Solving $9-8(3r-2) = -8r + 40$ A Step-by-Step Guide

Introduction

Hey guys! Today, we're diving deep into solving a linear equation. If you've ever felt a bit puzzled by these, don't worry – we're going to break it down step by step. We're tackling the equation $9 - 8(3r - 2) = -8r + 40$. Linear equations might seem intimidating at first, but with a systematic approach, they become much easier to handle. Understanding how to solve these types of equations is crucial for various fields, from basic algebra to more advanced mathematical concepts. So, let's roll up our sleeves and get started! We'll go through each step meticulously, ensuring you grasp the underlying principles. Our main goal here is to isolate the variable r and find its value. This involves using algebraic properties such as the distributive property, combining like terms, and performing inverse operations. By the end of this guide, you’ll not only know how to solve this specific equation but also gain a solid foundation for tackling similar problems. Remember, practice makes perfect, so feel free to try out other equations on your own once you understand the process. Let's jump into the first step and see how we can simplify this equation to make it more manageable. We’ll focus on distributing the terms and then combining the like terms to get a clearer picture of what we're dealing with. So, grab a pen and paper, and let's get started on this mathematical journey together!

Step 1: Distribute the -8

The first hurdle in our equation, $9 - 8(3r - 2) = -8r + 40$, is the term -8(3r - 2). To simplify this, we need to apply the distributive property. What's the distributive property, you ask? It's a fundamental rule in algebra that allows us to multiply a single term by each term inside a set of parentheses. In simpler terms, we're going to multiply -8 by both 3r and -2. This step is crucial because it helps us eliminate the parentheses and makes the equation easier to manage. When we distribute -8 to 3r, we get -8 * 3r = -24r. Next, we distribute -8 to -2, which gives us -8 * -2 = 16. Remember, multiplying two negative numbers results in a positive number. So, our equation now looks like this: $9 - 24r + 16 = -8r + 40$. Notice how the parentheses are gone, and we have a series of terms that we can now combine and rearrange. This step is often where mistakes can happen, so it’s essential to be careful with your signs. Make sure you’re distributing the negative sign along with the number. Getting this right sets the stage for the rest of the solution. Now that we've successfully distributed the -8, the next step is to combine like terms on each side of the equation. This will further simplify the equation and bring us closer to isolating the variable r. So, let's move on to the next step and see how we can tidy up both sides of the equation.

Step 2: Combine Like Terms

Now that we've distributed the -8, our equation stands as $9 - 24r + 16 = -8r + 40$. The next logical step is to combine the like terms on each side of the equation. What exactly are "like terms"? They're terms that have the same variable raised to the same power. In our case, we have constant terms (numbers without variables) and terms with the variable r. On the left side of the equation, we have two constant terms: 9 and 16. We can combine these by simply adding them together: $9 + 16 = 25$. So, the left side of the equation now simplifies to $25 - 24r$. On the right side of the equation, we have -8r + 40. There are no like terms to combine here, so we'll leave it as it is. Our equation now looks much cleaner: $25 - 24r = -8r + 40$. Combining like terms is a crucial step in solving equations because it reduces the number of terms and makes the equation easier to work with. By simplifying each side, we're getting closer to isolating the variable r. This step also helps in avoiding common mistakes that can occur when dealing with more complex equations. Now that we've combined like terms, the next step is to move all the terms with r to one side of the equation and the constant terms to the other side. This will help us isolate r and find its value. So, let's proceed to the next step and learn how to rearrange the equation to get r by itself.

Step 3: Move Terms with 'r' to One Side

We've reached the point where our equation is $25 - 24r = -8r + 40$. To solve for r, we need to gather all the terms containing r on one side of the equation. A common strategy is to move the terms with r to the side that will result in a positive coefficient for r. In this case, we have -24r on the left side and -8r on the right side. To eliminate -24r from the left side, we can add 24r to both sides of the equation. This maintains the balance of the equation, which is crucial in algebra. Adding 24r to both sides gives us: $25 - 24r + 24r = -8r + 40 + 24r$. Simplifying this, we get $25 = 16r + 40$. Notice how the -24r and +24r on the left side cancel each other out, leaving us with just 25. On the right side, -8r + 24r combines to give us 16r. Moving terms in this way is a fundamental technique in solving equations. It helps us consolidate the variable terms and the constant terms, making it easier to isolate the variable. This step is all about strategic manipulation to simplify the equation and bring us closer to the solution. Now that we have all the r terms on one side, the next step is to move the constant terms to the other side. This will further isolate the r term and allow us to solve for r. So, let's move on to the next step and see how we can separate the constant terms from the variable terms.

Step 4: Move Constant Terms to the Other Side

Our equation is now $25 = 16r + 40$. The next step in isolating r is to move all the constant terms to one side of the equation. We currently have the constant term 40 on the right side, along with the term 16r. To eliminate 40 from the right side, we subtract 40 from both sides of the equation. This ensures that we maintain the balance of the equation, which is a golden rule in algebra. Subtracting 40 from both sides gives us: $25 - 40 = 16r + 40 - 40$. Simplifying this, we get $-15 = 16r$. Notice how the 40 and -40 on the right side cancel each other out, leaving us with just 16r. On the left side, 25 - 40 gives us -15. This step is crucial because it brings us one step closer to isolating r. By moving the constant terms to one side, we've effectively separated the variable term from the constants, making it easier to solve for r. It's essential to perform the same operation on both sides of the equation to maintain balance. This principle is the backbone of algebraic manipulation. Now that we have the equation in the form -15 = 16r, the final step is to divide both sides by the coefficient of r to solve for r. So, let's move on to the last step and find the value of r.

Step 5: Solve for 'r'

We've arrived at the equation $-15 = 16r$. The final step to solve for r is to isolate r by dividing both sides of the equation by its coefficient, which is 16. This will give us the value of r that satisfies the original equation. Dividing both sides by 16 gives us: $\frac{-15}{16} = \frac{16r}{16}$. Simplifying this, we get $r = \frac{-15}{16}$. And there you have it! We've successfully solved for r. The value of r that makes the equation true is $\frac{-15}{16}$. This final step is the culmination of all the previous steps. It's where we finally isolate the variable and find its value. Dividing by the coefficient is the inverse operation of multiplication, and it's the key to unlocking the value of the variable. Remember, it's always a good idea to check your answer by plugging it back into the original equation to make sure it holds true. This helps prevent errors and ensures that you've found the correct solution. Now that we've solved for r, let's take a moment to recap the steps we took and highlight the key concepts involved in solving linear equations.

Conclusion

Alright, guys! We've successfully navigated through the equation $9 - 8(3r - 2) = -8r + 40$ and found that $r = \frac{-15}{16}$. We started by distributing the -8, which helped us eliminate the parentheses. Then, we combined like terms to simplify the equation. After that, we moved all the r terms to one side and the constant terms to the other side. Finally, we divided by the coefficient of r to isolate r and find its value. Solving linear equations is a fundamental skill in algebra, and it's all about following a systematic approach. Remember, the key is to maintain the balance of the equation by performing the same operations on both sides. It might seem a bit challenging at first, but with practice, it becomes second nature. The steps we followed today are applicable to a wide range of linear equations. So, the next time you encounter a similar equation, remember these steps, and you'll be well-equipped to solve it. Keep practicing, and you'll become a pro at solving linear equations in no time! If you ever get stuck, don't hesitate to review these steps or seek help. Math is a journey, and every step you take brings you closer to mastery. Keep up the great work, and happy solving!