Simplify 1/9(5x + 3y) - 1/3(x-y) Step-by-Step Solution
Hey guys! Today, we're diving into a common algebraic problem: simplifying the expression 1/9(5x + 3y) - 1/3(x-y). If you've ever felt a little lost when dealing with fractions and variables, don't worry, you're in the right place. We're going to break this down step by step, making it super easy to understand. By the end of this guide, you'll not only know how to solve this specific problem but also have a solid foundation for tackling similar algebraic challenges. So, let's get started and turn those confusing expressions into simple solutions!
Understanding the Basics of Algebraic Expressions
Before we jump into the solution, let's quickly recap the basics. Algebraic expressions are combinations of variables (like x and y), constants (numbers), and operations (addition, subtraction, multiplication, division). Simplifying these expressions means making them as concise and easy to work with as possible. This usually involves combining like terms and getting rid of unnecessary clutter. Think of it like decluttering your room – you want to organize things so you can find them easily and use them effectively. In algebra, a simplified expression is easier to understand and use for further calculations. For instance, imagine you have a complex expression representing the total cost of items you're buying. Simplifying it helps you quickly see the actual cost without getting bogged down in the details. So, remember, the goal is to make things simpler and clearer. This foundation is key to mastering more complex algebraic problems, and it all starts with understanding how to manipulate and combine terms effectively. By grasping these basics, you'll find that even the most daunting expressions become manageable and less intimidating. So let's keep these fundamentals in mind as we move forward with our simplification journey!
Step 1: Distribute the Fractions
The first step in simplifying our expression is to distribute the fractions. This means we'll multiply the fractions outside the parentheses by each term inside. For the first part, 1/9(5x + 3y), we'll multiply 1/9 by both 5x and 3y. This gives us (1/9 * 5x) + (1/9 * 3y), which simplifies to 5x/9 + 3y/9. Now, let's tackle the second part: -1/3(x - y). We'll multiply -1/3 by both x and -y. Remember, a negative times a negative is a positive! So, we get (-1/3 * x) + (-1/3 * -y), which simplifies to -x/3 + y/3. Distributing the fractions is crucial because it allows us to remove the parentheses and start combining like terms. It's like unpacking your groceries – you need to get everything out of the bags before you can put it away. This step sets the stage for the rest of the simplification process. By carefully distributing the fractions, we're breaking down the expression into smaller, more manageable pieces. So, let's make sure we've got this down before moving on, because it's the foundation for the next steps!
Step 2: Combine Like Terms
Now that we've distributed the fractions, our expression looks like this: 5x/9 + 3y/9 - x/3 + y/3. The next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our case, 5x/9 and -x/3 are like terms because they both have x, and 3y/9 and y/3 are like terms because they both have y. To combine them, we need to have a common denominator. Let's start with the x terms. We have 5x/9 and -x/3. The least common denominator for 9 and 3 is 9. So, we need to convert -x/3 to have a denominator of 9. To do this, we multiply both the numerator and the denominator by 3: (-x/3) * (3/3) = -3x/9. Now we can combine the x terms: 5x/9 - 3x/9 = (5x - 3x)/9 = 2x/9. Next, let's combine the y terms. We have 3y/9 and y/3. Again, we need a common denominator, which is 9. We convert y/3 to have a denominator of 9 by multiplying both the numerator and the denominator by 3: (y/3) * (3/3) = 3y/9. Now we can combine the y terms: 3y/9 + 3y/9 = (3y + 3y)/9 = 6y/9. Combining like terms is like sorting your laundry – you group the socks together, the shirts together, and so on. It simplifies the expression and makes it easier to see the final result. This step is essential for getting to the simplest form of our expression. So, by identifying and combining like terms, we're one step closer to our final answer!
Step 3: Simplify the Fractions
After combining like terms, our expression looks like this: 2x/9 + 6y/9. The final step is to simplify the fractions, if possible. Looking at our expression, we can see that 2x/9 is already in its simplest form because 2 and 9 don't have any common factors other than 1. However, 6y/9 can be simplified further. Both 6 and 9 are divisible by 3. So, we can divide both the numerator and the denominator by 3: (6y/9) ÷ (3/3) = 2y/3. Now our expression looks like this: 2x/9 + 2y/3. This is the simplest form of our expression. Simplifying fractions is like editing a piece of writing – you remove any unnecessary words or phrases to make it clearer and more concise. In this case, we're reducing the fractions to their lowest terms, which makes the expression easier to understand and work with. This step is crucial for getting to the final, most simplified answer. By simplifying the fractions, we're ensuring that our expression is as clean and straightforward as possible. So, let's take a moment to appreciate how far we've come and how much simpler our expression looks now!
Final Result: 2x/9 + 2y/3
So, after all the steps, we've simplified the expression 1/9(5x + 3y) - 1/3(x-y) to 2x/9 + 2y/3. Isn't that awesome? We started with a seemingly complex expression and, by following a few simple steps, we've broken it down into a much more manageable form. This final result is not only simpler but also easier to use in further calculations or applications. It's like solving a puzzle – each step brings you closer to the final picture, and the feeling of accomplishment when you get there is fantastic. Remember, the key to simplifying algebraic expressions is to take it one step at a time. Distribute, combine like terms, and simplify fractions. By mastering these steps, you'll be able to tackle any algebraic expression that comes your way. And don't forget, practice makes perfect! The more you work with these types of problems, the more comfortable and confident you'll become. So, keep practicing, keep simplifying, and keep rocking those algebraic expressions! You've got this!
Tips and Tricks for Simplifying Algebraic Expressions
To really master simplifying algebraic expressions, let's dive into some extra tips and tricks. First off, always double-check your work, especially when dealing with negative signs. It's super easy to make a small mistake with a negative, and it can throw off your whole answer. Another handy trick is to rewrite subtraction as addition of a negative. For example, instead of thinking of x - y, think of it as x + (-y). This can help you keep track of the signs more easily. When combining like terms, it can be helpful to physically group them together. You can use different colors or underline them in different ways. This visual aid can make it easier to see which terms go together. Also, remember that the order of operations (PEMDAS/BODMAS) is your friend. Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. Following this order ensures you're tackling the expression in the correct sequence. Another great tip is to practice regularly. The more you practice, the more these steps will become second nature. Try working through a variety of problems, from simple to more complex. This will help you build your skills and confidence. And finally, don't be afraid to ask for help! If you're stuck on a problem, reach out to a teacher, a tutor, or a friend. Sometimes a fresh perspective is all you need to break through a challenge. So, keep these tips in mind as you continue your algebraic journey. With practice and the right strategies, you'll be simplifying expressions like a pro in no time!
Common Mistakes to Avoid
Hey, everyone makes mistakes, but knowing the common pitfalls in simplifying algebraic expressions can help you avoid them! One of the biggest mistakes is forgetting to distribute negative signs correctly. Remember, when you're distributing a negative, it changes the sign of every term inside the parentheses. For example, -(x - y) becomes -x + y, not -x - y. Another common mistake is combining unlike terms. You can only combine terms that have the same variable raised to the same power. For instance, you can combine 3x and 5x, but you can't combine 3x and 5x². Also, be careful when simplifying fractions. Make sure you're dividing both the numerator and the denominator by the same factor. It's easy to accidentally simplify just one part of the fraction. Another mistake is skipping steps in your work. While it might seem faster to skip steps, it also increases the chances of making an error. It's better to write out each step clearly, especially when you're first learning. Don't forget the order of operations (PEMDAS/BODMAS). Doing operations in the wrong order can lead to incorrect answers. For instance, if you add before multiplying, you'll get a different result. Another common mistake is not simplifying the final answer completely. Always check to see if you can reduce fractions or combine any remaining like terms. And finally, don't be afraid to double-check your work. It's always a good idea to go back and review your steps to catch any errors. By being aware of these common mistakes, you can minimize your chances of making them and improve your accuracy in simplifying algebraic expressions. So, keep these pitfalls in mind and keep striving for algebraic excellence!
Simplify the expression 1/9(5x + 3y) - 1/3(x-y) and show the steps.
Simplify 1/9(5x + 3y) - 1/3(x-y) Step-by-Step Solution
