Hey everyone! Let's dive into a fun math problem today. We're going to evaluate an algebraic expression, which basically means we'll find its value when we know the values of the variables involved. Specifically, we're tackling the expression 5y - z given that y = 1/5 and z = 3/4. Sounds like a plan? Awesome! This is a common type of problem in algebra, and mastering it will build a solid foundation for more complex concepts. So, grab your pencils and let’s get started!
Step-by-Step Solution
Alright, let's break this down step-by-step so it's super clear. The key to evaluating algebraic expressions is simply substituting the given values for the variables and then performing the arithmetic operations. It’s like a recipe – you just follow the steps in the right order and you’ll get the correct result. First things first, we have the expression 5y - z. We know that y is equal to 1/5 and z is equal to 3/4. The first step is substituting these values into the expression. So, wherever we see a y, we'll replace it with 1/5, and wherever we see a z, we'll replace it with 3/4. This gives us 5 * (1/5) - (3/4). Notice how I put the 1/5 in parentheses? This is just to make it visually clear that we're multiplying 5 by the fraction 1/5. It's a good habit to use parentheses in these situations to avoid any confusion, especially when you're dealing with more complex expressions. Now that we've substituted the values, the expression looks a lot less abstract and more like a straightforward arithmetic problem. Next, we need to follow the order of operations, which you might remember by the acronym PEMDAS (or BODMAS): Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). In our expression, 5 * (1/5) - (3/4), we have multiplication and subtraction. According to PEMDAS, we need to do the multiplication first. So, let’s tackle 5 * (1/5). Remember that multiplying a whole number by a fraction involves multiplying the whole number by the numerator (the top number) and keeping the same denominator (the bottom number). In this case, we can rewrite 5 as 5/1 to make it even clearer. So, 5 * (1/5) is the same as (5/1) * (1/5). Multiplying the numerators, we get 5 * 1 = 5. Multiplying the denominators, we get 1 * 5 = 5. So, (5/1) * (1/5) = 5/5. And what is 5/5? It’s simply equal to 1. Great! We've simplified the first part of the expression. Now we have 1 - (3/4). This is a subtraction problem involving a whole number and a fraction. To subtract a fraction from a whole number, we need to express the whole number as a fraction with the same denominator as the fraction we're subtracting. In this case, our fraction has a denominator of 4, so we need to express 1 as a fraction with a denominator of 4. We know that any number divided by itself is equal to 1, so 1 is the same as 4/4. Now we can rewrite our expression as (4/4) - (3/4). Subtracting fractions with the same denominator is easy – we simply subtract the numerators and keep the same denominator. So, (4/4) - (3/4) is equal to (4 - 3) / 4, which simplifies to 1/4. And there you have it! We've evaluated the expression 5y - z when y = 1/5 and z = 3/4. The final answer is 1/4.
Detailed Breakdown of Each Step
Let's delve even deeper into each step to make sure we've got a rock-solid understanding. This is where we really solidify the concepts and ensure that we can apply them to other problems. Remember, math isn't just about getting the right answer; it's about understanding why the answer is correct. So, let’s break it down! Step 1: Substitution This is where we replace the variables (y and z) with their given values (1/5 and 3/4 respectively). This step is absolutely crucial because it transforms the abstract algebraic expression into a concrete arithmetic problem that we can actually solve. It's like translating from one language to another; we're taking the algebraic language and translating it into the language of numbers. The original expression is 5y - z. When we substitute y = 1/5 and z = 3/4, we get 5 * (1/5) - (3/4). Notice the careful use of parentheses here. While it might seem like a small detail, it's incredibly important for clarity. The parentheses make it perfectly clear that we are multiplying 5 by the fraction 1/5. Without the parentheses, it might be misread as 51/5, which is totally different! So, always be mindful of parentheses, especially when dealing with multiplication and fractions. This simple act of substitution is the foundation upon which the rest of the solution is built. If we make a mistake here, the entire problem will be thrown off. So, double-check your substitutions and make sure you've replaced each variable with its correct value. Step 2: Multiplication According to the order of operations (PEMDAS/BODMAS), we need to perform multiplication before subtraction. So, we focus on the term 5 * (1/5). This is where our understanding of fraction multiplication comes into play. Remember, when we multiply a whole number by a fraction, we can think of the whole number as a fraction with a denominator of 1. So, 5 can be written as 5/1. Now we have (5/1) * (1/5). To multiply fractions, we multiply the numerators (the top numbers) together and multiply the denominators (the bottom numbers) together. So, (5/1) * (1/5) = (5 * 1) / (1 * 5) = 5/5. This gives us the fraction 5/5. Now, what does 5/5 mean? It means 5 divided by 5, which is simply equal to 1. This is a really important simplification. We've taken the multiplication 5 * (1/5) and boiled it down to the single number 1. This significantly simplifies the expression and makes it easier to work with. This step highlights the power of understanding fractions and their relationship to whole numbers. By recognizing that 5/5 is just another way of writing 1, we can make the problem much more manageable. Step 3: Subtraction Now that we've performed the multiplication, our expression has been simplified to 1 - (3/4). This is a subtraction problem involving a whole number and a fraction. To subtract a fraction from a whole number, we need to express the whole number as a fraction with the same denominator as the fraction we're subtracting. In our case, the fraction we're subtracting, 3/4, has a denominator of 4. So, we need to express the whole number 1 as a fraction with a denominator of 4. We know that any number divided by itself is equal to 1. Therefore, 1 can be written as 4/4. This is a crucial step because it allows us to perform the subtraction. We can't directly subtract 3/4 from 1 unless they have the same denominator. By rewriting 1 as 4/4, we've created a common denominator and made the subtraction possible. Now our expression looks like this: (4/4) - (3/4). To subtract fractions with the same denominator, we simply subtract the numerators and keep the denominator the same. So, (4/4) - (3/4) = (4 - 3) / 4 = 1/4. This gives us the final answer: 1/4. We've successfully subtracted the fraction from the whole number and arrived at the solution. Each of these steps is a building block, and understanding each one thoroughly is essential for success in algebra and beyond.
Common Mistakes to Avoid
Hey, we all make mistakes, especially when we're learning something new! The important thing is to learn from them so we don't repeat them. When evaluating expressions like this, there are a few common pitfalls that students sometimes fall into. Let's talk about those so you can steer clear of them! Forgetting the Order of Operations (PEMDAS/BODMAS) This is probably the most common mistake in algebra. Remember, the order of operations is like the recipe for math – you have to follow the steps in the right order to get the correct result. PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) tells us the order in which we need to perform operations. If you don't follow this order, you're likely to get the wrong answer. In our problem, 5y - z, we needed to do the multiplication (5 * (1/5)) before the subtraction. If someone mistakenly subtracted 3/4 from 5 first, they'd end up with a completely different answer. So, always double-check the order of operations and make sure you're performing the steps in the correct sequence. A handy way to remember it is using the mnemonic PEMDAS or BODMAS. Incorrectly Multiplying Fractions Multiplying fractions might seem simple, but it's easy to make a slip-up if you're not careful. Remember, to multiply fractions, you multiply the numerators together and multiply the denominators together. For example, to multiply (2/3) * (3/4), you'd multiply 2 * 3 to get 6 and 3 * 4 to get 12, resulting in 6/12. In our problem, we multiplied 5 * (1/5). It's crucial to remember that we can think of 5 as 5/1. Then, we multiply (5/1) * (1/5) to get 5/5, which simplifies to 1. A common mistake is to only multiply the whole number by the numerator and forget about the denominator. Incorrectly Subtracting Fractions Subtracting fractions requires a common denominator. This means that the fractions need to have the same bottom number before you can subtract them. If the fractions don't have a common denominator, you need to find one. In our problem, we had to subtract 3/4 from 1. To do this, we rewrote 1 as 4/4 so that both numbers had the same denominator. Then we could subtract the numerators: 4/4 - 3/4 = 1/4. A common mistake is to try to subtract the numerators without having a common denominator, which will lead to an incorrect answer. Forgetting to Substitute Values Correctly This might sound obvious, but it's a mistake that can easily happen, especially when you're working quickly or dealing with more complex expressions. Make sure you substitute the correct values for the variables. Double-check that you've replaced each variable with its assigned value. In our problem, we had to substitute y = 1/5 and z = 3/4. If you accidentally swapped these values or made a typo, the whole solution would be off. Not Simplifying Fractions Always simplify your fractions to their lowest terms. This means dividing both the numerator and the denominator by their greatest common factor. For example, if you ended up with 2/4, you should simplify it to 1/2. Simplifying fractions makes the answer cleaner and easier to understand. It's also often required in math tests and assignments. By being aware of these common mistakes, you can be more careful and avoid them. Math is all about precision, so taking your time and double-checking your work can make a big difference! Remember to always follow the order of operations, be careful with fractions, and double-check your substitutions. With practice and attention to detail, you'll become a pro at evaluating expressions!
Practice Problems
Okay, now that we've walked through the solution and discussed common mistakes, it's time to put your knowledge to the test! The best way to master evaluating expressions is to practice, practice, practice. So, let's try a few similar problems to solidify your understanding. Remember, the goal isn't just to get the right answer, but also to understand the process and build your problem-solving skills. Here are a few problems for you to try:
- Evaluate 3x + y if x = 2/3 and y = 1/2.
- Evaluate a - 2b if a = 5/4 and b = 1/8.
- Evaluate 4m - n/2 if m = 1/4 and n = 3/5.
- Evaluate 2p + 3q - 1 if p = 1/2 and q = 2/3.
Try working through these problems on your own. Follow the same steps we used in the example problem: substitute the given values for the variables, follow the order of operations (PEMDAS/BODMAS), and simplify your answer. Don't be afraid to make mistakes – that's how we learn! If you get stuck, go back and review the steps we discussed earlier. Remember, math is a journey, and each problem you solve makes you a little bit stronger. Working through these practice problems will help you build confidence and develop a deeper understanding of evaluating algebraic expressions. So, grab your pencil and paper, and let's get to work! Good luck, and remember to have fun with it!
Conclusion
Alright guys, we've reached the end of our journey into evaluating the expression 5y - z when y = 1/5 and z = 3/4. We’ve covered a lot of ground, from the initial substitution to the final simplified answer of 1/4. We've broken down each step in detail, talked about common mistakes to avoid, and even tackled some practice problems. Hopefully, you now feel much more confident in your ability to evaluate algebraic expressions. Remember, the key to success in math is understanding the underlying concepts and practicing regularly. Evaluating expressions is a fundamental skill in algebra, and it’s a building block for more advanced topics. So, mastering this skill will set you up for success in your future math endeavors. Think of it like learning the alphabet before you can read – it's an essential foundation. Don’t be afraid to revisit this explanation or work through the practice problems again if you need a refresher. And most importantly, don’t give up! Math can be challenging, but it’s also incredibly rewarding. Every problem you solve is a victory, and every mistake you learn from makes you stronger. So, keep practicing, keep asking questions, and keep exploring the wonderful world of math. You've got this! If you have any more questions or want to explore other math topics, feel free to ask. Keep up the great work, and happy calculating!