Solving Linear Equations: A Step-by-Step Guide

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Hey guys! Let's dive into the world of algebra and learn how to solve linear equations, specifically focusing on the equation 2x - 11 = 5(2 - x). Understanding how to solve these types of equations is super important because they pop up all over the place, from basic math problems to more complex stuff later on. This guide will break down the process step-by-step, making it easy to follow along. We'll go through each stage carefully, making sure you grasp every concept. Ready to get started? Let's go!

Understanding Linear Equations

Alright, before we get our hands dirty with the equation, let's quickly review what a linear equation actually is. Basically, it's an equation where the highest power of the variable (usually 'x') is 1. This means you won't see any x² or x³ hanging around. Linear equations always result in a straight line when graphed, hence the name! They generally take the form of ax + b = c, where 'a', 'b', and 'c' are constants, and 'x' is the variable we're trying to solve for. Our equation, 2x - 11 = 5(2 - x), fits this description. We're looking for the value of 'x' that makes the equation true. Solving a linear equation means finding the value of the variable that makes the equation true. It’s like finding the missing piece of a puzzle; once you find it, everything fits perfectly! The goal is to isolate the variable on one side of the equation and have the constant on the other side. This is done by applying inverse operations to both sides of the equation, ensuring the equation remains balanced. We'll do this using a methodical approach to ensure that you all understand the core principles involved. Trust me; once you grasp the basics, solving linear equations becomes a piece of cake. This process involves several key steps. We'll explore these meticulously in the following sections.

The Importance of Order of Operations

Before we start, a quick reminder about the order of operations is super helpful. You might know it as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This tells you the order in which to solve different parts of the equation. First, we tackle what's inside parentheses. Next, we look for exponents. Then, we handle multiplication and division (from left to right), and finally, we deal with addition and subtraction (also from left to right). Getting the order right is critical; otherwise, you'll end up with the wrong answer! So, let’s keep PEMDAS in mind as we work through our equation. This structured approach helps ensure accuracy and efficiency in your calculations. Following PEMDAS ensures that we solve the equation correctly. This is particularly crucial when dealing with more complex equations. So, let’s get started. Make sure you remember this rule, as it will be your guide through all calculations.

Solving 2x - 11 = 5(2 - x): Step-by-Step

Now, let's solve the equation 2x - 11 = 5(2 - x) step-by-step. I will show you how to do it. Follow along, and you'll become a pro in no time! Here’s how we'll solve it, breaking down each stage for you:

Step 1: Expand the Parentheses

The first thing we need to do is get rid of those parentheses. We do this by distributing the 5 across the terms inside the parentheses. So, we multiply 5 by both 2 and -x. This gives us:

  • 5 * 2 = 10
  • 5 * -x = -5x

So, our equation becomes: 2x - 11 = 10 - 5x

Make sure you carefully distribute the number outside the parentheses to each term inside. This is a common place where mistakes can happen, so take your time and double-check your work! The initial step, expanding the parentheses, sets the stage for the following steps. It involves multiplying the number outside the parentheses by each term inside. We transform the equation into a more manageable format, making it easier to solve for 'x'. This is crucial because it simplifies the equation and allows us to isolate the variable. By eliminating parentheses, we remove a layer of complexity and pave the way for a more direct path to the solution. The transformation is critical for proceeding with further steps, allowing you to combine like terms and isolate the variable.

Step 2: Combine Like Terms

Next, we need to get all the 'x' terms on one side of the equation and the constant terms on the other side. Let's add 5x to both sides of the equation to move the -5x from the right side to the left side:

  • 2x - 11 + 5x = 10 - 5x + 5x
  • This simplifies to: 7x - 11 = 10

Now we have all the 'x' terms on the left side. Then, we want to isolate the constant terms, so we'll add 11 to both sides:

  • 7x - 11 + 11 = 10 + 11
  • This simplifies to: 7x = 21

Combining like terms involves grouping similar terms together. Our goal is to consolidate terms containing the variable 'x' on one side. This process involves applying arithmetic operations to both sides of the equation. We add 5x to both sides. By isolating the x terms on one side, we bring the equation closer to a solution. The aim is to simplify the equation to the point where the value of 'x' can be directly computed. This strategic grouping enables us to create an equation that can be easily solved. By combining similar elements, we aim to streamline and reduce the equation's complexity. You can now move on to the next step, where you isolate the variable.

Step 3: Isolate the Variable

We're almost there! Now, to isolate 'x', we need to get rid of the 7 that's multiplied by it. We do this by dividing both sides of the equation by 7:

  • 7x / 7 = 21 / 7
  • This simplifies to: x = 3

And there you have it! The solution to the equation 2x - 11 = 5(2 - x) is x = 3. Now, we've found the value of x that makes the original equation true. The process of isolating the variable is a fundamental step in solving equations. By applying the inverse operation, we can effectively remove any coefficients or constants that are next to the variable. Dividing both sides of the equation by 7 helps to get a value for 'x'. This step ensures that we have the variable by itself. This crucial step is the key to determining the value of 'x'. We isolate the variable, which simplifies the equation and brings us closer to the final solution. This step is usually a straightforward process, but it is important to perform the same operation on both sides to maintain the balance of the equation.

Step 4: Verify the Solution

It’s always a good idea to check your answer. To make sure our solution is correct, we'll substitute x = 3 back into the original equation:

  • 2(3) - 11 = 5(2 - 3)
  • 6 - 11 = 5(-1)
  • -5 = -5

Since the equation is true, we know our solution, x = 3, is correct! Always do this, guys; it can save you from a lot of headaches! Verification is a vital process in solving equations. It involves substituting the value we found for 'x' back into the original equation. By doing so, we confirm whether our solution is correct. This step is essential in validating the accuracy of the result and ensuring that it meets the equation's criteria. It's a method to cross-check the answer. By substituting the found value back into the original equation, we can determine the equality of both sides. This ensures that the solution is precise and satisfies the given conditions. This step is simple but effective, helping you catch any errors. The ultimate goal is to ascertain whether our calculation of 'x' is correct. This is not only a good practice but also enhances your understanding of the underlying principles of the equation.

Key Takeaways and Tips

Alright, let’s summarize what we’ve learned, and I'll give you some useful tips to help you solve these types of equations. You’ve now learned how to solve linear equations, but here's a few key points to keep in mind:

  • PEMDAS: Always remember the order of operations. This is non-negotiable! This is super important to follow.
  • Combine Like Terms: Group similar terms together. Keep 'x' terms on one side and constants on the other.
  • Isolate the Variable: Get 'x' by itself by performing the inverse operations on both sides of the equation.
  • Check Your Answer: Always substitute your solution back into the original equation to verify your work. This helps ensure accuracy.

Common Mistakes to Avoid

Here are some common mistakes to avoid: Failing to distribute the number outside the parentheses correctly, forgetting to change signs when moving terms across the equals sign, and not following the order of operations. Watch out for these pitfalls and you’ll do great!

Practice Makes Perfect

Solving linear equations is all about practice. The more you work through problems, the more comfortable and confident you'll become. So, grab some practice problems and get to work! Keep practicing, and you will become a master of solving linear equations in no time! With continuous effort, you will develop the ability to solve these equations swiftly and confidently. Regular exercises and repeated problem-solving will significantly improve your skills and understanding.

Conclusion

Awesome work, everyone! You've successfully learned how to solve the linear equation 2x - 11 = 5(2 - x)! Remember to practice and review these steps. Keep these steps in mind, and you'll be solving equations like a pro. This guide has given you a strong foundation. This will help you ace your math tests and feel confident in your algebra skills. Keep up the great work! You have the tools and knowledge. You're now well-equipped to tackle more challenging problems! Now that you have learned the material, continue learning and keep solving problems!