Solving For X: A Step-by-Step Guide To 11(x-1)+(10-12x)=0

by Andrew McMorgan 58 views

Hey Plastik Magazine readers! Ever find yourself staring at a math problem and feeling totally lost? Don't worry, we've all been there. Today, we're going to break down a common type of algebraic equation and show you exactly how to solve it. We'll be tackling the equation 11(x-1)+(10-12x)=0, so buckle up and let's get started!

Understanding the Equation

Before we dive into the solution, let's quickly understand what the equation is asking us. In essence, we need to find the value of 'x' that makes the entire equation true. This involves manipulating the equation using mathematical rules until we isolate 'x' on one side. Think of it like a puzzle – we're just rearranging the pieces until we see the solution clearly. To solve this, we'll use a combination of the distributive property, combining like terms, and inverse operations. These are the fundamental tools in our algebraic toolkit, and mastering them will make solving equations like this a breeze. Don't be intimidated by the parentheses or the multiple terms; we'll take it one step at a time and make sure everything is crystal clear. Remember, math is a language, and once you understand the rules, you can decipher any equation! This particular equation involves linear expressions, which means the highest power of 'x' is 1. This simplifies our task because we don't have to worry about quadratic formulas or other advanced techniques. Linear equations have a straightforward solution process, making them a great starting point for building your algebra skills.

Step-by-Step Solution

Okay, let's get our hands dirty and solve this equation! We'll break it down into manageable steps so you can follow along easily. Remember, the key is to stay organized and apply the rules of algebra consistently.

Step 1: Distribute

The first thing we need to do is get rid of those parentheses. We can do this by using the distributive property. This means we multiply the number outside the parentheses by each term inside. In our case, we have 11(x-1), so we'll multiply 11 by both 'x' and '-1'.

  • 11 * x = 11x
  • 11 * -1 = -11

So, 11(x-1) becomes 11x - 11. Let's rewrite the entire equation with this change:

11x - 11 + (10 - 12x) = 0

Now, we can remove the remaining parentheses since there's no coefficient multiplying them. This gives us:

11x - 11 + 10 - 12x = 0

Step 2: Combine Like Terms

Next up, we need to combine like terms. This means grouping together the terms with 'x' and the constant terms (the numbers without 'x'). In our equation, we have 11x and -12x as 'x' terms, and -11 and 10 as constant terms. Let's combine them:

  • 11x - 12x = -1x (which we can simply write as -x)
  • -11 + 10 = -1

Now, our equation looks much simpler:

-x - 1 = 0

See how much cleaner it is? Combining like terms is a crucial step in simplifying equations and making them easier to solve. It's like organizing your workspace before starting a project – it helps you focus on the essential parts.

Step 3: Isolate the Variable

Our goal is to get 'x' all by itself on one side of the equation. To do this, we'll use inverse operations. This means performing the opposite operation to undo what's being done to 'x'. Currently, we have '-x - 1 = 0'. The '-1' is being subtracted from '-x', so we need to add 1 to both sides of the equation to cancel it out.

-x - 1 + 1 = 0 + 1

This simplifies to:

-x = 1

We're almost there! We have '-x', but we want 'x'. Remember, '-x' is the same as '-1 * x'. So, to isolate 'x', we need to divide both sides of the equation by -1:

-x / -1 = 1 / -1

This gives us:

x = -1

Step 4: Check Your Solution

It's always a good idea to check your solution to make sure it's correct. To do this, we'll substitute our answer (x = -1) back into the original equation and see if it holds true.

Original equation: 11(x-1)+(10-12x)=0

Substitute x = -1: 11(-1-1)+(10-12(-1))=0

Simplify: 11(-2)+(10+12)=0

Continue simplifying: -22 + 22 = 0

0 = 0

Our solution checks out! This means that x = -1 is indeed the correct answer. Checking your work is a fantastic habit to develop in math. It gives you confidence in your answer and helps you catch any small errors you might have made along the way. Think of it as double-checking your directions before a road trip – it can save you from a lot of frustration!

Common Mistakes to Avoid

While solving equations, it's easy to make small mistakes that can lead to the wrong answer. Let's look at some common pitfalls to watch out for:

  • Distributing incorrectly: Make sure to multiply the term outside the parentheses by every term inside. For example, in 11(x-1), you need to multiply 11 by both 'x' and '-1'.
  • Combining unlike terms: You can only combine terms that have the same variable and exponent. For example, you can combine 11x and -12x, but you can't combine 11x and -11.
  • Forgetting the negative sign: Pay close attention to negative signs, especially when distributing or combining like terms. A missed negative sign can completely change the answer.
  • Not performing the same operation on both sides: Remember, whatever you do to one side of the equation, you must do to the other side to maintain balance. This is the golden rule of equation solving!
  • Skipping steps: It's tempting to try and do things in your head, but skipping steps increases the risk of making a mistake. Write out each step clearly, especially when you're first learning.

Practice Makes Perfect

The best way to get comfortable with solving equations is to practice, practice, practice! The more you work through problems, the more natural the process will become. Start with simple equations and gradually work your way up to more complex ones. There are tons of resources available online and in textbooks where you can find practice problems. Don't be afraid to make mistakes – they're a part of the learning process. When you do make a mistake, take the time to understand why you made it so you can avoid it in the future.

Conclusion

So, there you have it! We've successfully solved the equation 11(x-1)+(10-12x)=0 and found that x = -1. We walked through each step carefully, highlighting the key concepts and common mistakes to avoid. Remember, solving equations is a fundamental skill in algebra, and with a little practice, you'll become a pro in no time. Keep practicing, keep asking questions, and most importantly, keep having fun with math! You got this, guys! Math might seem intimidating at first, but breaking it down into manageable steps and understanding the underlying principles makes it much less daunting. Think of each equation as a puzzle waiting to be solved, and enjoy the process of finding the solution. And remember, the more equations you solve, the better you'll become at it. So, keep challenging yourself, and don't be afraid to tackle even the trickiest problems. You might surprise yourself with what you can achieve! Now, go forth and conquer those equations!