Solving Linear Equations: A Step-by-Step Guide

Hey guys! Let's dive into the world of linear equations and tackle the equation: $9(2x - 8) - 2(4x - 19) = 26$. Don't worry, it might look a bit intimidating at first, but we'll break it down into simple, manageable steps. By the end of this, you'll be a pro at solving these types of problems. This equation is a classic example of a linear equation with one variable, 'x'. Our goal is to find the value of 'x' that makes the equation true. We'll use basic algebraic principles to isolate 'x' on one side of the equation. So, grab your pencils and let's get started! Understanding how to solve these equations is fundamental in algebra and opens the door to more complex mathematical concepts. It's like learning the alphabet before you start writing a novel; it's the foundation upon which everything else is built. The key to solving these equations is to keep the equation balanced. Anything you do to one side, you must do to the other. This ensures that the equality remains valid throughout the solution process. We'll be using the distributive property, combining like terms, and isolating the variable. These are the tools of the trade when it comes to solving these kinds of equations. Remember, practice makes perfect, so don't be discouraged if it takes a few tries to get the hang of it. Each step brings you closer to mastering the art of algebra, and with each solved equation, your confidence will soar. So, let's roll up our sleeves and get to work – it's going to be a fun journey of discovery! The initial step involves applying the distributive property to both sets of parentheses. This means multiplying the number outside the parentheses by each term inside the parentheses. This is a crucial step as it simplifies the equation and allows us to combine like terms later on. Remember that this process will make the equation cleaner and easier to solve. The distributive property is one of those mathematical rules that seems simple, but is essential for manipulating and solving equations. By applying it correctly, we're ensuring that we're following the proper order of operations and maintaining the integrity of the equation. Don't worry, it's not as scary as it sounds. We'll go step by step, and it will become second nature in no time. Once you get the hang of it, you'll be able to quickly apply this property and move on to the next steps of solving the equation. Remember, practice is the key. The more you work through these problems, the faster and more comfortable you'll become with this fundamental skill.

Step-by-Step Solution

Step 1: Distribute and Simplify

Alright, let's kick things off by applying the distributive property. We have: $9(2x - 8) - 2(4x - 19) = 26$ First, distribute the 9 to both terms inside the first parentheses: $9 * 2x = 18x$ $9 * -8 = -72$ This gives us: $18x - 72$ Next, distribute the -2 to both terms inside the second parentheses: $-2 * 4x = -8x$ $-2 * -19 = 38$ This gives us: $-8x + 38$ Now, let's put it all together. Our equation becomes: $18x - 72 - 8x + 38 = 26$ Wow, we've already made some significant progress, guys! We've successfully expanded both sets of parentheses and are one step closer to solving for 'x'. It's important to keep track of the signs (+ and -), as they can make a big difference. Remember, a negative times a negative equals a positive. Also, a positive times a negative equals a negative. Keep these rules in mind as you work through the distributive property. Take your time, double-check your work, and you'll do great! We’re now ready to move onto the next step, which involves combining like terms. This will further simplify the equation, allowing us to isolate the variable 'x' more easily. Keep in mind that accuracy is the key here. Small mistakes can lead to the wrong answer. So, take your time and don’t rush the process. If you’re ever unsure, always double-check your calculations to ensure everything is correct. It's a great habit to develop as you move forward in mathematics!

Step 2: Combine Like Terms

Okay, let's tidy things up by combining like terms. In our simplified equation: $18x - 72 - 8x + 38 = 26$, we can see that we have two 'x' terms and two constant terms. Let's combine them separately. Combine the 'x' terms: $18x - 8x = 10x$ Combine the constant terms: $-72 + 38 = -34$ Now, rewrite the equation with the combined terms: $10x - 34 = 26$ See how much cleaner that looks? Combining like terms is all about simplifying the equation to make it easier to solve. We’re essentially grouping similar elements together. This step reduces the number of terms we need to work with, making the overall process less cluttered and more straightforward. You will start to appreciate how crucial it is to simplify things as you get into more complex math. The idea is to make the equation as easy to read and understand as possible. You should always simplify whenever possible, to avoid unnecessary complications. Also, remember that combining like terms involves adding or subtracting coefficients of the same variable. Constant terms are also added or subtracted. Remember to pay attention to the signs (+ and -). If you are adding a negative number, it's the same as subtracting. And if you are subtracting a negative number, it's the same as adding. It’s a great idea to practice with different examples. The more you do it, the more comfortable you'll become. And we are very close to the final steps now, so keep up the good work!

Step 3: Isolate the Variable

Time to get 'x' all by itself! Our current equation is: $10x - 34 = 26$ To isolate the term with 'x', we need to get rid of the -34. We do this by adding 34 to both sides of the equation. Remember, anything we do to one side, we must do to the other to keep things balanced. So: $10x - 34 + 34 = 26 + 34$ This simplifies to: $10x = 60$ We're almost there! Isolating the variable is like giving 'x' its own space on one side of the equation. We’re aiming to get 'x' by itself on one side, with a number on the other side. This is achieved by performing inverse operations. Since -34 is being subtracted from 10x, we use the inverse operation, which is addition, to eliminate it from the left side. Don’t forget to do the same operation on both sides of the equation. The goal is to isolate 'x', ensuring that it equals a single numerical value. Remember, that this is the core of solving any equation. As you tackle more complex problems, this fundamental step will always be present, and it is a good idea to master it.

Step 4: Solve for x

Alright, the final step! We have: $10x = 60$ To solve for 'x', we need to get rid of the 10 that's being multiplied by 'x'. We do this by dividing both sides of the equation by 10: $rac{10x}{10} = rac{60}{10}$ This simplifies to: $x = 6$ And there you have it, guys! We've solved for 'x'! The value of x that satisfies the original equation is 6. This step is about getting 'x' completely alone. If 'x' is being multiplied by a number, we divide both sides by that number. This is the inverse operation, and it isolates 'x'. Once you get to this step, you're just one small division away from the answer! Remember, this method is universal to all the equations. Whatever coefficient is next to 'x', you just need to divide both sides by that value. Then, you'll have your solution. When you're dividing, make sure you don't make any errors in your calculations. Double check your math to ensure you’re getting the right value. This simple step can make the difference between a correct and incorrect solution. We are at the finish line now!

Conclusion: Verifying Your Answer

Let's verify our solution, to ensure that we've done everything correctly! It’s always a good idea to check your answer by substituting the value of 'x' back into the original equation. Our original equation was: $9(2x - 8) - 2(4x - 19) = 26$ We found that $x = 6$, so let's plug that in: $9(26 - 8) - 2(46 - 19) = 26$ $9(12 - 8) - 2(24 - 19) = 26$ $9(4) - 2(5) = 26$ $36 - 10 = 26$ $26 = 26$ It checks out! This means our solution, x = 6, is correct. Verifying your answer is a crucial step in the problem-solving process. It allows you to confirm that your solution is correct and to catch any mistakes you might have made along the way. By substituting the value of 'x' back into the original equation, you can ensure that the equation holds true. This provides you with confidence in your answer. Moreover, it helps you develop good habits of critical thinking. Make it a practice to always check your answers, especially as the problems become more complex. You’ll be able to identify and correct any mistakes, helping you understand the underlying concepts better. It’s a great way to improve your math skills! Well done, guys! You've successfully solved a linear equation. Keep up the great work, and you'll be acing these problems in no time! Remember to practice regularly, and you'll become more confident in your abilities. Feel free to try more examples and to challenge yourself with different types of linear equations. You've got this!