Simplify Expressions: Combining Terms & Proving Identities

Hey guys! Let's dive into the world of algebra and tackle some expression simplification. We'll be focusing on combining like terms, simplifying expressions, and even proving identities. So, grab your pencils, and let's get started!

Combining Like Terms: 16 – 2x – 27 + 3x

Okay, so you've got this expression: 16 – 2x – 27 + 3x. The main goal here is to combine the like terms. What does that even mean, right? Well, like terms are those that have the same variable raised to the same power. In our case, we've got '-2x' and '+3x' as our variable terms, and '16' and '-27' as our constant terms. To effectively combine like terms, think of it like grouping similar objects together. You wouldn't mix apples and oranges, would you? It's the same idea here.

First, let's focus on the 'x' terms. We have '-2x' and '+3x'. Imagine you owe someone 2 'x' (that's the '-2x') and then you gain 3 'x' (the '+3x'). What do you have in the end? You've got 1 'x' left, or simply 'x'. Mathematically, this is expressed as -2x + 3x = x. This step is crucial because it directly reduces the complexity of the expression. By adding the coefficients of the 'x' terms, we are essentially condensing the expression into a more manageable form.

Now, let's move on to the constant terms: '16' and '-27'. This is a straightforward arithmetic operation. You have 16, and you subtract 27 from it. That gives you -11. So, 16 - 27 = -11. Handling the constants is as important as dealing with the variable terms. Constants determine the base value of the expression, and their accurate calculation ensures the overall correctness of the simplification.

Putting it all together, we combine our simplified variable term ('x') with our simplified constant term ('-11'). This gives us the final simplified expression: x – 11. And that's it! We've successfully combined the like terms and simplified the original expression. This resulting expression, 'x - 11', is much cleaner and easier to work with in further calculations or problem-solving scenarios.

Simplifying Expressions: 1.5(x – 4) + 1.2(8 – x) when x = 5

Next up, we're simplifying the expression 1.5(x – 4) + 1.2(8 – x) and finding its value when x = 5. This involves a couple of steps: the distributive property and then substitution. Think of the distributive property as spreading the love (or the multiplication, in this case) across the terms inside the parentheses.

First, let's apply the distributive property to 1.5(x – 4). We multiply 1.5 by both 'x' and '-4'. This gives us 1.5 * x = 1.5x and 1.5 * -4 = -6. So, 1.5(x – 4) becomes 1.5x – 6. This distribution is key because it removes the parentheses, making the expression easier to combine with other terms. Each term inside the parentheses is affected by the multiplier outside, ensuring the expression remains equivalent.

Now, let's do the same for 1.2(8 – x). We multiply 1.2 by both '8' and '-x'. This gives us 1.2 * 8 = 9.6 and 1.2 * -x = -1.2x. So, 1.2(8 – x) becomes 9.6 – 1.2x. Just like before, distributing 1.2 across the terms inside the parentheses helps to linearize the expression, setting the stage for combining like terms.

Now, let's put the pieces together. Our expression is now 1.5x – 6 + 9.6 – 1.2x. See how much simpler it looks already? It's like decluttering a room! Now, we combine like terms again. We've got '1.5x' and '-1.2x' as our variable terms, and '-6' and '9.6' as our constant terms. Combining '1.5x' and '-1.2x' gives us 1.5x - 1.2x = 0.3x. This step is crucial for simplifying the expression, as it reduces multiple 'x' terms into a single term, making it easier to manage and evaluate.

Combining '-6' and '9.6' gives us -6 + 9.6 = 3.6. Constants are the backbone of the expression, providing a fixed value that doesn't depend on the variable. Accurate combination of constants is vital for the correct evaluation of the expression.

So, our simplified expression is 0.3x + 3.6. But we're not done yet! We need to find the value of this expression when x = 5. This is where substitution comes in. We replace 'x' with '5' in our simplified expression. So, we have 0.3 * 5 + 3.6. Now, 0.3 * 5 = 1.5, and adding 3.6 to that gives us 1.5 + 3.6 = 5.1. Therefore, the value of the expression 1.5(x – 4) + 1.2(8 – x) when x = 5 is 5.1. This final evaluation step ties everything together, providing a numerical answer that represents the value of the entire expression under the given condition.

Simplifying Expressions: 4(-0.2a – 1) – 0.5(6a – 4)

Alright, let's tackle another simplification: 4(-0.2a – 1) – 0.5(6a – 4). This one looks a bit more intense, but don't worry, we'll break it down step by step. Just like before, we'll start with the distributive property. Remember, we're spreading the multiplication across the terms inside the parentheses.

First, let's distribute the '4' in 4(-0.2a – 1). We multiply 4 by both '-0.2a' and '-1'. This gives us 4 * -0.2a = -0.8a and 4 * -1 = -4. So, 4(-0.2a – 1) becomes -0.8a – 4. The distributive property is a cornerstone of algebraic simplification, allowing us to remove parentheses and rearrange terms into a more manageable format.

Next, we distribute the '-0.5' in – 0.5(6a – 4). Notice the negative sign! It's super important to include that when you distribute. We multiply -0.5 by both '6a' and '-4'. This gives us -0.5 * 6a = -3a and -0.5 * -4 = 2. So, -0.5(6a – 4) becomes -3a + 2. Attention to signs is critical in algebraic manipulations. A misplaced negative can completely alter the result, so always double-check your work.

Now, let's combine our results. Our expression is now -0.8a – 4 – 3a + 2. Time to combine like terms again! We've got '-0.8a' and '-3a' as our variable terms, and '-4' and '2' as our constant terms. Combining '-0.8a' and '-3a' gives us -0.8a - 3a = -3.8a. Like terms must be combined accurately to simplify the expression effectively. This involves summing their coefficients while keeping the variable part the same.

Combining '-4' and '2' gives us -4 + 2 = -2. Constants are the numerical anchors of the expression, and combining them correctly ensures the expression's baseline value is accurate.

So, our simplified expression is -3.8a – 2. See? Not so scary after all! We've successfully simplified a complex-looking expression by carefully applying the distributive property and combining like terms. This simplified form is not only easier to understand but also more straightforward to use in further calculations or problem-solving contexts.

Proving Identities: 15x – (-3(7x + 30)) = 6(6x + 15)

Last but not least, let's prove the identity 15x – (-3(7x + 30)) = 6(6x + 15). Proving an identity means showing that the expressions on both sides of the equation are equal, no matter what value we plug in for 'x'. It's like showing that two different paths lead to the same destination.

To prove this identity, we'll simplify both sides of the equation separately until we (hopefully!) get the same expression on both sides. Let's start with the left side: 15x – (-3(7x + 30)).

First, we need to deal with those parentheses. Let's distribute the '-3' in (-3(7x + 30)). We multiply -3 by both '7x' and '30'. This gives us -3 * 7x = -21x and -3 * 30 = -90. So, -3(7x + 30) becomes -21x – 90. Distributing correctly is crucial here, as it sets the stage for further simplification. The correct application of the distributive property ensures that the integrity of the expression is maintained.

Now, our left side looks like this: 15x – (-21x – 90). Notice the double negative! Subtracting a negative is the same as adding a positive. So, we can rewrite this as 15x + 21x + 90. Handling double negatives correctly is a fundamental skill in algebra. Failing to do so can lead to incorrect simplification and an inability to prove the identity.

Now, let's combine like terms. We've got '15x' and '21x' as our variable terms, and '90' as our constant term. Combining '15x' and '21x' gives us 15x + 21x = 36x. So, the left side simplifies to 36x + 90. The goal of combining like terms is to consolidate the expression into its simplest form, making it easier to compare with the other side of the equation.

Okay, let's move on to the right side: 6(6x + 15). We need to distribute the '6' here. We multiply 6 by both '6x' and '15'. This gives us 6 * 6x = 36x and 6 * 15 = 90. So, 6(6x + 15) becomes 36x + 90. The distributive property ensures that every term inside the parentheses is correctly affected by the multiplier, maintaining the equivalence of the expression.

Hey, look at that! The right side simplified to 36x + 90. And the left side also simplified to 36x + 90. Both sides are the same! That means we've proven the identity. We've shown that 15x – (-3(7x + 30)) is indeed equal to 6(6x + 15) for any value of 'x'. Proving an identity is like solving a puzzle, where you manipulate both sides of the equation until they match, demonstrating their inherent equivalence.

Conclusion

So, there you have it! We've tackled simplifying expressions by combining like terms, using the distributive property, and even proving identities. Remember, guys, practice makes perfect! The more you work with these concepts, the more comfortable you'll become with them. Keep up the great work, and you'll be algebra aces in no time!