution.
- Incorrectly Combining Like Terms: Make sure you're only combining terms that have the same variable and exponent. For example, you can combine 3x and 4x, but not 3x and 4x². Pay close attention to signs (positive and negative) when combining terms.
- Making Arithmetic Errors: Simple arithmetic mistakes, like adding or subtracting numbers incorrectly, can lead to wrong answers. Double-check your calculations, especially when dealing with fractions and negative numbers.
- Not Following the Order of Operations: Remember PEMDAS/BODMAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Follow the correct order of operations to avoid errors.
- Skipping Verification: Always check your solution by substituting it back into the original equation. This is a crucial step to catch any mistakes you might have made.
Avoiding these common mistakes will significantly improve your accuracy and confidence in solving linear equations. It's like having a map that highlights the danger zones, helping you navigate the problem-solving terrain more safely.
Real-World Applications
You might be thinking, “Okay, solving equations is cool, but where will I ever use this in real life?” Well, you’d be surprised! Linear equations are incredibly versatile and pop up in numerous real-world scenarios. Understanding them is like having a Swiss Army knife in your mathematical toolkit – you can use them for a wide range of tasks.
- Finance: Calculating interest, loan payments, and investments often involves linear equations. For example, you can use them to figure out how long it will take to pay off a loan or how much interest you’ll earn on a savings account.
- Physics: Many physics problems, such as those involving motion at a constant speed, can be modeled using linear equations. They help in determining distances, velocities, and times.
- Engineering: Engineers use linear equations to design structures, circuits, and systems. These equations help ensure that designs are safe, efficient, and effective.
- Everyday Life: From figuring out the cost of a taxi ride based on distance to calculating the total bill at a restaurant after splitting it with friends, linear equations are at play. They’re the unsung heroes of our daily calculations!
So, the next time you’re faced with a real-world problem, remember that linear equations might just be the key to solving it. It's like discovering a hidden superpower that you can use to tackle everyday challenges.
Conclusion
So, guys, we've successfully solved the equation (1/2)x + 9 = (-2/3)x and found that x = -54/7. We tackled fractions, combined like terms, and isolated 'x' step-by-step. Remember, the key to mastering these equations is practice, practice, practice! The more you solve, the more comfortable and confident you’ll become. Keep these skills sharp, and you'll be well-equipped to handle any linear equation that comes your way. Keep up the great work, and happy solving!