Simplifying (2x - 3) - (x - 1): A Step-by-Step Guide

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Let's dive into the world of algebra! In this article, we will break down the process of simplifying the algebraic expression (2x - 3) - (x - 1). This is a common type of problem in mathematics, and understanding how to solve it is crucial for building a strong foundation in algebra. Whether you are a student tackling homework or just brushing up on your math skills, this guide will provide you with a clear and easy-to-follow explanation.

Understanding the Basics of Algebraic Expressions

Before we jump into solving the specific expression, let's quickly review some fundamental concepts. Algebraic expressions are combinations of variables (like 'x'), constants (numbers), and mathematical operations (addition, subtraction, multiplication, division). The key to simplifying these expressions lies in combining like terms and following the order of operations. Like terms are terms that have the same variable raised to the same power. For instance, in the expression 2x + 3x - 1, 2x and 3x are like terms because they both contain the variable 'x' raised to the power of 1. The constant -1 is not a like term because it does not have a variable.

When simplifying, we aim to reduce the expression to its simplest form, making it easier to understand and work with. This often involves removing parentheses, combining like terms, and performing any necessary arithmetic operations. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), guides us in which order to perform these operations. Now that we have a grasp of the basics, let's move on to tackling the problem at hand.

Step-by-Step Solution for Simplifying (2x - 3) - (x - 1)

Our main task is to simplify the expression (2x - 3) - (x - 1). Let's break this down into manageable steps:

Step 1: Distribute the Negative Sign

The first step is to deal with the subtraction of the entire expression (x - 1). Subtracting an expression is the same as adding the negative of that expression. So, we need to distribute the negative sign (which is essentially multiplying by -1) to each term inside the second set of parentheses. This means we change the sign of each term within (x - 1). The expression then becomes:

2x - 3 - x + 1

Notice how the '- x' term comes from distributing the negative sign to the 'x' inside the parentheses, and the '+ 1' term comes from distributing the negative sign to the '- 1'. This step is crucial because it allows us to remove the parentheses and combine like terms in the next step. Make sure you understand this distribution process, as it's a common technique used throughout algebra.

Step 2: Identify and Combine Like Terms

Now that we've removed the parentheses, we can focus on combining like terms. Remember, like terms are those that have the same variable raised to the same power. In our expression 2x - 3 - x + 1, the like terms are:

  • 2x and -x (both have the variable 'x' to the power of 1)
  • -3 and +1 (both are constants)

To combine like terms, we simply add or subtract their coefficients (the numbers in front of the variables) and add or subtract the constants. So, we combine 2x and -x: 2x - x = 1x, which is commonly written as just x. Next, we combine the constants: -3 + 1 = -2. After combining the like terms, our expression now looks like this:

x - 2

This simplified form is much cleaner and easier to work with than the original expression. We've successfully reduced the expression by combining the terms that share a common variable or are constants.

Step 3: The Simplified Result

After distributing the negative sign and combining like terms, we arrive at our simplified expression: x - 2. This is the final answer. There are no more like terms to combine, and the expression is in its simplest form. Therefore, (2x - 3) - (x - 1) simplifies to x - 2.

So, the answer is B. x - 2. Understanding these steps is essential for mastering algebraic simplification. Let's recap the key takeaways from this process.

Key Takeaways and Common Mistakes to Avoid

Let's summarize the key steps we took to simplify the expression (2x - 3) - (x - 1):

  1. Distribute the negative sign: This involves multiplying -1 to each term inside the parentheses being subtracted. Remember, subtracting an entire expression is the same as adding its negative. This step is crucial for correctly removing the parentheses and preparing the expression for further simplification.
  2. Identify like terms: Look for terms that have the same variable raised to the same power (e.g., 2x and -x) and constant terms (e.g., -3 and +1). Identifying like terms is a critical step because only like terms can be combined.
  3. Combine like terms: Add or subtract the coefficients of the like terms and add or subtract the constant terms. This reduces the expression to its simplest form. Make sure to pay attention to the signs (positive or negative) of the terms when combining them.

Common Mistakes to Avoid

While the process is straightforward, there are some common mistakes that students often make when simplifying algebraic expressions. Being aware of these pitfalls can help you avoid them and ensure accurate solutions:

  • Forgetting to distribute the negative sign: This is one of the most frequent errors. Always remember that when you subtract an entire expression in parentheses, you need to change the sign of every term inside those parentheses. Failing to do so will lead to an incorrect answer.
  • Incorrectly combining like terms: Only combine terms that have the same variable raised to the same power. For example, you cannot combine x and x², or a constant with a term containing a variable. Mixing up these terms will result in an incorrect simplification.
  • Ignoring the order of operations: PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) is essential. Always perform operations in the correct order to ensure accurate results. In our example, the distribution (which is a form of multiplication) must be done before combining like terms (addition and subtraction).
  • Sign errors: Pay close attention to the signs (positive or negative) of each term when combining them. A simple sign error can change the entire outcome of the simplification.

By understanding these common mistakes, you can be more careful in your approach and minimize the chances of making errors. Practice is key to mastering these concepts, so let's explore some additional examples to solidify your understanding.

Additional Examples and Practice Problems

To further enhance your understanding of simplifying algebraic expressions, let's work through a few more examples. These examples will reinforce the steps we've discussed and help you develop confidence in your skills.

Example 1: Simplify 3(y + 2) - (2y - 1)

  1. Distribute:
    • Distribute the 3 to the terms inside the first parentheses: 3 * y + 3 * 2 = 3y + 6
    • Distribute the negative sign to the terms inside the second parentheses: -1 * 2y + (-1) * (-1) = -2y + 1
    • The expression becomes: 3y + 6 - 2y + 1
  2. Identify like terms:
    • Like terms are: 3y and -2y (terms with the variable y)
    • 6 and 1 (constants)
  3. Combine like terms:
    • Combine the 'y' terms: 3y - 2y = y
    • Combine the constants: 6 + 1 = 7
    • Simplified expression: y + 7

Example 2: Simplify 4x² - 2x + 5 - (x² + 3x - 2)

  1. Distribute:
    • Distribute the negative sign: -1 * x² + (-1) * 3x + (-1) * (-2) = -x² - 3x + 2
    • The expression becomes: 4x² - 2x + 5 - x² - 3x + 2
  2. Identify like terms:
    • Like terms are: 4x² and -x² (terms with x²)
    • -2x and -3x (terms with x)
    • 5 and 2 (constants)
  3. Combine like terms:
    • Combine the 'x²' terms: 4x² - x² = 3x²
    • Combine the 'x' terms: -2x - 3x = -5x
    • Combine the constants: 5 + 2 = 7
    • Simplified expression: 3x² - 5x + 7

Now that you've seen these examples, try working through some practice problems on your own. The more you practice, the more comfortable you'll become with simplifying algebraic expressions. Remember to focus on distributing correctly, identifying like terms, and combining them accurately. Consider trying these practice problems:

  1. Simplify: 2(a - 4) + (3a + 2)
  2. Simplify: 5b² - (2b² - b + 3)
  3. Simplify: -3(x + 1) - 2(x - 4)

By tackling these problems, you'll not only reinforce your understanding but also develop your problem-solving skills in algebra. Don't hesitate to review the steps we've discussed if you encounter any difficulties. Now, let's look at how these simplification skills apply to real-world applications.

Real-World Applications of Simplifying Algebraic Expressions

Simplifying algebraic expressions might seem like an abstract mathematical exercise, but it has numerous practical applications in various real-world scenarios. Understanding how to simplify expressions can help you solve problems in fields ranging from engineering and physics to finance and computer science. Let's explore some specific examples:

1. Engineering and Physics

In engineering and physics, algebraic expressions are used to model physical phenomena, design structures, and calculate forces. For example, when calculating the trajectory of a projectile, engineers use equations that involve variables for initial velocity, angle of launch, and gravitational acceleration. Simplifying these equations can make it easier to predict the projectile's path and ensure accuracy in their calculations. Similarly, in circuit analysis, electrical engineers use algebraic expressions to describe the behavior of circuits. Simplifying these expressions helps them determine voltage, current, and resistance values, which are crucial for designing efficient and safe electrical systems.

2. Finance

In the world of finance, simplifying algebraic expressions is essential for calculating investments, interest rates, and loan payments. For instance, the formula for compound interest involves an exponential expression that can be simplified to analyze different investment scenarios. Financial analysts also use algebraic expressions to model stock prices and predict market trends. By simplifying these expressions, they can gain insights into potential investment opportunities and make informed decisions. Understanding how to manipulate and simplify these expressions can provide a competitive edge in the financial industry.

3. Computer Science

Computer science relies heavily on algebraic expressions for developing algorithms, writing code, and optimizing performance. For example, when designing search algorithms, computer scientists use algebraic expressions to represent the complexity of the algorithm. Simplifying these expressions helps them compare different algorithms and choose the most efficient one. In graphics programming, algebraic expressions are used to transform and manipulate objects in 3D space. Simplifying these expressions can improve the performance of the graphics engine and create smoother visual effects. The ability to simplify algebraic expressions is a fundamental skill for anyone pursuing a career in computer science.

4. Everyday Life

Beyond these specialized fields, simplifying algebraic expressions can also be useful in everyday life. For example, when planning a budget, you might use algebraic expressions to represent your income and expenses. Simplifying these expressions can help you identify areas where you can save money and make informed financial decisions. Similarly, when cooking, you might need to adjust recipe quantities based on the number of servings you want to make. Simplifying algebraic expressions can help you scale recipes accurately and avoid over- or under-producing food. These everyday applications highlight the practical value of understanding algebraic simplification.

In conclusion, simplifying algebraic expressions is not just an abstract mathematical concept; it's a powerful tool that can be applied in a wide range of fields and everyday situations. By mastering the techniques we've discussed, you'll be well-equipped to tackle complex problems and make informed decisions in various aspects of your life. Remember, practice is key, so keep working on simplifying expressions to build your skills and confidence. If you want to delve deeper into algebra and related topics, consider exploring resources like Khan Academy's Algebra Section, which offers comprehensive lessons and practice exercises.