Solving The Algebraic Expression (−3.4x + 4.7) Minus (3 + 2.9x)
This article delves into solving a fundamental algebraic problem: finding the difference between two expressions. Specifically, we will explore the steps involved in subtracting (3 + 2.9x) from (−3.4x + 4.7). This type of problem is common in introductory algebra and requires a solid understanding of combining like terms and the distributive property. We will break down the process step-by-step, making it easy to follow along and grasp the underlying concepts. By the end of this article, you'll not only know the correct answer but also understand the methodology behind solving similar algebraic subtraction problems.
Understanding the Problem: Subtracting Algebraic Expressions
To begin, let's clarify what it means to subtract one algebraic expression from another. In mathematics, subtraction involves finding the difference between two quantities. When dealing with algebraic expressions, this means we need to subtract each term in the second expression from the corresponding term in the first expression. The key here is to correctly handle the signs and combine only the like terms. Like terms are terms that have the same variable raised to the same power. For instance, 3x and 2.9x are like terms because they both have the variable x raised to the power of 1. Similarly, constants like 4.7 and 3 are also like terms.
The expression we are working with is (−3.4x + 4.7) − (3 + 2.9x). The first step in solving this is to distribute the negative sign in front of the second parenthesis. This is crucial because it changes the sign of each term inside the parenthesis. Essentially, we are multiplying each term inside the second parenthesis by -1. This process is based on the distributive property, which states that a(b + c) = ab + ac. In our case, we are distributing -1 across the terms 3 and 2.9x.
After distributing the negative sign, the expression will look different, and we will be ready to combine the like terms. This involves adding or subtracting the coefficients of the like terms. The coefficient is the numerical part of a term. For example, in the term -3.4x, the coefficient is -3.4. By correctly identifying and combining like terms, we simplify the expression to its final form. This process ensures that we arrive at the correct difference between the two original expressions.
Step-by-Step Solution: Unpacking the Algebraic Subtraction
Now, let's walk through the step-by-step solution to the problem: (−3.4x + 4.7) − (3 + 2.9x). This methodical approach will help you understand each operation and why it is performed. Breaking down the problem into manageable steps makes it easier to follow and reduces the chance of making errors.
The first crucial step is to distribute the negative sign in front of the second parenthesis, (3 + 2.9x). This means we multiply each term inside the parenthesis by -1. So, the expression (3 + 2.9x) becomes -3 - 2.9x. This step is essential because it ensures that we are subtracting both the constant term and the term with the variable x. If we skip this step or perform it incorrectly, the entire solution will be flawed.
Once we distribute the negative sign, the original expression (−3.4x + 4.7) − (3 + 2.9x) transforms into −3.4x + 4.7 − 3 − 2.9x. Notice how the signs of both 3 and 2.9x have changed. Now we have a single expression with four terms, and we are ready to combine the like terms. This is the next key step in simplifying the expression.
To combine like terms, we group together terms that have the same variable and terms that are constants. In our expression, −3.4x and −2.9x are like terms because they both contain the variable x. Similarly, 4.7 and −3 are like terms because they are both constants. The process of combining like terms involves adding or subtracting their coefficients. We will add the coefficients of the x terms and then add the constant terms separately. This ensures that we simplify the expression correctly and arrive at the final answer.
Combining Like Terms: Simplifying the Expression
After distributing the negative sign, we have the expression: −3.4x + 4.7 − 3 − 2.9x. Now, we need to combine the like terms to simplify this expression. This involves grouping the terms with the variable x together and the constant terms together. Combining like terms is a fundamental skill in algebra, and it is essential for simplifying expressions and solving equations. It allows us to reduce the complexity of an expression and make it easier to work with.
Let's start by combining the terms with the variable x: −3.4x and −2.9x. To do this, we add their coefficients. The coefficients are -3.4 and -2.9. Adding these gives us -3.4 + (-2.9) = -6.3. Therefore, when we combine these terms, we get −6.3x. This new term represents the combined effect of the two original x terms. It is a simplified form that is easier to manage in further calculations.
Next, we combine the constant terms: 4.7 and −3. To do this, we simply add them together: 4.7 + (-3) = 1.7. So, the combined constant term is 1.7. This is a straightforward addition operation that results in a single constant value.
By combining the like terms, we have simplified the expression to −6.3x + 1.7. This is the final simplified form of the original expression after performing the subtraction. It consists of a single term with the variable x and a constant term. This simplified expression is much easier to interpret and use in further mathematical operations.
Final Answer and Options: Identifying the Correct Solution
After combining like terms, we arrived at the simplified expression: −6.3x + 1.7. This is the final result of subtracting (3 + 2.9x) from (−3.4x + 4.7). Now, we need to compare this result with the given options to identify the correct answer. This step is crucial to ensure that we select the right solution from the provided choices.
The options given are:
- A)
−0.5x + 1.7 - B)
−6.3x − 1.7 - C)
−0.5x − 1.7 - D)
−6.3x + 1.7
By comparing our simplified expression, −6.3x + 1.7, with the options, we can see that it matches option D exactly. Option D also has the term −6.3x and the constant term +1.7. This confirms that option D is the correct solution to the problem.
Options A, B, and C are incorrect because they do not match our simplified expression. Option A has the wrong coefficient for the x term, option B has the correct coefficient for the x term but the wrong sign for the constant term, and option C has both the wrong coefficient for the x term and the wrong sign for the constant term. Therefore, by carefully comparing our result with the options, we can confidently select the correct answer, which is D) −6.3x + 1.7.
Common Mistakes to Avoid: Ensuring Accuracy in Algebraic Subtraction
When solving algebraic subtraction problems, there are several common mistakes that students often make. Understanding these pitfalls and how to avoid them can significantly improve accuracy and confidence in algebra. Let's explore some of these common errors and how to sidestep them.
One of the most frequent mistakes is failing to distribute the negative sign correctly. As we discussed earlier, when subtracting an expression in parentheses, you must multiply each term inside the parentheses by -1. For example, in the problem (−3.4x + 4.7) − (3 + 2.9x), the negative sign in front of (3 + 2.9x) must be distributed to both 3 and 2.9x. This means 3 becomes -3 and 2.9x becomes -2.9x. A common error is to only change the sign of the first term inside the parentheses or to forget to distribute the negative sign altogether. To avoid this, always double-check that you have correctly distributed the negative sign to each term inside the parentheses.
Another common mistake is incorrectly combining like terms. Like terms are terms that have the same variable raised to the same power. For example, 3x and 2.9x are like terms, but 3x and 2.9x² are not. When combining like terms, you add or subtract their coefficients, but the variable and its exponent remain the same. A frequent error is to add or subtract terms that are not like terms or to change the exponent of the variable when combining terms. To avoid this, carefully identify like terms and only combine those. Ensure that you are only adding or subtracting the coefficients and that the variable and its exponent stay the same.
Sign errors are also a common source of mistakes in algebraic subtraction. These errors can occur when distributing the negative sign, combining like terms, or performing arithmetic operations. A simple sign error can change the entire outcome of the problem. To prevent sign errors, it's crucial to pay close attention to the signs of each term and to double-check your work. Writing each step clearly and carefully can help reduce the likelihood of making sign errors.
Finally, skipping steps or trying to do too much mentally can lead to mistakes. Algebra problems often involve multiple steps, and it's essential to perform each step meticulously. Skipping steps can cause you to miss crucial operations or make errors in your calculations. To avoid this, write out each step clearly and take your time. This approach will help you stay organized and reduce the chance of making mistakes. By avoiding these common pitfalls, you can significantly improve your accuracy and confidence in solving algebraic subtraction problems.
Practice Problems: Strengthening Your Understanding
To solidify your understanding of algebraic subtraction, it's essential to practice with various problems. The more you practice, the more comfortable and confident you will become with the process. Here are a few practice problems that you can try. Working through these problems will help you reinforce the concepts we've discussed and develop your problem-solving skills.
Practice Problem 1: Simplify the expression (5x − 2) − (2x + 3). This problem is similar to the one we solved in the article, so it's a great starting point. Remember to distribute the negative sign correctly and then combine like terms. Pay close attention to the signs and ensure that you are only combining terms that have the same variable.
Practice Problem 2: Simplify the expression (−4.2y + 1.5) − (2.8y − 3.1). This problem involves decimals, which can sometimes make the calculations a bit trickier. However, the process is the same. Distribute the negative sign, combine like terms, and be careful with your arithmetic.
Practice Problem 3: Simplify the expression (7a − 4b + 2) − (3a + 2b − 1). This problem introduces two variables, a and b. The key here is to combine the like terms for each variable separately. Distribute the negative sign, then combine the a terms, the b terms, and the constant terms.
Practice Problem 4: Simplify the expression (1/2x + 3/4) − (1/4x − 1/2). This problem involves fractions, which can be challenging for some students. Remember the rules for adding and subtracting fractions. You'll need to find a common denominator before combining the like terms.
After you've attempted these problems, it's a good idea to check your answers. You can do this by comparing your solutions with the steps outlined in this article or by using online calculators or resources. If you encounter any difficulties, review the steps and explanations in the article. Practice is the key to mastering algebraic subtraction, so keep working at it, and you'll see improvement over time. Each problem you solve will build your skills and confidence in algebra.
Conclusion: Mastering Algebraic Subtraction
In conclusion, solving the algebraic problem (−3.4x + 4.7) − (3 + 2.9x) involves several key steps: distributing the negative sign, combining like terms, and simplifying the expression. We have walked through each of these steps in detail, providing clear explanations and examples to help you understand the process thoroughly. By correctly applying these steps, we arrived at the solution: −6.3x + 1.7, which corresponds to option D.
Algebraic subtraction is a fundamental skill in mathematics, and mastering it is essential for success in more advanced topics. This article has provided a comprehensive guide to solving this type of problem, from understanding the basic concepts to avoiding common mistakes. We have highlighted the importance of distributing the negative sign correctly, combining like terms accurately, and paying close attention to signs throughout the process. By following these guidelines, you can improve your accuracy and confidence in solving algebraic subtraction problems.
We have also discussed common mistakes that students often make and provided tips on how to avoid them. These include being careful with sign errors, not skipping steps, and ensuring that you are only combining like terms. By being aware of these potential pitfalls, you can proactively prevent them and improve your problem-solving skills.
Finally, we have included several practice problems to help you strengthen your understanding. Practice is crucial for mastering any mathematical skill, and algebraic subtraction is no exception. By working through these problems, you can reinforce the concepts we've covered and develop your ability to apply them in different contexts.
Remember, algebraic subtraction is not just about finding the right answer; it's also about understanding the underlying principles and developing a systematic approach to problem-solving. With practice and a clear understanding of the steps involved, you can master algebraic subtraction and build a strong foundation for your future studies in mathematics.
