Simplify -3x + 3 - (5 - 6x) A Step-by-Step Guide

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Understanding Algebraic Expressions

In the realm of mathematics, algebraic expressions form the bedrock of numerous concepts. These expressions are combinations of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. Simplifying these expressions is a fundamental skill that allows us to manipulate and solve equations, making complex problems more manageable. In this article, we will delve into the process of simplifying a specific algebraic expression: $-3x + 3 - (5 - 6x)$.

The Importance of Simplification

Before we dive into the step-by-step simplification, it's crucial to understand why we simplify expressions. Simplification serves several key purposes:

1. Clarity: A simplified expression is easier to understand and interpret. It reduces the complexity and allows us to see the core components more clearly.
2. Efficiency: Simplified expressions are more efficient to work with. They reduce the number of operations required to evaluate or manipulate the expression.
3. Problem Solving: In many mathematical problems, simplifying expressions is a necessary step towards finding a solution. It allows us to isolate variables, combine like terms, and ultimately solve for unknowns.

Now that we understand the importance of simplification, let's proceed with the task at hand.

Step-by-Step Simplification

The expression we aim to simplify is: $-3x + 3 - (5 - 6x)$.

Step 1: Distribute the Negative Sign

The first step in simplifying this expression involves dealing with the parentheses. Specifically, we need to distribute the negative sign that precedes the parentheses. This means multiplying each term inside the parentheses by -1. Let's break it down:

$-3x + 3 - (5 - 6x) = -3x + 3 + (-1)(5) + (-1)(-6x)$

When we multiply -1 by 5, we get -5. And when we multiply -1 by -6x, we get +6x. So, the expression becomes:

$-3x + 3 - 5 + 6x$

This step is crucial because it removes the parentheses and allows us to combine like terms in the subsequent steps.

Step 2: Identify Like Terms

Like terms are terms that have the same variable raised to the same power. In our expression, $-3x + 3 - 5 + 6x$, we can identify two types of like terms:

• Terms with the variable x: -3x and 6x
• Constant terms: 3 and -5

Identifying like terms is a critical step because we can only combine terms that are alike. This is a fundamental rule in algebraic simplification.

Step 3: Combine Like Terms

Now that we have identified the like terms, we can combine them. This involves adding or subtracting the coefficients of the like terms.

Let's start with the terms containing the variable x: -3x and 6x.

$-3x + 6x = (-3 + 6)x = 3x$

Next, we combine the constant terms: 3 and -5.

$3 - 5 = -2$

By combining like terms, we have simplified the expression significantly.

Step 4: Write the Simplified Expression

Finally, we write the simplified expression by combining the results from the previous step. We have:

• Combined x terms: 3x
• Combined constant terms: -2

So, the simplified expression is:

$3x - 2$

This is the most simplified form of the original expression. It is concise, clear, and easy to work with.

Common Mistakes to Avoid

Simplifying algebraic expressions can be tricky, and it's easy to make mistakes if you're not careful. Here are some common mistakes to avoid:

1. Forgetting to Distribute the Negative Sign: This is a common error when dealing with parentheses preceded by a negative sign. Always remember to multiply each term inside the parentheses by -1.
2. Combining Unlike Terms: You can only combine terms that have the same variable raised to the same power. Don't try to combine terms like 3x and -2, as they are not like terms.
3. Incorrectly Adding or Subtracting Coefficients: When combining like terms, make sure you add or subtract the coefficients correctly. Pay attention to the signs (positive or negative) of the coefficients.
4. Skipping Steps: It's tempting to try to simplify expressions in your head, but this can lead to errors. Write out each step clearly to minimize the risk of mistakes.

By being aware of these common mistakes, you can improve your accuracy and confidence in simplifying algebraic expressions.

Practice Makes Perfect

Like any mathematical skill, simplifying expressions requires practice. The more you practice, the more comfortable and confident you will become.

Here are some additional expressions you can try simplifying:

1. $2(x + 3) - 4x$
2. $-5y + 7 - 2(3y - 1)$
3. $4a - 2b + 3(a + b)$

Work through these examples step by step, applying the principles we discussed earlier. Check your answers to ensure you're on the right track.

Conclusion

Simplifying algebraic expressions is a fundamental skill in mathematics. It allows us to make complex expressions more manageable, solve equations, and tackle a wide range of mathematical problems. In this article, we walked through the step-by-step process of simplifying the expression $-3x + 3 - (5 - 6x)$. We learned how to distribute the negative sign, identify like terms, combine like terms, and write the simplified expression. We also discussed common mistakes to avoid and emphasized the importance of practice.

By mastering the art of simplification, you'll be well-equipped to tackle more advanced mathematical concepts and problems. So, keep practicing, and you'll find that simplifying expressions becomes second nature.

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To clarify the request, the keyword is a question asking to simplify the algebraic expression $-3x + 3 - (5 - 6x)$. The essence of the question is how to reduce the given expression into its simplest form by combining like terms and performing the necessary arithmetic operations. This process typically involves distributing any negative signs or coefficients across parentheses, identifying and grouping like terms (terms with the same variable and exponent), and then adding or subtracting the coefficients of these like terms. The goal is to arrive at an equivalent expression that is more concise and easier to understand.

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Simplifying algebraic expressions is a crucial skill in mathematics, acting as a cornerstone for solving more complex equations and problems. The expression $-3x + 3 - (5 - 6x)$ provides an excellent example to illustrate this process. This guide will take you through a detailed, step-by-step approach to simplify this expression, ensuring clarity and understanding at each stage. Mastering this technique will not only help you tackle similar problems but also build a solid foundation for advanced mathematical concepts.

Understanding the Basics of Algebraic Simplification

Before diving into the specifics of this expression, let's understand the fundamental principles of algebraic simplification. At its core, simplification involves reducing an expression to its most basic and understandable form without changing its value. This is typically achieved through a series of operations such as combining like terms, distributing multiplication over addition or subtraction, and removing parentheses. The goal is to make the expression easier to work with and interpret. Algebraic expressions are the building blocks of equations, and simplifying them is often the first step in solving any mathematical problem.

The primary tools for simplification include:

• Distributive Property: This allows you to multiply a single term by each term inside a set of parentheses. For example, a(b + c) becomes ab + ac. This property is crucial when dealing with expressions that have terms enclosed in parentheses.
• Combining Like Terms: Like terms are those that have the same variable raised to the same power (e.g., 3x and -5x are like terms, but 3x and 3x^2 are not). You can combine like terms by adding or subtracting their coefficients. This step reduces the number of terms in the expression, making it simpler.
• Order of Operations (PEMDAS/BODMAS): This ensures that mathematical operations are performed in the correct order: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Following this order is critical to avoid errors in simplification.

By understanding and applying these principles, you can systematically simplify a wide variety of algebraic expressions.

Step-by-Step Simplification of $-3x + 3 - (5 - 6x)$

Now, let's apply these principles to simplify the expression $-3x + 3 - (5 - 6x)$. We will break the process down into manageable steps.

Step 1: Distribute the Negative Sign

The first step in simplifying this expression involves dealing with the parentheses. The negative sign in front of the parentheses acts as a -1 that needs to be distributed across the terms inside. This means multiplying each term within the parentheses by -1. The expression then becomes:

$-3x + 3 - (5 - 6x) = -3x + 3 + (-1 * 5) + (-1 * -6x)$

Multiplying -1 by 5 gives -5, and multiplying -1 by -6x gives +6x. Thus, the expression is now:

$-3x + 3 - 5 + 6x$

This step is essential because it removes the parentheses, allowing us to combine like terms in the subsequent steps. Failing to distribute the negative sign correctly is a common mistake that can lead to an incorrect final answer.

Step 2: Identify Like Terms

The next step is to identify the like terms in the expression. Like terms are terms that contain the same variable raised to the same power or are constants (numbers without variables). In our expression, we can identify two groups of like terms:

• Terms with the variable x: -3x and +6x
• Constant terms: +3 and -5

Recognizing like terms is crucial because you can only combine terms that are alike. This is a fundamental rule of algebra. Mixing unlike terms will lead to incorrect simplification.

Step 3: Combine Like Terms

Now that we've identified the like terms, we can combine them. This involves adding or subtracting the coefficients (the numbers in front of the variables) of the like terms and adding or subtracting the constant terms.

First, let's combine the terms with the variable x:

$-3x + 6x = (-3 + 6)x = 3x$

Next, we combine the constant terms:

$3 - 5 = -2$

By combining these like terms, we've reduced the expression significantly.

Step 4: Write the Simplified Expression

Finally, we write the simplified expression by combining the results from the previous step. We have:

• Combined x terms: 3x
• Combined constant terms: -2

Putting these together, the simplified expression is:

$3x - 2$

This is the most simplified form of the original expression. It's clear, concise, and easier to understand and work with in further mathematical operations.

Common Pitfalls and How to Avoid Them

Simplifying algebraic expressions can be challenging, and there are several common mistakes to watch out for. Recognizing these pitfalls and knowing how to avoid them can greatly improve your accuracy.

1. Incorrect Distribution of the Negative Sign: As mentioned earlier, failing to distribute the negative sign across all terms inside the parentheses is a frequent error. Always remember to multiply each term by -1.
2. Combining Unlike Terms: Only like terms can be combined. For instance, you cannot combine 3x and -2 because one is a variable term, and the other is a constant. Ensure you are only adding or subtracting terms with the same variable and exponent.
3. Arithmetic Errors: Simple addition or subtraction errors can lead to an incorrect answer. Double-check your calculations, especially when dealing with negative numbers.
4. Ignoring the Order of Operations: Ensure you follow the correct order of operations (PEMDAS/BODMAS). Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. Performing operations in the wrong order will result in an incorrect simplification.

To avoid these mistakes, it's helpful to write out each step clearly and double-check your work as you go. Practice is key to mastering these skills and avoiding errors.

The Importance of Practice

Like any mathematical skill, mastering the simplification of algebraic expressions requires practice. The more you practice, the more comfortable and proficient you will become. Start with simple expressions and gradually work your way up to more complex ones. This incremental approach will build your confidence and understanding.

To enhance your practice, try simplifying the following expressions:

1. $4(2x - 1) + 3x$
2. $-2y + 5 - (3 - 4y)$
3. $6a - 2b + 2(a + 3b)$

Work through these examples step by step, applying the principles discussed in this guide. Check your solutions to verify your understanding. Online resources and textbooks often provide additional practice problems and solutions.

Real-World Applications of Simplifying Algebraic Expressions

The ability to simplify algebraic expressions is not just an academic exercise; it has practical applications in various fields. Simplifying expressions is a fundamental skill in engineering, physics, computer science, and economics. In these fields, complex equations and models are often used to describe real-world phenomena. Being able to simplify these equations makes them easier to analyze and solve.

For example:

• Engineering: Simplifying expressions is essential for designing structures, circuits, and systems. Engineers often use algebraic equations to model physical systems and must simplify these equations to optimize designs.
• Physics: Physicists use algebraic expressions to describe motion, forces, and energy. Simplifying these expressions allows them to make predictions and solve problems in mechanics, thermodynamics, and electromagnetism.
• Computer Science: Simplifying expressions is important for writing efficient code and optimizing algorithms. Computer scientists use algebraic techniques to analyze and improve the performance of software and hardware.
• Economics: Economists use algebraic models to analyze economic behavior and make predictions. Simplifying these models is crucial for understanding economic trends and developing policies.

By mastering the simplification of algebraic expressions, you gain a valuable tool that can be applied in a wide range of disciplines.

Conclusion: Mastering the Art of Simplification

In conclusion, simplifying algebraic expressions is a fundamental skill in mathematics with broad applications. This guide has provided a comprehensive, step-by-step approach to simplifying the expression $-3x + 3 - (5 - 6x)$, emphasizing the importance of distributing the negative sign, identifying and combining like terms, and writing the simplified expression. We also discussed common pitfalls and how to avoid them, highlighting the significance of practice and the real-world applications of this skill.

By mastering the art of simplification, you not only enhance your mathematical abilities but also gain a valuable tool that can be applied in various fields. Keep practicing, and you'll find that simplifying expressions becomes an intuitive and essential part of your mathematical toolkit. With a solid understanding of these principles, you will be well-prepared to tackle more complex mathematical challenges.