Simplifying Algebraic Expressions Step By Step Solutions

Hey guys! 👋 Feeling a bit lost with algebraic expressions? No worries, you're not alone! Algebra can seem like a daunting beast at first, but trust me, once you break it down into smaller steps, it becomes super manageable. In this article, we're going to tackle some common algebraic simplification problems. We'll go through each one step-by-step, so you can follow along and really understand the process. So, grab a pen and paper, and let's dive into the world of simplifying algebraic expressions! Our main focus here is to help you not just get the answers, but to truly grasp the why behind each step. We'll be using concepts like the distributive property, combining like terms, and the rules of exponents. Think of this as your friendly guide to conquering algebraic simplifications! Let's get started, shall we? We're about to turn those confusing expressions into something crystal clear. Remember, the key to mastering algebra is practice, so don't be afraid to try these problems on your own first. If you get stuck, no sweat! That's what this guide is for. We're here to help you every step of the way. Now, let's get those algebraic gears turning and unlock the secrets of simplification!

Problem 1 Simplify (20x – 4y) / (-4)

In this first problem, simplifying algebraic expressions is the name of the game, guys. We're starting with the expression (20x – 4y) / (-4). The key here is the distributive property in reverse. Think of it like dividing each term inside the parentheses by -4. This is a crucial concept in algebra, and mastering it will make simplifying expressions a breeze. So, let's break it down step by step. First, we divide 20x by -4. Remember the rules of signs: a positive divided by a negative is a negative. So, 20x / -4 equals -5x. Got it? Great! Now, let's move on to the second term. Next, we divide -4y by -4. Here, we have a negative divided by a negative, which gives us a positive result. So, -4y / -4 equals +y. Fantastic! We're almost there. Now, we simply combine the results. We have -5x from the first division and +y from the second. Put them together, and we get -5x + y. And that, my friends, is our simplified expression! See? It wasn't so scary after all. Remember, the key is to take it one step at a time and focus on the rules of signs. This problem highlights the importance of the distributive property and how it applies to division as well as multiplication. It's a fundamental skill in algebra, so make sure you've got a good grasp of it. We're just getting started, though. There are more expressions to simplify, and each one will give us a chance to practice and learn even more. So, keep your thinking caps on, and let's tackle the next challenge! We're building a solid foundation in algebra here, and the more we practice, the more confident we'll become. Let's move on!

Step-by-step solution:

1. (20x – 4y) / (-4) = (20x / -4) + (-4y / -4)
2. = -5x + y

Problem 2 Simplify (5a – 8b) + 3(-a + 2b)

Okay, guys, let's jump into the second problem! This time, we're tackling the expression (5a – 8b) + 3(-a + 2b). This problem introduces us to another important concept: combining like terms, but before we can do that, we need to deal with the parentheses. And how do we do that? You guessed it – the distributive property! The distributive property is like the secret weapon of algebra. It allows us to multiply a term outside the parentheses by each term inside. So, let's focus on that 3(-a + 2b) part first. We need to multiply 3 by both -a and 2b. 3 times -a is -3a, and 3 times 2b is 6b. So, 3(-a + 2b) simplifies to -3a + 6b. Awesome! Now, let's rewrite the entire expression with this simplification: (5a – 8b) + (-3a + 6b). Now we're getting somewhere! This looks much more manageable. Here comes the fun part – combining like terms. Like terms are terms that have the same variable raised to the same power. In our expression, 5a and -3a are like terms, and -8b and 6b are like terms. To combine like terms, we simply add or subtract their coefficients (the numbers in front of the variables). So, let's combine the 'a' terms first: 5a + (-3a) = 2a. And now the 'b' terms: -8b + 6b = -2b. Finally, we put those simplified terms together, and we get 2a - 2b. And there you have it! We've successfully simplified the expression. This problem really emphasizes the importance of two key concepts: the distributive property and combining like terms. These are fundamental skills in algebra, and you'll use them constantly. So, make sure you're comfortable with them. We're building our algebraic toolbox here, one problem at a time. Let's keep the momentum going and move on to the next challenge!

Step-by-step solution:

1. (5a – 8b) + 3(-a + 2b) = (5a – 8b) + (-3a + 6b)
2. = 5a – 8b – 3a + 6b
3. = (5a – 3a) + (-8b + 6b)
4. = 2a – 2b

Problem 3 Simplify 5(x + 3y) – 4(2x – 2y)

Alright, guys, let's tackle problem number three! This one is a fantastic combination of the skills we've already worked on, so it's a great way to solidify our understanding. The expression we're simplifying is 5(x + 3y) – 4(2x – 2y). Notice anything familiar? Yep, we've got the distributive property knocking on our door again! This time, we have two sets of parentheses, so we'll need to apply the distributive property twice. No sweat, we've got this! First, let's distribute the 5 in 5(x + 3y). 5 times x is 5x, and 5 times 3y is 15y. So, 5(x + 3y) simplifies to 5x + 15y. Great start! Now, let's tackle the second part: -4(2x – 2y). It's crucial to remember that we're distributing -4, not just 4. This is where those pesky sign errors can creep in, so pay close attention! -4 times 2x is -8x, and -4 times -2y is +8y (remember, a negative times a negative is a positive!). So, -4(2x – 2y) simplifies to -8x + 8y. Awesome! We've conquered the parentheses. Now, let's rewrite the entire expression with our simplified terms: 5x + 15y – 8x + 8y. Time for the next step: combining like terms! We know the drill by now. We need to identify the terms with the same variable and add or subtract their coefficients. Our 'x' terms are 5x and -8x, and our 'y' terms are 15y and 8y. Let's combine the 'x' terms: 5x - 8x = -3x. And now the 'y' terms: 15y + 8y = 23y. Finally, we put it all together, and we get our simplified expression: -3x + 23y. Fantastic job! This problem really demonstrates how the distributive property and combining like terms work together to simplify complex expressions. It's like a dynamic duo of algebraic simplification! The more you practice these types of problems, the more natural these steps will become. You'll be simplifying expressions like a pro in no time! Let's keep sharpening those algebraic skills and move on to the next challenge. We're on a roll!

Step-by-step solution:

1. 5(x + 3y) – 4(2x – 2y) = 5x + 15y – 8x + 8y
2. = (5x – 8x) + (15y + 8y)
3. = -3x + 23y

Problem 4 Simplify 7x × 4y

Okay, guys, let's switch gears a bit for problem number four. This time, we're dealing with multiplication: 7x × 4y. This one might seem simpler than the previous problems, but it's still important to understand the underlying principles. When we multiply algebraic terms, we're essentially combining the coefficients and the variables. The key here is to remember that multiplication is commutative and associative. That means we can change the order of the terms and group them in different ways without changing the result. So, we can rewrite 7x × 4y as 7 × x × 4 × y. Now, let's rearrange the terms so that the coefficients are together and the variables are together: 7 × 4 × x × y. Much better! Now, we can multiply the coefficients: 7 × 4 = 28. And we can simply write the variables next to each other: x × y = xy. Putting it all together, we get 28xy. And that's our simplified expression! See? Sometimes simplification is as straightforward as multiplying the numbers and writing the variables next to each other. This problem highlights the importance of understanding the properties of multiplication in algebra. It's a reminder that we can rearrange and regroup terms to make the simplification process easier. This is a valuable tool to have in your algebraic arsenal. It might seem like a small thing, but mastering these fundamental principles is what allows us to tackle more complex problems later on. So, even though this problem was relatively simple, it reinforced an important concept. Let's keep building on these foundations and move on to the next challenge! We're expanding our algebraic knowledge with each problem we solve.

Step-by-step solution:

1. 7x × 4y = 7 × 4 × x × y
2. = 28xy

Problem 5 Simplify 3a² × (-2a)

Alright, guys, let's dive into problem number five! This one is going to introduce us to the exciting world of exponents! We're simplifying the expression 3a² × (-2a). Notice that 'a²' means 'a' multiplied by itself (a * a). So, we have a variable raised to a power, which means we'll need to remember the rules of exponents. Just like in the previous problem, we can rearrange the terms: 3 × a² × (-2) × a. Let's group the coefficients together: 3 × (-2) = -6. Now, let's focus on the variables: a² × a. This is where the exponent rules come into play. Remember the rule that says when you multiply terms with the same base, you add the exponents? In this case, the base is 'a'. The exponent of a² is 2, and the exponent of 'a' (which is the same as a¹) is 1. So, a² × a¹ = a^(2+1) = a³. Now we have all the pieces! We have -6 from multiplying the coefficients and a³ from multiplying the variables. Putting them together, we get our simplified expression: -6a³. And there you have it! We've successfully simplified an expression involving exponents. This problem highlights the importance of remembering the rules of exponents, especially the one about adding exponents when multiplying terms with the same base. These rules are essential for simplifying algebraic expressions and will come up time and time again in your algebraic journey. It's like learning a new language – the more you use the rules, the more fluent you become. So, make sure you have a good understanding of the exponent rules. We're building our algebraic vocabulary here, and each rule we learn makes us more confident and capable. Let's keep expanding our knowledge and move on to the next problem!

Step-by-step solution:

1. 3a² × (-2a) = 3 × (-2) × a² × a
2. = -6a^(2+1)
3. = -6a³

Problem 6 Simplify (-9x)²

Hey guys, let's jump into problem number six! This one is all about squaring a term, and it's a fantastic opportunity to reinforce our understanding of exponents and negative signs. We're simplifying the expression (-9x)². Remember that squaring something means multiplying it by itself. So, (-9x)² is the same as (-9x) × (-9x). Now, let's break this down step by step. First, let's think about the coefficients: -9 × -9. A negative times a negative is a positive, so -9 × -9 = 81. Great! Now, let's think about the variables: x × x. This is the same as x², right? Perfect! Now, we just put the pieces together. We have 81 from the coefficients and x² from the variables. So, (-9x)² simplifies to 81x². And that's it! We've successfully squared an algebraic term. This problem highlights the importance of paying close attention to the signs, especially when dealing with exponents. Squaring a negative number always results in a positive number, which is a crucial point to remember. This is like a little algebraic gem – a simple rule that can make a big difference in your calculations. It's also a reminder that exponents apply to everything inside the parentheses. The exponent 2 applies to both the -9 and the x. So, we square both of them. This is a common mistake that students make, so it's worth emphasizing. We're building our algebraic attention to detail here, which is just as important as knowing the rules themselves. The more we practice these types of problems, the more automatic these steps will become. Let's keep our focus sharp and move on to the next challenge! We're mastering these algebraic skills one problem at a time.

Step-by-step solution:

1. (-9x)² = (-9x) × (-9x)
2. = (-9 × -9) × (x × x)
3. = 81x²

Problem 7 Simplify (-16a²) / 4a

Okay, guys, let's tackle problem number seven! This one brings us back to division, and it's a great opportunity to practice our skills with coefficients, variables, and exponents. We're simplifying the expression (-16a²) / 4a. Just like with multiplication, we can think of division in terms of coefficients and variables separately. Let's start with the coefficients: -16 / 4. A negative divided by a positive is a negative, so -16 / 4 = -4. Great! Now, let's move on to the variables: a² / a. This is where the exponent rules come in handy again. Remember the rule that says when you divide terms with the same base, you subtract the exponents? The base is 'a', the exponent of a² is 2, and the exponent of 'a' (which is the same as a¹) is 1. So, a² / a¹ = a^(2-1) = a¹. And a¹ is simply 'a'. Now we have all the pieces! We have -4 from dividing the coefficients and 'a' from dividing the variables. Putting them together, we get our simplified expression: -4a. And that's it! We've successfully simplified an expression involving division and exponents. This problem highlights the importance of remembering the exponent rule for division: subtracting the exponents when dividing terms with the same base. It's like the inverse of the multiplication rule, where we added the exponents. These rules are two sides of the same coin, and understanding both is crucial for simplifying algebraic expressions. We're adding more tools to our algebraic toolbox, and the more tools we have, the more confidently we can tackle any problem that comes our way. Let's keep practicing and solidifying these skills. We're on a journey of algebraic mastery, and each problem we solve brings us closer to our goal. Let's move on to the next challenge!

Step-by-step solution:

1. (-16a²) / 4a = (-16 / 4) × (a² / a)
2. = -4a^(2-1)
3. = -4a

Problem 8 Simplify 4x² / 6x × 3y

Alright, guys, let's jump into the final problem! This one is a bit of a combination of everything we've learned so far, so it's a fantastic way to test our overall understanding. We're simplifying the expression 4x² / 6x × 3y. Notice that we have both division and multiplication in this expression. The key here is to remember the order of operations. We perform division and multiplication from left to right. So, let's tackle the division first: 4x² / 6x. We can simplify the coefficients: 4/6 can be simplified to 2/3. Now, let's simplify the variables: x² / x. Using our exponent rule for division, we subtract the exponents: x^(2-1) = x¹. So, x² / x simplifies to x. Putting it together, 4x² / 6x simplifies to (2/3)x. Great! Now, let's bring in the multiplication: (2/3)x × 3y. We can multiply the coefficients: (2/3) × 3 = 2. And we can multiply the variables: x × y = xy. Putting it all together, we get our simplified expression: 2xy. And that's it! We've conquered all eight problems! You guys have done an amazing job working through these simplifications. This problem really highlights the importance of following the order of operations and applying all the skills we've learned: simplifying fractions, using exponent rules, and multiplying variables. It's like a final exam for our algebraic simplification skills! We've built a solid foundation in simplifying algebraic expressions, and you should feel proud of the progress you've made. Remember, the key to mastering algebra is practice, so keep working at it, and you'll become even more confident and skilled. We've reached the end of this guide, but your algebraic journey is just beginning! Keep exploring, keep practicing, and keep simplifying! You've got this!

Step-by-step solution:

1. 4x² / 6x × 3y = (4x² / 6x) × 3y
2. = (2/3)x × 3y
3. = 2xy

I hope this comprehensive guide has helped you understand how to simplify algebraic expressions. Remember to practice regularly, and you'll master these skills in no time! Good luck, guys! 😉