Simplifying Algebraic Expressions: A Step-by-Step Guide

Hey guys! Let's dive into simplifying algebraic expressions. Today, we're tackling the expression: $-5 "." 5 + 500 + 0 "." 9t + (\frac{11}{2} - 1 "." 9t)$. Don't worry, it's not as scary as it looks! We'll break it down step by step, so you can follow along and understand exactly how to simplify such expressions. Understanding algebraic expressions is super useful, not just in math class but also in real-life problem-solving. So, grab a pen and paper, and let's get started!

Understanding the Basics

Before we jump into the simplification, let's make sure we're all on the same page with the basic concepts. Algebraic expressions are combinations of numbers, variables, and mathematical operations. Variables are symbols (usually letters like x, y, or t) that represent unknown values. In our expression, t is the variable. Mathematical operations include addition, subtraction, multiplication, and division. To simplify an expression, we combine like terms and perform the operations in the correct order (PEMDAS/BODMAS - Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction).

Order of Operations

The order of operations is crucial. It tells us the sequence in which we should perform the calculations. Remember PEMDAS/BODMAS: Parentheses/Brackets first, then Exponents/Orders, followed by Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right). This ensures that we always arrive at the correct simplified form. For example, in our expression, we'll first handle the multiplication and then deal with the addition and subtraction. Ignoring the order of operations can lead to incorrect results, so always keep it in mind!

Combining Like Terms

Combining like terms is another key concept. Like terms are terms that have the same variable raised to the same power. For instance, 3t and 5t are like terms, but 3t and 5t² are not. To combine like terms, we simply add or subtract their coefficients (the numbers in front of the variables). For example, 3t + 5t = 8t. In our expression, we'll identify and combine the terms that contain the variable t. This will help us reduce the expression to its simplest form. Understanding how to combine like terms is fundamental to simplifying any algebraic expression.

Step-by-Step Simplification

Okay, let's get our hands dirty and simplify the expression: $-5 "." 5 + 500 + 0 "." 9t + (\frac{11}{2} - 1 "." 9t)$. We'll go through it step by step, so you can see exactly how it's done.

Step 1: Perform the Multiplication

First, we'll perform the multiplication operations. We have $-5 "." 5$ and $0 "." 9t$.

• $-5 "." 5 = -25$
• $0 "." 9t = 0$

So our expression now looks like: $-25 + 500 + 0 + (\frac{11}{2} - 1 "." 9t)$.

Step 2: Simplify Inside the Parentheses

Next, let's simplify inside the parentheses. We have $(\frac{11}{2} - 1 "." 9t)$. We can rewrite $\frac{11}{2}$ as $5.5$, so the expression inside the parentheses becomes $(5.5 - 1.9t)$. Our entire expression now is: $-25 + 500 + 0 + 5.5 - 1.9t$.

Step 3: Combine Constants

Now, let's combine the constant terms (the numbers without any variables). We have $-25 + 500 + 0 + 5.5$. Adding these together:

• $-25 + 500 = 475$
• $475 + 0 = 475$
• $475 + 5.5 = 480.5$

So, the constant terms combine to give us $480.5$.

Step 4: Write the Simplified Expression

Finally, we write the simplified expression by combining the constant term and the term with the variable t. Our expression is now: $480.5 - 1.9t$.

So, the simplified form of the expression $-5 "." 5 + 500 + 0 "." 9t + (\frac{11}{2} - 1 "." 9t)$ is $480.5 - 1.9t$.

Common Mistakes to Avoid

When simplifying algebraic expressions, there are a few common mistakes that students often make. Let's go over these, so you can avoid them!

Forgetting the Order of Operations

One of the most common mistakes is forgetting the order of operations (PEMDAS/BODMAS). Always remember to perform operations in the correct order: Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction. Mixing up the order can lead to incorrect results.

Incorrectly Combining Like Terms

Another mistake is incorrectly combining like terms. Remember, you can only combine terms that have the same variable raised to the same power. For example, you can combine 3x and 5x, but you cannot combine 3x and 5x². Make sure to pay close attention to the variables and their exponents when combining terms.

Sign Errors

Sign errors are also common. Be careful when dealing with negative signs. Make sure to distribute negative signs correctly when removing parentheses. For example, $-(x - 2)$ becomes $-x + 2$, not $-x - 2$. Always double-check your signs to avoid these errors.

Calculation Errors

Simple calculation errors can also throw off your answer. Double-check your arithmetic to make sure you haven't made any mistakes in addition, subtraction, multiplication, or division. It's always a good idea to write out your steps clearly and double-check each one.

Practice Problems

To really nail down your understanding, let's try a few practice problems. These will help you apply what we've learned and build your confidence.

Practice Problem 1

Simplify the expression: $3(x + 2) - 5x + 7$

Practice Problem 2

Simplify the expression: $-2(y - 4) + 6y - 3$

Practice Problem 3

Simplify the expression: $4(2a + 1) - 3(a - 2)$

Try these problems on your own, and then check your answers. The more you practice, the better you'll become at simplifying algebraic expressions!

Conclusion

Alright, guys, we've covered a lot in this guide. We started with the basics of algebraic expressions, walked through a step-by-step simplification of the expression $-5 "." 5 + 500 + 0 "." 9t + (\frac{11}{2} - 1 "." 9t)$, discussed common mistakes to avoid, and even tackled a few practice problems. Simplifying algebraic expressions might seem tricky at first, but with practice and a good understanding of the basic concepts, you'll become a pro in no time. Remember to always follow the order of operations, combine like terms carefully, and double-check your work. Keep practicing, and you'll be simplifying expressions like a champ! Keep up the great work!