Evaluate Expressions: A=4, B=5 - Algebra Basics

Hey guys! Today, we're diving into the world of basic algebra with a fun and practical guide. We'll be tackling a set of expressions where we know the values of our variables. Specifically, we're given that a = 4 and b = 5. Our mission? To evaluate a series of expressions using these values. This is super important because it lays the foundation for more complex algebraic problems you'll encounter later on. So, let's roll up our sleeves and get started!

1. a + 8 =

Okay, let's kick things off with our first expression: a + 8. This one's pretty straightforward. We know that a = 4, so all we need to do is substitute that value into the expression. This gives us 4 + 8. Now, a simple addition problem tells us that 4 + 8 = 12. So, the value of the expression a + 8 when a = 4 is 12. See? That wasn't so bad, was it? This is the basic principle of substitution, which is a fundamental concept in algebra. You'll be using this technique all the time, so it's crucial to get comfortable with it now. Think of it like replacing a placeholder (a) with its actual value (4). Once you've done that, the rest is just arithmetic. We'll be using this same approach for all the expressions that follow, so make sure you're following along. Remember, the key is to identify the variable, substitute its value, and then perform the necessary operations. With a little practice, you'll be a substitution pro in no time!

2. 3b =

Next up, we have the expression 3b. Now, remember that in algebra, when a number is written directly next to a variable, it means multiplication. So, 3b really means 3 multiplied by b. We know that b = 5, so we can substitute that value into the expression, giving us 3 * 5. Now, we just need to do the multiplication. 3 multiplied by 5 equals 15. So, the value of the expression 3b when b = 5 is 15. This illustrates another important concept in algebra: the implied multiplication. It's a shorthand way of writing things, and it's something you'll see frequently. So, whenever you see a number right next to a variable, remember that it means multiplication. It's also worth noting that the order of operations still applies here. Even though we're substituting first, we still need to follow the rules of PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In this case, we only have multiplication, so it's pretty simple. But as expressions get more complex, keeping the order of operations in mind will be crucial to getting the correct answer.

3. 30 ÷ b =

Alright, let's tackle the third expression: 30 ÷ b. This one involves division. We know that b = 5, so we substitute that value into the expression, giving us 30 ÷ 5. Now, we just need to perform the division. 30 divided by 5 equals 6. So, the value of the expression 30 ÷ b when b = 5 is 6. Just like with multiplication, division is a fundamental operation in algebra. It's the inverse of multiplication, and it's used extensively in solving equations and simplifying expressions. In this case, we had a simple division problem, but sometimes you might encounter more complex division involving fractions or decimals. The key is to remember the basic principles of division and to apply them carefully. It's also important to note that division by zero is undefined. So, if you ever encounter an expression where you're dividing by a variable that could be zero, you need to be mindful of that possibility. In this case, b = 5, so we don't have to worry about that. But it's a good thing to keep in mind for future problems.

4. 2b + 3 =

Now, let's move on to a slightly more complex expression: 2b + 3. This one involves both multiplication and addition. Remember, we know that b = 5. So, let's substitute that value in. This gives us 2 * 5 + 3. Now, we need to remember the order of operations (PEMDAS). Multiplication comes before addition, so we need to do 2 * 5 first. 2 multiplied by 5 equals 10. So, now we have 10 + 3. Finally, 10 + 3 = 13. So, the value of the expression 2b + 3 when b = 5 is 13. This expression highlights the importance of the order of operations. If we had added 3 to 5 first, and then multiplied by 2, we would have gotten a completely different answer. So, always remember PEMDAS! This is a crucial rule in algebra, and it will help you avoid many common mistakes. When you're evaluating expressions, always look for parentheses, exponents, multiplication, division, addition, and subtraction, and perform the operations in that order. With practice, it will become second nature.

5. a + b =

Alright, let's get back to basics with our next expression: a + b. This one's super simple. We know that a = 4 and b = 5. So, we substitute those values into the expression, giving us 4 + 5. Now, we just add them up. 4 + 5 = 9. So, the value of the expression a + b when a = 4 and b = 5 is 9. This expression is a great reminder of the fundamental nature of variables in algebra. They're just placeholders for numbers. Once you know the value of the variable, you can substitute it in and perform the arithmetic. This is the essence of evaluating algebraic expressions. It's like having a puzzle piece that you can slot into the right place. Once you've done that, the rest is usually pretty straightforward. In this case, we had a simple addition problem, but the principle applies to more complex expressions as well. The key is to identify the variables, substitute their values, and then perform the operations in the correct order.

6. 5a - 2 =

Let's tackle the expression 5a - 2. We know that a = 4. Substituting this value gives us 5 * 4 - 2. Remember the order of operations! We need to do the multiplication before the subtraction. So, 5 multiplied by 4 equals 20. Now we have 20 - 2. Subtracting 2 from 20 gives us 18. Therefore, the value of the expression 5a - 2 when a = 4 is 18. This expression further reinforces the importance of PEMDAS. If we had subtracted 2 from 4 first, and then multiplied by 5, we would have arrived at the wrong answer. Always prioritize multiplication and division before addition and subtraction. This is a golden rule in algebra and will save you from many potential errors. As you encounter more complex expressions, the order of operations becomes even more crucial. So, make sure you have a firm grasp of it. Practice makes perfect, so keep working on these types of problems and you'll become a pro in no time!

7. a + b² =

Now, let's spice things up with an exponent! We have the expression a + b². We know that a = 4 and b = 5. So, substituting these values gives us 4 + 5². Now, what does 5² mean? It means 5 raised to the power of 2, which is the same as 5 multiplied by itself: 5 * 5. So, 5² = 25. Now our expression looks like this: 4 + 25. Adding these together, we get 29. Therefore, the value of the expression a + b² when a = 4 and b = 5 is 29. This introduces us to the concept of exponents, which are a way of expressing repeated multiplication. The exponent tells you how many times to multiply the base by itself. In this case, the base is 5 and the exponent is 2. Exponents are a fundamental part of algebra and will come up frequently in more advanced topics. So, it's important to understand how they work. Remember, exponents come before multiplication, division, addition, and subtraction in the order of operations (PEMDAS).

8. 10a =

Moving on, we have the expression 10a. Remember that when a number is written directly next to a variable, it implies multiplication. So, 10a means 10 multiplied by a. We know that a = 4. Substituting this value, we get 10 * 4. Now, we just multiply. 10 multiplied by 4 equals 40. So, the value of the expression 10a when a = 4 is 40. This is a straightforward example of the implied multiplication we talked about earlier. It's a common convention in algebra, and it's important to recognize it. Whenever you see a number right next to a variable, remember that it means multiplication. This will help you avoid confusion and ensure that you're evaluating expressions correctly. In this case, we had a simple multiplication problem, but the same principle applies to more complex expressions. The key is to identify the implied multiplication and perform the operation accordingly.

9. 5a² + 2b² =

Now, let's tackle the most complex expression in our list: 5a² + 2b². This one combines exponents, multiplication, and addition. We know that a = 4 and b = 5. So, substituting these values gives us 5 * 4² + 2 * 5². Remember the order of operations (PEMDAS)! We need to deal with the exponents first. We already know that 5² = 25. And 4² means 4 * 4, which equals 16. So, our expression now looks like this: 5 * 16 + 2 * 25. Next, we need to do the multiplication. 5 multiplied by 16 equals 80. And 2 multiplied by 25 equals 50. So, now we have 80 + 50. Finally, we add these together: 80 + 50 = 130. Therefore, the value of the expression 5a² + 2b² when a = 4 and b = 5 is 130. This expression is a great example of how important it is to follow the order of operations. If we had done the addition before the exponents or the multiplication, we would have gotten a completely different (and incorrect) answer. This is why PEMDAS is so crucial in algebra. It's a set of rules that ensures we're all performing operations in the same order, so we can arrive at the correct answer. With complex expressions like this one, it's often helpful to break it down into smaller steps, as we did here. This makes it easier to keep track of what you're doing and to avoid mistakes.

10. Discussion Category: Mathematics

This exercise falls squarely into the category of mathematics, specifically algebra. It deals with evaluating algebraic expressions by substituting given values for variables. These types of problems are fundamental to understanding more advanced mathematical concepts. They form the building blocks for solving equations, graphing functions, and much more. So, mastering these basics is essential for anyone pursuing further studies in mathematics or related fields. The skills you've practiced here – substitution, order of operations, and understanding exponents – are all key components of algebraic thinking. They're tools that you'll use again and again as you delve deeper into the world of math. So, keep practicing, keep exploring, and you'll be amazed at what you can achieve!

This exercise is important for students learning algebra, as it helps them practice substitution and the order of operations. By working through these problems, students can solidify their understanding of these fundamental concepts and build a strong foundation for more advanced topics in algebra and beyond. Remember, mathematics is a journey, not a destination. So, enjoy the process of learning, and don't be afraid to ask questions. There's a whole world of mathematical wonders waiting to be explored!