Solving Algebraic Expressions Finding The Difference

by THE IDEN 53 views

In the realm of algebra, mastering the manipulation of expressions is crucial. This article delves into the process of finding the difference between algebraic terms, focusing on combining like terms and simplifying expressions. We will dissect a series of examples, providing a comprehensive understanding of the underlying principles. These problems cover fundamental algebraic operations, including combining like terms with both positive and negative coefficients. By understanding these concepts, you will build a solid foundation for more advanced algebraic manipulations. Let's embark on this journey of algebraic exploration, where we'll unravel the nuances of combining terms and simplifying expressions.

6. -16a² - (-7a²)

Understanding the Problem: This problem involves subtracting a negative term from another term. The key here is to remember that subtracting a negative number is the same as adding its positive counterpart. This is a fundamental concept in algebra that is essential for simplifying expressions. When dealing with algebraic expressions, it's crucial to pay close attention to the signs of the terms. A simple sign error can lead to a completely different result. In this specific case, we are dealing with terms that have the same variable and exponent, which means they are like terms and can be combined. Combining like terms is a core skill in algebra, allowing us to simplify complex expressions into more manageable forms.

Step-by-Step Solution:

  1. Rewrite the subtraction of a negative as addition: -16a² - (-7a²) becomes -16a² + 7a²
    • This step is crucial because it transforms the subtraction operation into an addition operation, which is often easier to handle. By changing the signs appropriately, we avoid potential errors in the calculation. This transformation is based on the mathematical principle that subtracting a negative number is equivalent to adding its positive counterpart. Understanding this principle is fundamental to simplifying algebraic expressions involving negative numbers.
  2. Combine like terms: -16a² + 7a² = (-16 + 7)a²
    • Here, we combine the coefficients of the like terms. Like terms are terms that have the same variable raised to the same power. In this case, both terms have the variable 'a' raised to the power of 2, making them like terms. To combine like terms, we simply add or subtract their coefficients. The coefficient is the numerical factor that multiplies the variable. In this step, we are essentially factoring out the common term 'a²' and adding the coefficients. This process is a fundamental technique in simplifying algebraic expressions.
  3. Simplify the coefficients: (-16 + 7)a² = -9a²
    • This step involves performing the addition within the parentheses. Adding -16 and 7 gives us -9. The result is then multiplied by the variable term 'a²'. This final simplification provides the answer in its most concise form. The ability to perform arithmetic operations with negative numbers is essential for success in algebra. This step demonstrates the importance of accurately applying the rules of addition and subtraction to arrive at the correct answer.

Final Answer: -9a²

This result represents the simplified form of the original expression. By understanding the steps involved in combining like terms and handling negative signs, you can confidently tackle similar algebraic problems.

7. 9m³ - 14m³

Initial Assessment: This problem requires subtracting one term from another, both of which contain the variable 'm' raised to the power of 3. These are like terms, meaning they can be combined directly. The challenge lies in performing the subtraction of the coefficients, one of which is larger than the other, resulting in a negative value. Understanding the concept of like terms is crucial here. Like terms are terms that have the same variable raised to the same power. Only like terms can be combined through addition or subtraction. This problem provides a straightforward example of how to combine like terms with different coefficients.

Step-by-Step Solution:

  1. Identify like terms: 9m³ and -14m³ are like terms.
    • This step is fundamental because it establishes that the two terms can be combined. Both terms have the same variable 'm' raised to the power of 3, satisfying the condition for like terms. Recognizing like terms is the first step in simplifying algebraic expressions. This identification allows us to proceed with combining the coefficients, which is the next step in the solution.
  2. Combine the coefficients: 9 - 14 = -5
    • Here, we perform the subtraction operation on the coefficients of the like terms. Subtracting 14 from 9 results in -5. This step is a simple arithmetic operation, but it's crucial for obtaining the correct answer. The ability to perform arithmetic operations accurately is essential in algebra. This step demonstrates the importance of paying attention to the signs of the numbers involved in the calculation.
  3. Write the result with the variable term: -5m³
    • This step combines the result of the coefficient subtraction with the variable term. The -5 becomes the new coefficient of the m³ term. This step completes the simplification process, providing the final answer. The final answer represents the simplified form of the original expression. This step highlights the importance of maintaining the variable term throughout the calculation and attaching it to the final coefficient.

Final Answer: -5m³

The difference between 9m³ and 14m³ is -5m³. This problem highlights the importance of correctly subtracting coefficients, especially when dealing with positive and negative numbers.

8. -15x - 8x

Problem Overview: This problem involves subtracting two terms, both containing the variable 'x'. The presence of negative coefficients requires careful attention to the rules of integer subtraction. The core concept here is combining like terms, which are terms that share the same variable raised to the same power. This problem provides an excellent opportunity to practice combining like terms with negative coefficients. Mastering this skill is crucial for simplifying more complex algebraic expressions.

Detailed Solution:

  1. Recognize like terms: -15x and -8x are like terms.
    • The first step is to identify that both terms have the same variable 'x' raised to the power of 1. This confirms that they are like terms and can be combined. Recognizing like terms is the foundation of simplifying algebraic expressions. This identification allows us to proceed with combining the coefficients, which is the next step in the solution.
  2. Combine coefficients: -15 - 8 = -23
    • In this step, we subtract the coefficients of the like terms. Subtracting 8 from -15 results in -23. This is a straightforward arithmetic operation, but it's essential to get it right. The ability to perform arithmetic operations accurately is crucial in algebra. This step demonstrates the importance of paying attention to the signs of the numbers involved in the calculation.
  3. Form the final term: -23x
    • The final step is to combine the result from the previous step with the variable 'x'. This gives us the simplified term -23x. This is the final answer to the problem. The final answer represents the simplified form of the original expression. This step highlights the importance of maintaining the variable term throughout the calculation and attaching it to the final coefficient.

Solution: -23x

The simplified expression is -23x. This example reinforces the rule of adding negative numbers when subtracting.

9. 8y - (-9y)

Problem Setup: This problem presents a subtraction involving a negative term. The key to solving this is understanding that subtracting a negative is equivalent to adding a positive. This transformation is crucial for simplifying the expression correctly. The problem also involves like terms, as both terms contain the variable 'y' raised to the power of 1. This allows us to combine the coefficients directly after handling the negative sign.

Step-by-Step Solution:

  1. Rewrite subtraction of a negative as addition: 8y - (-9y) becomes 8y + 9y
    • This step is crucial because it transforms the subtraction operation into an addition operation. Subtracting a negative number is the same as adding its positive counterpart. This transformation simplifies the expression and reduces the chance of errors. This step is based on a fundamental mathematical principle that is essential for algebraic manipulation.
  2. Identify like terms: 8y and 9y are like terms.
    • Both terms have the same variable 'y' raised to the same power (1), making them like terms. Recognizing like terms is the first step in simplifying algebraic expressions. This identification allows us to proceed with combining the coefficients, which is the next step in the solution.
  3. Combine the coefficients: 8 + 9 = 17
    • Here, we add the coefficients of the like terms. Adding 8 and 9 results in 17. This is a simple arithmetic operation, but it's crucial for obtaining the correct answer. The ability to perform arithmetic operations accurately is essential in algebra. This step demonstrates the importance of paying attention to the signs of the numbers involved in the calculation.
  4. Write the final term: 17y
    • This step combines the result from the previous step with the variable 'y'. This gives us the simplified term 17y. This is the final answer to the problem. The final answer represents the simplified form of the original expression. This step highlights the importance of maintaining the variable term throughout the calculation and attaching it to the final coefficient.

Final Result: 17y

The simplified expression is 17y. This problem effectively demonstrates the rule of subtracting a negative and its conversion to addition.

10. -n - 6n

Initial Analysis: This problem involves subtracting one term from another, both of which contain the variable 'n'. The coefficients are both negative, which requires careful attention to the rules of integer subtraction. This problem is a good example of how to combine like terms when both terms have negative coefficients. Mastering this skill is crucial for simplifying more complex algebraic expressions.

Detailed Solution Steps:

  1. Identify like terms: -n and -6n are like terms.
    • Both terms have the same variable 'n' raised to the power of 1. This confirms that they are like terms and can be combined. Recognizing like terms is the foundation of simplifying algebraic expressions. This identification allows us to proceed with combining the coefficients, which is the next step in the solution.
  2. Rewrite -n as -1n: -n - 6n becomes -1n - 6n
    • This step is helpful for clarity, as it explicitly shows the coefficient of the first term as -1. This can prevent errors in the next step. This step is based on the understanding that a variable without a visible coefficient has an implied coefficient of 1. This clarification can be particularly helpful when dealing with negative coefficients.
  3. Combine coefficients: -1 - 6 = -7
    • In this step, we subtract the coefficients of the like terms. Subtracting 6 from -1 results in -7. This is a straightforward arithmetic operation, but it's essential to get it right. The ability to perform arithmetic operations accurately is crucial in algebra. This step demonstrates the importance of paying attention to the signs of the numbers involved in the calculation.
  4. Form the final term: -7n
    • The final step is to combine the result from the previous step with the variable 'n'. This gives us the simplified term -7n. This is the final answer to the problem. The final answer represents the simplified form of the original expression. This step highlights the importance of maintaining the variable term throughout the calculation and attaching it to the final coefficient.

Final Answer: -7n

The simplified expression is -7n. This demonstrates combining negative coefficients in subtraction.

By working through these examples, you've gained valuable practice in simplifying algebraic expressions. Remember to always combine like terms and pay close attention to the signs of the coefficients. With consistent practice, you'll master these fundamental algebraic skills and be well-prepared for more advanced concepts.