Solve 2x - 10 + 5x + 23 = 48: Step-by-Step Guide

Hey guys! Today, we are diving deep into a common yet crucial topic in algebra: solving linear equations. Specifically, we'll be tackling the equation $2x - 10 + 5x + 23 = 48$. If you've ever felt a bit lost when faced with equations like this, don't worry! We're going to break it down step by step, making sure you not only understand the how but also the why behind each move. Think of this as your friendly guide to mastering linear equations. We’ll cover everything from the basic principles to the advanced techniques, ensuring you’re well-equipped to solve similar problems on your own. So, let’s jump right in and turn equation-solving from a daunting task into a piece of cake!

Understanding the Basics of Linear Equations

Before we get into the nitty-gritty of solving $2x - 10 + 5x + 23 = 48$, let's take a moment to understand the fundamental concepts behind linear equations. A linear equation is, at its core, an equation where the highest power of the variable (in our case, x) is 1. This means you won’t see any $x^2$, $x^3$, or other higher powers. These equations represent a straight line when graphed, hence the name “linear.” The beauty of linear equations lies in their simplicity and predictability, making them a cornerstone of algebra and many real-world applications.

The main goal when solving a linear equation is to isolate the variable on one side of the equation. This means getting x by itself, so we know exactly what value it represents. To do this, we use a series of algebraic operations that maintain the equality of the equation. Remember, an equation is like a balanced scale: whatever you do to one side, you must do to the other to keep it balanced. This principle is the golden rule of equation solving, and it’s what allows us to manipulate equations while ensuring the solution remains valid. Understanding this balance is crucial, and it's a concept we'll revisit throughout our solution process. So, as we move forward, keep in mind that every step we take is about maintaining this balance while getting closer to isolating x. With a solid grasp of these basics, you'll find solving linear equations becomes much more intuitive and less intimidating!

Step-by-Step Solution of $2x - 10 + 5x + 23 = 48$

Now, let’s dive into the step-by-step solution of our equation: $2x - 10 + 5x + 23 = 48$. Grab your pencils, and let's get started! Our journey begins with simplifying the equation. The first thing we want to do is combine like terms. Like terms are terms that contain the same variable raised to the same power (or are just constants). In our equation, we have two terms with x ($2x$ and $5x$) and two constant terms (-10 and +23). Combining these like terms is like grouping similar items together to make them easier to handle.

So, let’s combine $2x$ and $5x$. When we add these together, we get $7x$. Next, we combine the constant terms, -10 and +23. Adding these gives us +13. Now, our equation looks simpler: $7x + 13 = 48$. See how much cleaner that is? This simplification is a critical step because it reduces the complexity of the equation, making it easier to work with. The next step is to isolate the term with x on one side of the equation. To do this, we need to get rid of the +13. Remember our golden rule? Whatever we do to one side, we must do to the other. So, we subtract 13 from both sides of the equation. This gives us $7x + 13 - 13 = 48 - 13$, which simplifies to $7x = 35$. We're getting closer! Now, the final step is to isolate x completely. Currently, x is being multiplied by 7. To undo this multiplication, we divide both sides of the equation by 7. This gives us $7x / 7 = 35 / 7$, which simplifies to $x = 5$. And there you have it! We’ve successfully solved the equation. The solution is $x = 5$. This step-by-step approach is crucial for tackling any linear equation. By simplifying, isolating, and applying the golden rule, you can confidently find the value of the variable.

Detailed Breakdown of Each Step

Let's break down each step of solving $2x - 10 + 5x + 23 = 48$ even further. This detailed breakdown will help solidify your understanding and ensure you can apply these techniques to other equations. We'll start with the initial equation: $2x - 10 + 5x + 23 = 48$. The first key move here is combining like terms. This involves identifying terms that have the same variable (in this case, x) and terms that are constants (numbers without a variable). Think of it like sorting your laundry – you put all the shirts together, all the socks together, and so on. In our equation, we have $2x$ and $5x$ as terms with the variable x, and we have -10 and +23 as constants.

To combine these, we simply add the coefficients (the numbers in front of the variables) of the x terms: $2x + 5x = 7x$. Then, we add the constants: $-10 + 23 = 13$. This simplifies our equation to $7x + 13 = 48$. This step is crucial because it reduces the complexity of the equation, making it more manageable. Now, we move on to isolating the term with x. This means we want to get $7x$ by itself on one side of the equation. To do this, we need to eliminate the +13. We use the principle of inverse operations – we perform the opposite operation to what's currently being done. Since 13 is being added, we subtract 13 from both sides of the equation. Remember, the golden rule! What we do to one side, we must do to the other. So, we subtract 13 from both sides: $7x + 13 - 13 = 48 - 13$. This simplifies to $7x = 35$. We're one step closer to solving for x! The final step is to isolate x completely. Currently, x is being multiplied by 7. To undo this multiplication, we divide both sides of the equation by 7. This gives us $7x / 7 = 35 / 7$. On the left side, the 7s cancel out, leaving us with just x. On the right side, 35 divided by 7 is 5. So, our final answer is $x = 5$. Each of these steps is a building block, and understanding them individually helps you tackle more complex equations with confidence. By breaking down the process into smaller, manageable steps, you can see how each operation contributes to the final solution.

Common Mistakes to Avoid

When solving linear equations, it’s easy to make mistakes if you’re not careful. Let's talk about some common pitfalls to avoid, so you can solve equations accurately and confidently. One frequent error is not combining like terms correctly. Remember, you can only add or subtract terms that have the same variable and exponent, or are constants. For example, in the equation $2x - 10 + 5x + 23 = 48$, you can combine $2x$ and $5x$ because they both have x, and you can combine -10 and +23 because they are constants. However, you can't combine $2x$ with -10, as they are not like terms. Mixing up like and unlike terms can lead to incorrect simplifications and, ultimately, the wrong answer.

Another common mistake is forgetting to apply the same operation to both sides of the equation. This is the golden rule of equation solving – whatever you do to one side, you must do to the other. If you subtract 13 from the left side of the equation to isolate the x term, you must also subtract 13 from the right side. Failing to do so throws off the balance of the equation and results in an incorrect solution. It’s like only adding weight to one side of a scale; it won’t stay balanced. Sign errors are also a common source of mistakes. When moving terms across the equals sign, remember to change their sign. For instance, if you have $+13$ on one side and you move it to the other side, it becomes $-13$. Similarly, if you have $-10$ and move it, it becomes $+10$. Overlooking this sign change can lead to errors in your calculations. Additionally, be mindful of the order of operations. While solving equations, it’s crucial to follow the reverse order of operations (PEMDAS/BODMAS) when undoing operations. This means you typically address addition and subtraction before multiplication and division. If you mix up the order, you might end up performing the wrong operation and getting an incorrect result. Lastly, always double-check your work. After you’ve solved an equation, plug your solution back into the original equation to verify that it’s correct. This simple step can help you catch any mistakes you might have made along the way. By being aware of these common mistakes and taking steps to avoid them, you can significantly improve your accuracy and confidence in solving linear equations.

Real-World Applications of Linear Equations

Linear equations aren't just abstract math problems; they have tons of practical uses in the real world. Understanding how they apply to everyday situations can make learning them even more relevant and interesting. One common application is in finance. For example, you might use a linear equation to calculate the total cost of a loan. Let’s say you borrow money with a fixed interest rate. The equation could help you figure out how much you’ll pay in total, including both the principal amount and the interest. This kind of calculation is essential for budgeting and financial planning. In business, linear equations are used for things like cost analysis and revenue projections. A company might use a linear equation to model the relationship between the number of products they sell and their total revenue. This helps them forecast sales and make informed decisions about pricing and production. Understanding these relationships can be critical for a business's success.

Another area where linear equations shine is in physics. They are used to describe motion, such as the distance an object travels over time at a constant speed. For instance, if you know a car is traveling at 60 miles per hour, you can use a linear equation to calculate how far it will travel in a certain amount of time. This kind of calculation is vital in fields like transportation and engineering. Linear equations are also essential in fields like engineering and architecture. Engineers use them to calculate structural loads and stresses, ensuring that buildings and bridges are safe and stable. Architects use them to design spaces and plan layouts, optimizing the use of space and resources. In everyday life, you might use linear equations without even realizing it. For example, if you're trying to figure out how much time it will take to drive somewhere, given your average speed, you’re essentially using a linear equation. Similarly, if you’re calculating the cost of a taxi ride based on a fixed rate per mile, you’re using a linear equation. These real-world examples highlight the versatility and importance of linear equations. They are a fundamental tool for problem-solving in many different contexts, making it well worth the effort to master them.

Practice Problems and Further Learning

Now that we've gone through the steps and discussed common mistakes and real-world applications, let's talk about how you can further improve your skills in solving linear equations. The key to mastering any math concept is practice, practice, practice! Working through a variety of problems helps you solidify your understanding and build confidence. Start with simpler equations and gradually move on to more complex ones. Look for problems that require you to combine like terms, isolate variables, and apply the golden rule. The more you practice, the more comfortable you’ll become with the process. There are tons of resources available to help you find practice problems. Textbooks are an excellent source, as they usually have a wide range of exercises at the end of each section. Online resources, such as websites and educational platforms, also offer a wealth of practice problems, often with step-by-step solutions so you can check your work.

Don't be afraid to seek out additional learning materials if you find yourself struggling with certain concepts. There are many online tutorials and videos that can provide alternative explanations and examples. Sometimes, seeing the same concept explained in a different way can make all the difference. Tutoring services can also be a valuable resource. A tutor can provide personalized instruction and help you address specific challenges you’re facing. They can offer individualized attention and guidance, tailoring their approach to your learning style. Group study sessions can also be beneficial. Working with classmates or friends allows you to discuss concepts, share strategies, and learn from each other's mistakes. Explaining a concept to someone else is a great way to reinforce your own understanding. Remember, learning math is a journey, and it’s okay to make mistakes along the way. The important thing is to keep practicing, keep asking questions, and keep pushing yourself to learn more. With consistent effort and the right resources, you can master linear equations and build a strong foundation for more advanced math topics. So, grab some practice problems and start solving! You’ve got this!

By following these strategies and continuously practicing, you’ll not only master linear equations but also develop a deeper understanding of algebra and its applications. Keep up the great work!