Solving Algebraic Expressions: A Complete Guide

Hey guys! Let's dive into the world of algebra and learn how to solve some cool expressions. I'll walk you through each problem step-by-step, making it super easy to understand. Ready to get started? Let's go!

Understanding the Basics of Algebraic Expressions

Before we jump into the problems, let's quickly recap what algebraic expressions are all about. Basically, they're mathematical phrases that contain numbers, variables (like a, b, x, y), and operations (+, -, *, /). Our goal is usually to simplify these expressions or find their value when we know the values of the variables. Think of it like a puzzle; we're putting the pieces together to find the solution. The core idea is to combine like terms. Like terms are terms that have the same variables raised to the same powers. For example, 3x and 5x are like terms, but 3x and 3x² are not. We can add or subtract like terms by simply adding or subtracting their coefficients (the numbers in front of the variables). For instance, 3x + 5x = 8x. We'll be using this concept a lot in the following examples. We're also going to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). This ensures we solve the expressions in the correct order, which is crucial for getting the right answer. We need to substitute values into variables. This means replacing the variable with its given numerical value and then performing the indicated operations. Be extra careful with negative numbers and exponents. Remember, a negative number raised to an even power becomes positive, while a negative number raised to an odd power remains negative. When dealing with fractions or decimals, don't let them intimidate you; the principles are the same. Use your calculator if needed, but always show your work step by step to understand the process.

Example 1: Breakdown

• Key Concept: Combining like terms, substitution.
• Focus: Using the proper order of operation, recognizing like terms, correctly substituting values.

Solving the Expressions

Let's start solving these algebraic expressions. I'll break down each one, so you can follow along easily. Don't worry if it seems tricky at first; with practice, it'll become second nature! We will solve the expressions by first simplifying them by combining like terms. After simplification, we substitute the provided values for the variables and then perform the arithmetic operations to find the final answer. Take note that there may be negative numbers involved. For instance, when substituting the variable with a negative value, remember the rules for multiplying negative and positive values. Then we need to apply the order of operations, prioritizing multiplication and division before addition and subtraction. We also must be mindful of any exponents. Always double-check your calculations. If you are unsure, work through each step slowly and verify with a calculator. Be patient and remember that mistakes are a part of the learning process. It's important to learn from them.

Expression 1: a³b - 5a³b + 2a³b where a = -6 and b = -1

First, let's simplify the expression by combining like terms. Notice that all the terms have a³b, so we can combine them: a³b - 5a³b + 2a³b = (1 - 5 + 2)a³b = -2a³b. Now, let's substitute the given values for a and b: -2 (-6)³ (-1). Remember that (-6)³ = -216 because a negative number raised to an odd power is negative. So, we have: -2 * (-216) * (-1) = -432. Thus, the value of the expression is -432.

Expression 2: -xy + 6xy - 4xy where x = -7 and y = 1

Combine the like terms: -xy + 6xy - 4xy = (-1 + 6 - 4)xy = 1xy = xy. Now substitute the values of x and y: (-7) * (1) = -7. So the value of this expression is -7.

Expression 3: 0.17ab² - ab² + 0.73ab² where a = -5 and b = -2

Combine the like terms: 0.17ab² - ab² + 0.73ab² = (0.17 - 1 + 0.73)ab² = 0ab² = -0.1ab². Substitute the values of a and b: -0.1 * (-5) * (-2)². Remember that (-2)² = 4. So we have -0.1 * (-5) * 4 = 2. The value is 2.

Expression 4: 4.05x³y² - 5x³y² + 1.95x³y² where x = 2 and y = -2

Combine the like terms: 4.05x³y² - 5x³y² + 1.95x³y² = (4.05 - 5 + 1.95)x³y² = 1x³y² = x³y². Substitute the values of x and y: (2)³ * (-2)². Remember that (2)³ = 8 and (-2)² = 4. So, we have: 8 * 4 = 32. The result is 32.

Tips for Success

• Practice Regularly: The more you practice, the better you'll become at solving these expressions. Try working through different examples and problems.
• Master the Basics: Ensure you have a solid understanding of integers, fractions, decimals, and the order of operations.
• Show Your Work: Always write down each step. This helps you avoid careless errors and makes it easier to identify mistakes.
• Use a Calculator Wisely: Use a calculator to check your work, but don't rely on it too much. Make sure you understand the steps involved.
• Ask for Help: If you get stuck, don't hesitate to ask your teacher, a friend, or look for online resources.

Conclusion

Great job, guys! You've successfully solved a few algebraic expressions. Remember to practice consistently, and you'll become a pro in no time. Algebra might seem intimidating at first, but it becomes much more manageable with practice and a clear understanding of the concepts. Remember to focus on combining like terms, substituting the correct values, and following the order of operations. Keep up the fantastic work, and you'll conquer algebra! If you're still struggling, don't worry; it's normal. Keep practicing, and you'll improve. Good luck, and happy solving!