Solving (x-1)-(3x-4) A Step-by-Step Guide

Let's break down how to solve the algebraic expression (x-1)-(3x-4). This guide is designed to walk you through each step, making it super easy to understand. Whether you're tackling homework or just brushing up on your math skills, we've got you covered! So, grab your pencil and paper, and let's dive in!

Understanding the Basics of Algebraic Expressions

Before we get started, let's quickly recap what algebraic expressions are. Algebraic expressions are mathematical phrases that combine variables (like x), constants (like 1 and 4), and operations (like addition, subtraction, multiplication, and division). Solving these expressions often involves simplifying them to a more manageable form, which is exactly what we're going to do today. Understanding the fundamentals of algebraic expressions is crucial for anyone delving into mathematics. These expressions serve as the building blocks for more complex mathematical concepts and problem-solving techniques. At their core, algebraic expressions are combinations of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. Variables, often represented by letters like x, y, or z, stand for unknown values that can vary. Constants, on the other hand, are fixed numerical values like 1, 2, or 3. Mathematical operations dictate how these variables and constants interact with each other. To effectively solve and simplify algebraic expressions, it's essential to grasp the order of operations, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order ensures that calculations are performed consistently and accurately. For instance, expressions within parentheses should always be evaluated first, followed by exponents, multiplication and division (from left to right), and finally, addition and subtraction (also from left to right). Mastering these basics not only helps in simplifying individual expressions but also lays a solid foundation for tackling more advanced algebraic equations and problems. So, whether you're a student just starting out or someone looking to refresh your math skills, a thorough understanding of algebraic expressions is key to success in mathematics.

Step 1: Distribute the Negative Sign

The first step in solving (x-1)-(3x-4) is to distribute the negative sign in front of the parentheses. Think of it like multiplying the entire expression inside the parentheses by -1. So, we have:

-(3x-4) = -1 * (3x-4) = -3x + 4

Now, our expression looks like this:

(x-1) -3x + 4

Distributing the negative sign is a critical step in simplifying algebraic expressions, especially when dealing with expressions enclosed in parentheses. This process involves multiplying each term inside the parentheses by the negative sign (which is essentially -1). This step is crucial because it alters the signs of the terms within the parentheses, which directly impacts the subsequent steps of simplification. For instance, in the expression (x - 1) - (3x - 4), the negative sign before the second set of parentheses needs to be distributed across both 3x and -4. When we multiply -1 by 3x, we get -3x, and when we multiply -1 by -4, we get +4. This transformation changes the expression from (x - 1) - (3x - 4) to (x - 1) - 3x + 4. Neglecting to distribute the negative sign correctly can lead to significant errors in the final result. This is why it's essential to approach this step with careful attention to detail. Mastering the distribution of negative signs not only aids in accurately simplifying expressions but also builds a strong foundation for more complex algebraic manipulations. So, always remember to take your time and double-check your work when dealing with negative signs in algebraic expressions to ensure accurate and efficient problem-solving.

Step 2: Combine Like Terms

The next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our expression (x-1) - 3x + 4, the like terms are x and -3x, and the constants are -1 and 4. Let's group them together:

(x - 3x) + (-1 + 4)

Now, combine the coefficients of the x terms:

1x - 3x = -2x

And combine the constants:

-1 + 4 = 3

So, our expression simplifies to:

-2x + 3

Combining like terms is a fundamental technique in simplifying algebraic expressions. This process involves identifying and grouping terms that have the same variable raised to the same power, as well as constant terms. By combining like terms, we can reduce the complexity of an expression and make it easier to work with. In essence, like terms are those that can be added or subtracted directly because they represent the same type of quantity. For example, in the expression 3x + 2y - x + 5y, the like terms are 3x and -x, and 2y and 5y. To combine like terms, we simply add or subtract their coefficients (the numbers in front of the variables). So, 3x - x becomes 2x, and 2y + 5y becomes 7y. The simplified expression is then 2x + 7y. Similarly, constant terms, which are just numbers without any variables, can also be combined. For instance, in the expression 2x + 5 - x + 3, the constants are 5 and 3, which combine to give 8. Thus, the simplified expression would be x + 8. Combining like terms is crucial because it allows us to express algebraic expressions in their simplest form, which is essential for solving equations and understanding the relationships between variables. Mastering this skill is a key step in building proficiency in algebra and is applicable in a wide range of mathematical contexts.

Step 3: Final Simplified Expression

After combining like terms, we've arrived at our final simplified expression:

-2x + 3

This expression is now in its simplest form, meaning we can't simplify it any further. You did it! We've successfully solved the expression (x-1)-(3x-4). Awesome work!

Arriving at the final simplified expression is the ultimate goal in the process of simplifying algebraic expressions. Once like terms have been combined and all possible simplifications have been made, the expression is in its most concise and manageable form. This simplified form not only makes the expression easier to understand but also facilitates further mathematical operations, such as solving equations or evaluating the expression for specific values of the variable. The process of simplification is crucial because it reduces the complexity of the expression without changing its underlying mathematical meaning. For example, the expression 2(x + 3) - x + 1 can be simplified step-by-step. First, distribute the 2 across the parentheses to get 2x + 6 - x + 1. Next, combine the like terms 2x and -x to get x, and combine the constants 6 and 1 to get 7. The final simplified expression is x + 7. This simplified form is much easier to work with compared to the original expression. The final simplified expression is not only a result but also a tool. It allows for a clearer understanding of the relationships between variables and constants in the expression. Moreover, it serves as a solid foundation for more complex algebraic manipulations and problem-solving scenarios. So, achieving the final simplified expression is a significant milestone in algebra, showcasing the effectiveness of step-by-step simplification techniques and the power of mathematical precision.

Tips for Mastering Algebraic Expressions

• Practice Makes Perfect: The more you practice, the better you'll become at simplifying expressions. Try different problems and challenge yourself.
• Double-Check Your Work: Always review your steps to make sure you haven't made any mistakes, especially with negative signs.
• Understand the Rules: Make sure you understand the order of operations (PEMDAS) and how to combine like terms.
• Use Resources: There are tons of online resources, videos, and practice problems available to help you learn.

Mastering algebraic expressions is a journey that requires consistent practice and a solid understanding of fundamental concepts. Like any skill, proficiency in algebra comes with time and effort. One of the most effective strategies for improvement is to practice regularly. By tackling a variety of problems, you reinforce your understanding of different techniques and learn to apply them in various contexts. Start with simpler expressions and gradually move on to more complex ones as your confidence grows. Another crucial tip is to always double-check your work. Mathematical errors, especially with negative signs and distribution, are common pitfalls. By reviewing each step carefully, you can catch and correct mistakes before they lead to incorrect answers. It's also essential to have a clear understanding of the rules and conventions of algebra, such as the order of operations (PEMDAS/BODMAS) and the rules for combining like terms. Knowing these rules ensures that you approach problems in a systematic and logical manner. In addition to personal practice, make use of the wealth of resources available both online and offline. There are numerous websites, videos, and textbooks that offer explanations, examples, and practice problems. Don't hesitate to seek help from teachers, tutors, or classmates if you encounter difficulties. Collaborative learning can often provide new perspectives and clarify confusing concepts. By incorporating these tips into your study routine, you'll not only improve your ability to simplify algebraic expressions but also build a strong foundation for further mathematical studies.

Conclusion

And that's it, guys! You've now got a step-by-step guide to solving expressions like (x-1)-(3x-4). Remember to distribute the negative sign, combine like terms, and double-check your work. Keep practicing, and you'll be a math whiz in no time!

In conclusion, solving algebraic expressions is a fundamental skill in mathematics that can be mastered with a systematic approach and consistent practice. Throughout this guide, we've walked through the process step-by-step, starting from understanding the basics of algebraic expressions, such as variables, constants, and operations, to the final simplified form. The key steps involve distributing any negative signs, combining like terms, and ensuring each step is carefully reviewed. Mastering these techniques not only simplifies algebraic expressions but also lays a solid foundation for more advanced mathematical concepts. The ability to manipulate and simplify expressions is crucial for solving equations, understanding functions, and tackling various real-world problems that involve mathematical modeling. Moreover, the skills developed in this process, such as attention to detail, logical thinking, and problem-solving, are valuable in numerous academic and professional contexts. To reinforce your understanding and build confidence, it's essential to practice regularly with a variety of problems. Don't hesitate to revisit the steps outlined in this guide whenever you encounter difficulties. Remember, mathematics is a subject that builds upon itself, so a strong foundation in basic skills like simplifying algebraic expressions is invaluable for future success. By following the tips and strategies discussed, you can enhance your algebraic abilities and approach mathematical challenges with greater confidence and competence.