Solving $3(x-5)+7x=65$: Justifying Each Step

Hey guys! Today, we're diving deep into solving the equation $3(x-5) + 7x = 65$. But we're not just going to find the answer; we're going to break down every single step and justify why we can do what we do. Think of it as understanding the math rules of the game. So, which table shows the correct methods used to justify the solution steps? Let's find out together!

Understanding the Equation

Before we jump into solving, let's take a good look at our equation: $3(x-5) + 7x = 65$. It looks a bit complex, but don't worry, we'll tackle it piece by piece. The key here is to remember the order of operations (PEMDAS/BODMAS) and the properties of equality. We'll be using these tools throughout the solution. Understanding these fundamental concepts is crucial because they form the backbone of algebraic problem-solving. When you grasp the underlying principles, equations become less intimidating and more like puzzles waiting to be solved. Always remember, math isn't about memorizing steps; it's about understanding why those steps work. So, let's embark on this journey of unraveling the equation, step by step, with a clear understanding of the justifications behind each operation.

The Importance of Justification

Why do we even need to justify our steps? Well, in mathematics, precision is everything. Each step we take must be logically sound and based on established rules and properties. Justification ensures that our solution is not only correct but also valid. It's like building a house – you need a strong foundation and a clear plan for each stage of construction. In math, the justifications are our foundation, ensuring that our solution stands firm. Plus, understanding the justifications helps us apply the same principles to other problems. If you know why a step works, you can adapt it to different scenarios. So, as we solve this equation, we'll focus not just on the how but also on the why, solidifying our understanding and enhancing our problem-solving skills. Think of it as learning the language of mathematics, where each step is a sentence, and the justification is the grammar that makes it correct and meaningful.

Laying the Groundwork

Before we start manipulating the equation, let's identify the key components we're working with. We have variables (x), constants (numbers), and operations (multiplication, subtraction, addition). Our goal is to isolate the variable 'x' on one side of the equation. To do this, we'll use inverse operations and the properties of equality. Remember, whatever we do to one side of the equation, we must do to the other to maintain balance. It's like a see-saw; if you add weight to one side, you need to add the same weight to the other to keep it level. This principle of balance is fundamental to solving equations. We'll also be using the distributive property to simplify expressions within parentheses. So, with our tools in hand and our strategy in mind, let's get started on the first step of solving this equation. We're setting the stage for a clear and logical solution, where each step is a deliberate move toward our final answer. Understanding the groundwork ensures that we proceed with confidence and accuracy, turning a seemingly complex equation into a manageable and solvable challenge.

Step-by-Step Solution with Justifications

Okay, let's get our hands dirty and solve this equation! We'll break it down into manageable steps, justifying each one along the way.

Step 1: Distribute

The first thing we need to do is tackle those parentheses. We have $3(x-5)$, which means we need to distribute the 3 to both the x and the -5 inside the parentheses. This is using the distributive property, which states that a(b + c) = ab + ac. So, $3(x - 5)$ becomes $3 * x - 3 * 5$, which simplifies to $3x - 15$. Remember, the distributive property is like sharing – the number outside the parentheses gets multiplied by each term inside. This step is crucial because it simplifies the equation and allows us to combine like terms later on. Ignoring the distributive property would be like skipping a vital step in a recipe – the final dish wouldn't turn out right. So, let's apply this property diligently and set the stage for the next steps in our solution.

Step 2: Rewrite the Equation

Now that we've distributed, let's rewrite our equation with the simplified expression. Our equation now looks like this: $3x - 15 + 7x = 65$. Notice how we've replaced $3(x-5)$ with its simplified form, $3x - 15$. This step is about clarity and organization. It allows us to see the equation in a more manageable form, making it easier to identify the next steps. It's like decluttering your workspace before starting a project – it helps you focus and work more efficiently. Rewriting the equation after each significant step is a good habit to develop, as it minimizes errors and ensures that you're always working with the most simplified version of the problem. So, with our equation neatly rewritten, we're ready to move on to the next stage of the solution, feeling confident and organized.

Step 3: Combine Like Terms

Next up, we need to combine the like terms on the left side of the equation. Like terms are terms that have the same variable raised to the same power. In our equation, we have $3x$ and $7x$. We can combine these by adding their coefficients (the numbers in front of the variables). So, $3x + 7x$ equals $10x$. Our equation now looks like this: $10x - 15 = 65$. This step is based on the commutative and associative properties of addition, which allow us to rearrange and group terms as needed. Combining like terms simplifies the equation further, bringing us closer to isolating the variable 'x'. It's like consolidating your resources before launching an attack – you group similar forces to maximize your impact. By combining like terms, we're streamlining our equation, making it more manageable and paving the way for the final steps of the solution.

Step 4: Isolate the Variable Term

Our goal is to get 'x' by itself on one side of the equation. Right now, we have $10x - 15 = 65$. To isolate the $10x$ term, we need to get rid of the -15. We can do this by adding 15 to both sides of the equation. This is based on the addition property of equality, which states that if you add the same value to both sides of an equation, the equation remains balanced. So, adding 15 to both sides gives us: $10x - 15 + 15 = 65 + 15$, which simplifies to $10x = 80$. Remember, the addition property of equality is like maintaining balance on a scale – whatever you add to one side, you must add to the other to keep it level. This step is crucial for isolating the variable term and moving closer to the final solution. We're strategically eliminating the constant term, paving the way for us to solve for 'x' directly.

Step 5: Solve for x

We're almost there! We now have $10x = 80$. To solve for 'x', we need to get rid of the 10 that's multiplying it. We can do this by dividing both sides of the equation by 10. This is based on the division property of equality, which states that if you divide both sides of an equation by the same non-zero value, the equation remains balanced. So, dividing both sides by 10 gives us: $10x / 10 = 80 / 10$, which simplifies to $x = 8$. Congratulations, we've solved for x! The division property of equality is our final tool in isolating the variable and finding its value. It's like the final cut in sculpting – it reveals the true form of our solution. With this step, we've successfully navigated the equation and arrived at the answer, demonstrating our mastery of algebraic problem-solving techniques.

The Solution Table

Now, let's put all of these steps into a table format, making sure to include the justification for each step. This will help us clearly see the logical flow of our solution.

This table clearly outlines each step we took to solve the equation and the mathematical principle that justifies it. It's a great way to organize your work and ensure that your solution is both correct and logically sound. Remember, in mathematics, understanding the why is just as important as understanding the how. This table serves as a roadmap, guiding us through the solution process with clarity and precision. It's a testament to the power of structured problem-solving and the importance of justifying each step along the way.

Conclusion

So, there you have it! We've not only solved the equation $3(x-5) + 7x = 65$, but we've also broken down each step and justified it using mathematical properties. Remember, guys, math isn't just about finding the answer; it's about understanding the process. By understanding the justifications, you can apply these principles to solve a wide range of problems. Keep practicing, keep questioning, and you'll become a math whiz in no time! This journey through the solution process has highlighted the importance of precision, logical reasoning, and the application of fundamental mathematical principles. It's a reminder that every equation is a puzzle waiting to be solved, and with the right tools and understanding, we can confidently tackle any mathematical challenge. So, embrace the process, enjoy the journey, and keep exploring the fascinating world of mathematics! 🚀