Simplifying Algebraic Expressions Step By Step Solutions

Hey guys! Today, we're diving into the world of simplifying algebraic expressions. This is a crucial skill in mathematics, and mastering it will make your life so much easier when you tackle more complex problems. We'll break down each step, use a friendly tone, and ensure you understand the logic behind every operation. So, grab your pencils, notebooks, and let's get started!

Understanding Algebraic Expressions

Before we jump into the simplification process, let's make sure we're all on the same page about what algebraic expressions actually are. Algebraic expressions are combinations of variables (like x, y, or a), constants (numbers), and mathematical operations (like addition, subtraction, multiplication, and division). Think of them as mathematical phrases rather than full sentences (equations).

For example, 3x + 2y - 5 is an algebraic expression. Here, 3 and 2 are coefficients (the numbers multiplying the variables), x and y are variables, and -5 is a constant. Understanding these components is the first step in simplifying expressions effectively. When we talk about simplifying, we mean making the expression as concise as possible while keeping its value the same. This often involves combining like terms, distributing multiplication over parentheses, and performing other operations in the correct order.

Simplifying algebraic expressions is like decluttering your room. You want to group similar items together and get rid of any unnecessary extras. In math terms, this means identifying and combining like terms. Like terms are terms that have the same variable raised to the same power. For instance, 3x and 5x are like terms because they both have the variable x raised to the power of 1. On the other hand, 3x and 3x^2 are not like terms because the variables have different powers. Recognizing like terms is crucial because you can only add or subtract them directly. You can't combine x terms with y terms, or x terms with x^2 terms.

Another key concept is the distributive property. This property allows you to multiply a single term by each term inside a set of parentheses. For example, if you have 2(x + 3), the distributive property tells you to multiply 2 by both x and 3, resulting in 2x + 6. This is like giving everyone in a group a certain amount of something – you need to make sure everyone gets their fair share! The distributive property is incredibly useful when simplifying expressions that involve parentheses, and mastering it will significantly improve your ability to manipulate algebraic expressions. So, remember, distributing means multiplying the term outside the parentheses by each term inside, one at a time. This ensures you're applying the multiplication correctly across the entire expression, leading to a more simplified form.

Problem 8a: Simplifying 1/6 * (12a + 6b – 18)

Okay, let's tackle our first problem: 1/6 * (12a + 6b – 18). The goal here is to simplify this expression by applying the distributive property. Remember, this means we need to multiply 1/6 by each term inside the parentheses: 12a, 6b, and -18.

First, let's multiply 1/6 by 12a. This can be written as (1/6) * 12a. To simplify, we can think of 12 as a fraction, 12/1. So we have (1/6) * (12/1) * a. Multiplying the fractions gives us 12/6, which simplifies to 2. Thus, (1/6) * 12a becomes 2a. Remember, multiplying a fraction by a whole number is like finding a fraction of that number. In this case, we're finding one-sixth of 12a, which is 2a.

Next, we multiply 1/6 by 6b. This is (1/6) * 6b. Again, we can write 6 as 6/1, giving us (1/6) * (6/1) * b. Multiplying the fractions, we get 6/6, which simplifies to 1. So, (1/6) * 6b becomes 1b, or simply b. When a variable has a coefficient of 1, we usually don't write the 1 explicitly, so 1b is just b. This is a common practice in algebra to keep expressions as clean and simple as possible.

Finally, let's multiply 1/6 by -18. This is (1/6) * -18. We can think of this as (1/6) * (-18/1). Multiplying the fractions gives us -18/6, which simplifies to -3. So, (1/6) * -18 becomes -3. Remember to keep the negative sign in mind when dealing with negative numbers. Multiplying a positive fraction by a negative number will always result in a negative value. This is a fundamental rule in algebra, and keeping it in mind will help you avoid errors in your calculations.

Now, let's put it all together. We have 2a from (1/6) * 12a, b from (1/6) * 6b, and -3 from (1/6) * -18. Combining these terms gives us the simplified expression: 2a + b - 3. This is the simplest form of the original expression because we've distributed the 1/6 across all terms inside the parentheses and combined any like terms (in this case, there were no like terms to combine once we distributed). So, the final simplified form is 2a + b - 3.

Problem 8b: Simplifying (8x + 12y – 4) / 4

Now, let's move on to the second problem: (8x + 12y – 4) / 4. This problem involves dividing an entire expression by a constant. The key here is to remember that dividing by a number is the same as multiplying by its reciprocal. In this case, dividing by 4 is the same as multiplying by 1/4. So, we can rewrite the expression as (1/4) * (8x + 12y - 4). This makes it clear that we're going to use the distributive property again, similar to the previous problem.

We need to multiply 1/4 by each term inside the parentheses: 8x, 12y, and -4. First, let's multiply 1/4 by 8x. This is (1/4) * 8x. We can think of 8 as 8/1, so we have (1/4) * (8/1) * x. Multiplying the fractions gives us 8/4, which simplifies to 2. Thus, (1/4) * 8x becomes 2x. Remember, multiplying a fraction by a whole number is like finding a fraction of that number. In this case, we're finding one-quarter of 8x, which is 2x.

Next, let's multiply 1/4 by 12y. This is (1/4) * 12y. We can write 12 as 12/1, giving us (1/4) * (12/1) * y. Multiplying the fractions, we get 12/4, which simplifies to 3. So, (1/4) * 12y becomes 3y. Again, we're finding a fraction of a term – one-quarter of 12y, which is 3y. This step is similar to the previous one, just with a different coefficient and variable.

Finally, we multiply 1/4 by -4. This is (1/4) * -4. We can think of this as (1/4) * (-4/1). Multiplying the fractions gives us -4/4, which simplifies to -1. So, (1/4) * -4 becomes -1. Keep in mind the negative sign – multiplying a positive fraction by a negative number results in a negative value. This is a crucial detail to ensure you get the correct simplified expression.

Now, let's combine our results. We have 2x from (1/4) * 8x, 3y from (1/4) * 12y, and -1 from (1/4) * -4. Putting these together, the simplified expression is 2x + 3y - 1. This is the simplest form of the original expression because we've distributed the division (or multiplication by 1/4) across all terms inside the parentheses and there are no like terms to combine further. So, the final simplified form is 2x + 3y - 1. This result showcases the power of the distributive property and how it can be used to simplify complex-looking expressions into a much cleaner and manageable form.

Problem 8c: Simplifying 4x + 5 – (15x – 18) / 3

Alright, guys, let's dive into our last problem: 4x + 5 – (15x – 18) / 3. This one looks a bit more complex, but don't worry, we'll break it down step by step. The key here is to follow the order of operations (PEMDAS/BODMAS), which tells us to handle parentheses and division before addition and subtraction. So, our first step is to simplify the expression (15x – 18) / 3.

Similar to problem 8b, we can rewrite the division by 3 as multiplication by its reciprocal, 1/3. So, (15x – 18) / 3 becomes (1/3) * (15x – 18). Now we can use the distributive property to multiply 1/3 by each term inside the parentheses. This means we'll multiply 1/3 by 15x and 1/3 by -18.

First, let's multiply 1/3 by 15x. This is (1/3) * 15x. We can think of 15 as a fraction, 15/1, giving us (1/3) * (15/1) * x. Multiplying the fractions, we get 15/3, which simplifies to 5. Thus, (1/3) * 15x becomes 5x. Remember, multiplying a fraction by a whole number is like finding a fraction of that number. In this case, we're finding one-third of 15x, which is 5x. This step is a straightforward application of the distributive property, making the expression more manageable.

Next, let's multiply 1/3 by -18. This is (1/3) * -18. We can think of this as (1/3) * (-18/1). Multiplying the fractions gives us -18/3, which simplifies to -6. So, (1/3) * -18 becomes -6. Remember to keep the negative sign in mind when dealing with negative numbers. Multiplying a positive fraction by a negative number will always result in a negative value. This is crucial for getting the correct simplified expression.

Now, we've simplified (15x – 18) / 3 to 5x - 6. But we're not done yet! We need to substitute this back into the original expression: 4x + 5 – (15x – 18) / 3. Replacing (15x – 18) / 3 with 5x - 6 gives us 4x + 5 – (5x - 6). The parentheses here are still important because we're subtracting the entire expression (5x - 6). This means we need to distribute the negative sign across both terms inside the parentheses.

Distributing the negative sign is like multiplying by -1. So, –(5x - 6) becomes -1 * (5x - 6). Multiplying -1 by 5x gives us -5x, and multiplying -1 by -6 gives us +6. So, –(5x - 6) simplifies to -5x + 6. Remember, subtracting a negative number is the same as adding its positive counterpart. This is a key concept to keep in mind when simplifying expressions involving subtraction and parentheses.

Now our expression looks like this: 4x + 5 - 5x + 6. The final step is to combine like terms. We have two x terms: 4x and -5x. Combining them gives us 4x - 5x, which simplifies to -1x or simply -x. We also have two constant terms: 5 and 6. Adding them together gives us 5 + 6, which equals 11.

Putting it all together, we have -x + 11. This is the simplest form of the original expression. We've successfully handled the division, distributed the negative sign, and combined like terms. So, the final simplified form is -x + 11. This problem really highlights the importance of following the order of operations and carefully handling negative signs to avoid mistakes and arrive at the correct simplified expression.

Conclusion

And there you have it, guys! We've successfully simplified three different algebraic expressions. Remember, the key is to break down complex problems into smaller, manageable steps. Always follow the order of operations, distribute carefully, and combine like terms. With practice, you'll become a pro at simplifying algebraic expressions. Keep up the great work, and I'll see you in the next lesson!