Math Problems: Solving Expressions With Ease
Hey guys! Let's dive into some math problems today. We'll break down each one step by step to make sure everyone understands the process. Whether you're a math whiz or just starting out, these examples will help you get comfortable with algebraic expressions. Ready to get started? Let's go!
a) Unraveling the First Expression: (y² + 3x - 5) - (3y² + 6x + 10) =
Alright, let's start with this one: (y² + 3x - 5) - (3y² + 6x + 10) = ? Our goal here is to simplify this expression. Remember, when we subtract an entire expression, we're essentially subtracting each term within the parentheses. This means we'll need to pay close attention to the signs. The best way to handle this is to rewrite the problem, distributing the negative sign across the second set of parentheses. This changes the signs of each term inside those parentheses. So, we'll have: y² + 3x - 5 - 3y² - 6x - 10. Now that we've taken care of the subtraction, let's group our like terms. Like terms are those that have the same variables raised to the same powers. In our case, we have y² terms, x terms, and constant terms (the numbers without any variables). Let's group them together: (y² - 3y²) + (3x - 6x) + (-5 - 10). Now, we simply combine the like terms. For the y² terms, y² - 3y² = -2y². For the x terms, 3x - 6x = -3x. And for the constants, -5 - 10 = -15. Putting it all together, we get -2y² - 3x - 15. And that, my friends, is our simplified answer! See? It wasn't so bad, right? The key here is to distribute the negative sign carefully and then combine like terms. Always double-check your signs, and you'll be golden. Remember, practice makes perfect. The more you work through these problems, the easier they become. Don't be afraid to make mistakes; that's how we learn. Now, let's move on to the next one!
b) Conquering the Second Expression: 4a + b² + c - (-3a + 2b² + 3c - 5) =
Now, let's tackle the expression: 4a + b² + c - (-3a + 2b² + 3c - 5) = ? Just like before, the first step is to distribute that negative sign across the terms inside the second set of parentheses. This gives us: 4a + b² + c + 3a - 2b² - 3c + 5. Next, we group our like terms. We have 'a' terms, b² terms, 'c' terms, and a constant. Grouping them, we get: (4a + 3a) + (b² - 2b²) + (c - 3c) + 5. Now, combine the like terms. For the 'a' terms, 4a + 3a = 7a. For the b² terms, b² - 2b² = -b². For the 'c' terms, c - 3c = -2c. And the constant term remains as +5. So, putting it all together, we have 7a - b² - 2c + 5. There you have it! Another expression simplified. See how important it is to keep track of those signs? That's the most common place where errors can occur. Always take your time and double-check each step. With practice, you'll become a pro at this. Remember to write everything down. Don’t try to do it all in your head, because you are very likely to make a mistake. Make sure that when you rewrite the equation, you get all the signs right. Alright, let's keep the momentum going!
c) Tackling the Third Challenge: (4a³ + 5x + 11) + (7x⁴ - 3x³ + 7x) - (6x⁴ + 6x + 20) =
Okay, let's crank it up a notch with: (4a³ + 5x + 11) + (7x⁴ - 3x³ + 7x) - (6x⁴ + 6x + 20) = ? In this problem, we have addition and subtraction, but the principle remains the same. First, let's deal with the subtraction. We'll distribute the negative sign across the last set of parentheses: 4a³ + 5x + 11 + 7x⁴ - 3x³ + 7x - 6x⁴ - 6x - 20. Now, let's gather our like terms. We have x⁴ terms, x³ terms, x terms, a³ terms, and constant terms. Grouping them: 7x⁴ - 6x⁴ - 3x³ + 4a³ + 5x + 7x - 6x + 11 - 20. Combine the like terms. For the x⁴ terms, 7x⁴ - 6x⁴ = x⁴. The x³ term is just -3x³. The a³ term is just 4a³. For the x terms, 5x + 7x - 6x = 6x. And for the constants, 11 - 20 = -9. Putting it all together, our simplified expression is x⁴ - 3x³ + 4a³ + 6x - 9. That was a bit longer, but by breaking it down step by step, we made it manageable. Remember to focus on distributing signs correctly and then carefully combining like terms. Take your time, and don't rush through the process. It is important to stay organized. That will save you lots of headaches. Let’s move on!
d) Simplifying the Fourth Expression: (y² + 6z² + 6) - (-3y² + 5z² - 5) =
Let's get to work on: (y² + 6z² + 6) - (-3y² + 5z² - 5) = ? Again, we'll start by distributing the negative sign: y² + 6z² + 6 + 3y² - 5z² + 5. Next, group your like terms. We have y² terms, z² terms, and constant terms. Grouping them: (y² + 3y²) + (6z² - 5z²) + (6 + 5). Now, we combine the like terms. For the y² terms, y² + 3y² = 4y². For the z² terms, 6z² - 5z² = z². And for the constants, 6 + 5 = 11. So, our final answer is 4y² + z² + 11. Another one down! By now, you're getting pretty good at this. Remember to keep practicing and to pay close attention to the details. Always distribute the sign correctly. Stay organized, and break it into steps. You will be able to do this. Keep up the awesome work!
e) Multiplying the Fifth Expression: (2x + 6) . (-y) =
Okay, let's switch gears a bit and deal with some multiplication: (2x + 6) . (-y) = ? Here, we need to distribute the -y across the terms inside the parentheses. So, we multiply -y by both 2x and 6. This gives us: -y * 2x + (-y * 6). Simplifying, we get -2xy - 6y. And that's it! This is a good reminder that multiplication works differently. Here we aren’t combining terms; we are multiplying. Be mindful of those details. Just a quick and easy one to keep us on our toes. Make sure you understand the difference between addition/subtraction and multiplication. These are the building blocks of mathematics. Practice these basic concepts. Be aware of the signs. Let’s go to the next one!
f) Solving the Sixth Expression: (-5x) . (x - 2) =?
Last but not least, let's simplify this expression: (-5x) . (x - 2) = ? We're going to use the distributive property again. We need to multiply -5x by both x and -2. This gives us: (-5x * x) + (-5x * -2). Simplifying, we get -5x² + 10x. And that's the final answer. See? Another problem solved. Great job, guys! This wasn't so bad, right? Now you have more practice. You have seen different types of expressions and how to solve them. Keep up the awesome work!
Great job on working through these math problems with me! Remember, practice is key. The more you work with these types of problems, the easier they'll become. Don't hesitate to go back and review the steps if you need to. Keep up the awesome work, and keep exploring the wonderful world of math! And that's a wrap! Keep practicing, and you'll become a math master in no time! Remember to always stay positive and believe in yourself. You’ve got this!