Simplifying Algebraic Expressions A Step-by-Step Guide To -4(-6x-1)+x

Understanding the Expression

When dealing with algebraic expressions, simplification is a fundamental skill. Simplifying expressions makes them easier to understand and work with. Our main goal is to rewrite the expression in its most concise form, combining like terms and eliminating unnecessary parentheses. In this article, we'll explore the step-by-step process of simplifying the expression $-4(-6x - 1) + x$, ensuring that each step is clear and logically sound. Before we dive into the specifics, let's understand the basic principles of algebraic simplification. We often encounter expressions that can be made simpler by applying the distributive property and combining like terms. The distributive property states that $a(b + c) = ab + ac$, which is crucial when dealing with expressions with parentheses. Combining like terms involves adding or subtracting terms that have the same variable raised to the same power. For example, $3x$ and $5x$ are like terms, but $3x$ and $5x^2$ are not. By understanding these principles, we set the stage for simplifying complex algebraic expressions efficiently. Simplification is not just about arriving at a final answer; it's about understanding the underlying mathematical structure and applying the correct procedures. In the given expression, $-4(-6x - 1) + x$, we have a combination of distribution and combining like terms, which we will tackle systematically. Let's begin by addressing the distributive property, as it's the first operation we need to perform to remove the parentheses. We'll then focus on combining the like terms to achieve the simplest form of the expression. The step-by-step approach ensures clarity and minimizes the chances of errors. So, let's proceed with the simplification process, making sure each step is well-explained and easy to follow.

Step 1: Apply the Distributive Property

The first step in simplifying the expression $-4(-6x - 1) + x$ involves applying the distributive property. The distributive property is a fundamental algebraic rule that allows us to multiply a term outside the parentheses with each term inside the parentheses. In our case, we need to distribute $-4$ across both terms inside the parentheses, which are $-6x$ and $-1$. Let's break down this process further.

When we distribute $-4$ to $-6x$, we perform the multiplication $-4 * -6x$. A negative number multiplied by another negative number results in a positive number. Therefore, $-4 * -6x = 24x$. This is a crucial step because it transforms the expression inside the parentheses into a more manageable form. Next, we distribute $-4$ to $-1$. Again, we multiply $-4$ by $-1$. As before, the product of two negative numbers is positive, so $-4 * -1 = 4$. This completes the distribution process for the terms inside the parentheses. Now, we rewrite the expression after applying the distributive property. The expression $-4(-6x - 1)$ becomes $24x + 4$. Don't forget that we still have the $+ x$ part of the original expression to consider. Thus, after the distribution, our expression looks like $24x + 4 + x$. Applying the distributive property correctly is vital because it sets the stage for the next step: combining like terms. Mistakes in this step can propagate through the rest of the simplification process, leading to an incorrect final answer. To ensure accuracy, double-check each multiplication and pay close attention to the signs. With the distributive property successfully applied, the expression is now in a form where we can easily identify and combine the like terms. This step-by-step breakdown illustrates how each part of the distribution contributes to the overall simplification of the expression. Let's move on to the next step, where we will combine like terms to further simplify the expression.

Step 2: Combine Like Terms

After applying the distributive property, our expression is now $24x + 4 + x$. 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, $24x$ and $x$ are like terms because they both contain the variable $x$ raised to the power of 1. The term $4$ is a constant and does not have any variable, so it is not a like term with $24x$ or $x$. Combining like terms involves adding or subtracting their coefficients. The coefficient of a term is the number that multiplies the variable. In the term $24x$, the coefficient is 24. The term $x$ can be thought of as $1x$, so its coefficient is 1. To combine $24x$ and $x$, we add their coefficients: $24 + 1 = 25$. Therefore, $24x + x$ simplifies to $25x$. Now, let's rewrite the expression with the combined like terms. The expression $24x + 4 + x$ becomes $25x + 4$. The term $4$ remains unchanged because there are no other constant terms to combine it with. Combining like terms is a crucial step in simplifying algebraic expressions. It allows us to reduce the number of terms and present the expression in its most concise form. Misidentifying like terms or incorrectly adding their coefficients can lead to errors in the final result. To avoid mistakes, always double-check that the terms have the same variable and exponent before combining them. In our case, we have successfully combined the $x$ terms, resulting in a simplified expression. At this point, we have performed both the distributive property and the combination of like terms. The expression is now in its simplest form. Let's review the steps we have taken to ensure clarity and accuracy.

Final Simplified Expression

After applying the distributive property and combining like terms, we arrive at the final simplified expression. Let's recap the steps we took to get there. We started with the original expression: $-4(-6x - 1) + x$. First, we applied the distributive property by multiplying $-4$ with both $-6x$ and $-1$ inside the parentheses. This gave us $24x + 4 + x$. Next, we combined like terms. We identified $24x$ and $x$ as like terms and added their coefficients, resulting in $25x$. The constant term $4$ remained unchanged. Therefore, the simplified expression is $25x + 4$. This final form is the most concise representation of the original expression. It is easier to understand and work with in further algebraic manipulations, such as solving equations or evaluating the expression for specific values of $x$. The process of simplification not only reduces the complexity of the expression but also enhances our understanding of its underlying structure. The simplified expression $25x + 4$ consists of two terms: $25x$, which is a variable term, and $4$, which is a constant term. There are no more like terms to combine, and no further distribution is needed. Thus, we can confidently say that we have simplified the expression to its fullest extent. Simplification is a vital skill in algebra, and mastering it allows for efficient problem-solving and a deeper understanding of mathematical concepts. By following the steps of applying the distributive property and combining like terms, we can simplify a wide range of algebraic expressions. The journey from the original expression to the simplified form demonstrates the power of algebraic manipulation and the importance of a systematic approach. With the expression now in its simplest form, we have completed the simplification process.

Key Takeaways

Simplifying algebraic expressions is a fundamental skill in mathematics. To recap, we started with the expression $-4(-6x - 1) + x$ and simplified it to $25x + 4$. This process involved two key steps: applying the distributive property and combining like terms. The distributive property allowed us to remove the parentheses by multiplying $-4$ with both terms inside the parentheses, resulting in $24x + 4 + x$. Combining like terms then involved adding the coefficients of the terms with the same variable, which in this case were $24x$ and $x$, giving us $25x$. The constant term $4$ remained unchanged, leading to the final simplified expression $25x + 4$. The distributive property is crucial when dealing with expressions that contain parentheses. It ensures that each term inside the parentheses is properly multiplied by the term outside. This step is essential for rearranging the expression into a form that is easier to work with. Combining like terms is equally important as it helps to consolidate the expression into its most concise form. It reduces the number of terms and simplifies further algebraic manipulations. When combining like terms, it is important to ensure that the terms have the same variable raised to the same power. Errors in this step can lead to an incorrect final answer. The ability to simplify expressions effectively is valuable not only in mathematics but also in various fields that involve quantitative analysis. It allows for clearer understanding and efficient problem-solving. Mastery of these techniques builds a strong foundation for more advanced algebraic concepts. By consistently applying these steps, one can confidently simplify complex expressions and enhance their mathematical proficiency. Remember, the key to successful simplification lies in a systematic approach and attention to detail. Double-checking each step and understanding the underlying principles can prevent common errors. Simplifying expressions is not just about finding the correct answer; it's about developing a deeper understanding of algebraic structures and operations.

Practice Problems

To solidify your understanding of simplifying algebraic expressions, let's consider some practice problems. Working through these problems will help reinforce the concepts we've discussed and improve your problem-solving skills. Here are a few examples to get you started:

1. Simplify $3(2x + 5) - x$
2. Simplify $-2(4 - 3x) + 7x$
3. Simplify $5(x - 2) + 4(2x + 1)$

For the first problem, $3(2x + 5) - x$, you'll need to apply the distributive property first. Multiply $3$ by both $2x$ and $5$, which gives you $6x + 15$. Then, subtract $x$ from this expression. Identify the like terms and combine them to reach the final simplified form. In the second problem, $-2(4 - 3x) + 7x$, begin by distributing $-2$ across the terms inside the parentheses. This yields $-8 + 6x$. Then, add $7x$ to the resulting expression. Again, identify and combine the like terms to simplify the expression completely. The third problem, $5(x - 2) + 4(2x + 1)$, involves applying the distributive property twice. First, distribute $5$ across $x - 2$, which gives you $5x - 10$. Then, distribute $4$ across $2x + 1$, resulting in $8x + 4$. Combine the results: $5x - 10 + 8x + 4$. Finally, combine the like terms to arrive at the simplified expression. These practice problems cover the same principles we've discussed, including applying the distributive property and combining like terms. Working through these problems will not only enhance your understanding but also build confidence in your ability to simplify algebraic expressions. Remember to take a systematic approach, breaking down each problem into steps and double-checking your work. Practice is key to mastering any mathematical skill, and simplifying expressions is no exception. By consistently working through similar problems, you'll develop a strong intuition for algebraic manipulation and problem-solving.