Simplify 9 + 7 + N Step-by-Step Solution

When dealing with mathematical expressions, simplification is a fundamental concept. It involves reducing an expression to its simplest form, making it easier to understand and work with. In essence, we aim to rewrite the expression using fewer terms and operations, while maintaining its original value. This is particularly useful in algebra, where we often encounter expressions with variables and multiple terms. Simplifying expressions is not just a mathematical exercise; it's a skill that enhances problem-solving abilities and lays the foundation for more complex mathematical concepts. The ability to manipulate and simplify expressions efficiently is crucial for success in algebra and beyond.

At the heart of simplification lies the concept of combining like terms. Like terms are terms that have the same variable raised to the same power. For example, in the expression 3x + 2x - 5, the terms 3x and 2x are like terms because they both contain the variable x raised to the power of 1. The term -5 is a constant term and can only be combined with other constant terms. Combining like terms involves adding or subtracting their coefficients (the numerical part of the term) while keeping the variable part the same. In the given example, 3x + 2x simplifies to 5x, so the entire expression becomes 5x - 5. Understanding this basic principle is key to simplifying more complex expressions. Simplification not only makes expressions easier to read but also makes them easier to use in further calculations and problem-solving. By mastering the art of simplification, you'll gain a stronger grasp of algebraic manipulations and be well-prepared for tackling more advanced mathematical concepts. This skill is a cornerstone of mathematical fluency and is invaluable in various fields beyond academics, including engineering, finance, and computer science. The efficiency and accuracy gained through simplification techniques contribute significantly to problem-solving prowess.

In this section, we will walk through the simplification of the expression 9 + 7 + n step by step. This expression involves two constant terms (9 and 7) and a variable term (n). To simplify it, we need to combine the like terms, which in this case are the constant terms. The variable term 'n' remains separate as it cannot be combined with the constants without knowing its value. The process is straightforward but highlights the fundamental principle of combining like terms in algebraic simplification. By understanding this simple example, you can apply the same principle to more complex expressions.

First, identify the like terms in the expression 9 + 7 + n. As mentioned earlier, like terms are those that have the same variable raised to the same power. In this expression, 9 and 7 are like terms because they are both constants. The term n is a variable term and is different from the constants. The next step is to combine these like terms. Combining like terms involves adding or subtracting their coefficients. In this case, we need to add 9 and 7. 9 + 7 equals 16. So, we replace 9 + 7 with 16 in the expression. The expression now becomes 16 + n. This is the simplified form of the expression because there are no more like terms to combine. The constant terms have been added together, and the variable term remains separate. It's important to note that the order of terms does not affect the value of the expression due to the commutative property of addition. Therefore, 16 + n is equivalent to n + 16. This step-by-step approach ensures clarity and minimizes errors in simplification. By breaking down the process into smaller, manageable steps, you can confidently tackle more challenging algebraic expressions. This meticulous approach not only simplifies expressions effectively but also enhances your understanding of the underlying algebraic principles.

The correct simplification of the expression 9 + 7 + n is n + 16. This is achieved by combining the constant terms 9 and 7, which sum up to 16, and then adding the variable term n. The order of terms can be rearranged due to the commutative property of addition, making n + 16 equivalent to 16 + n. The result, n + 16, represents the simplest form of the original expression, where all possible like terms have been combined. This final form is both concise and mathematically accurate.

The options provided include: A. n + 16 B. n + 17 C. 16n

Option A, n + 16, is the correct answer, as it accurately reflects the sum of 9 and 7 (which is 16) combined with the variable n. Option B, n + 17, is incorrect because it suggests that 9 + 7 equals 17, which is false. Option C, 16n, is also incorrect because it implies multiplication between 16 and n, rather than addition. This mistake often occurs when students incorrectly try to combine constants and variables as if they were like terms suitable for multiplication. The key difference lies in recognizing that addition and multiplication are distinct operations. In n + 16, we are adding the constant 16 to the variable n, whereas 16n represents 16 times n. Understanding this distinction is critical for accurate algebraic manipulation. The correct answer, n + 16, maintains the appropriate operation (addition) and accurately combines the constant terms while keeping the variable term separate. This careful attention to detail ensures that the simplified expression is both correct and meaningful. By understanding why other options are incorrect, you reinforce your grasp of the fundamental principles of algebraic simplification.

When simplifying expressions, several common mistakes can lead to incorrect answers. Recognizing and avoiding these pitfalls is crucial for mastering algebraic manipulation. One frequent error is incorrectly combining unlike terms. Unlike terms are terms that do not have the same variable raised to the same power. For instance, in the expression 3x + 2y, 3x and 2y are unlike terms and cannot be combined. Similarly, in 4x^2 + 5x, 4x^2 and 5x are unlike terms because they have different powers of x. Students often mistakenly add or subtract the coefficients of unlike terms, leading to incorrect simplifications. To avoid this, always double-check that the terms you are combining have the exact same variable and exponent. This careful approach ensures accurate simplification.

Another common mistake is misapplying the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Failing to follow the correct order can drastically change the outcome of an expression. For example, in the expression 2 + 3 × 4, multiplication should be performed before addition. So, the correct simplification is 2 + 12 = 14, not (2 + 3) × 4 = 20. Ignoring the order of operations leads to a fundamentally different result. Therefore, always adhere to PEMDAS to ensure accurate calculations. Furthermore, errors in arithmetic, such as incorrect addition or subtraction, can also derail the simplification process. Even a small mistake in basic arithmetic can propagate through the entire expression, leading to a wrong final answer. It is essential to perform each arithmetic operation carefully and double-check your calculations to minimize these errors. Additionally, be mindful of signs, especially when dealing with negative numbers. For example, subtracting a negative number is equivalent to adding a positive number, and errors in sign manipulation are a common source of mistakes. By being vigilant about signs and arithmetic accuracy, you can significantly improve your simplification skills. By consciously avoiding these common mistakes, you will enhance your accuracy and confidence in simplifying algebraic expressions. This attention to detail is a hallmark of mathematical proficiency.

To solidify your understanding of simplifying expressions, it's essential to practice with various problems. Here are a few practice problems to test your skills. These problems are designed to reinforce the concepts discussed earlier, including combining like terms and following the order of operations. Working through these examples will help you identify any areas where you may need additional practice. Practice is key to mastering any mathematical skill, and simplifying expressions is no exception. By engaging with these problems, you'll build confidence and proficiency in algebraic manipulation.

1. Simplify the expression: 5x + 3 + 2x - 1
2. Simplify the expression: 10y - 4 + 3y + 2
3. Simplify the expression: 4(a + 2) - 3a
4. Simplify the expression: 7b + 2(b - 1)

For the first problem, 5x + 3 + 2x - 1, the first step is to identify and combine like terms. The like terms are 5x and 2x, which combine to give 7x. The constant terms are 3 and -1, which combine to give 2. Therefore, the simplified expression is 7x + 2. This problem highlights the importance of correctly identifying like terms and combining them accurately. For the second problem, 10y - 4 + 3y + 2, the like terms are 10y and 3y, which combine to give 13y. The constant terms are -4 and 2, which combine to give -2. So, the simplified expression is 13y - 2. This problem reinforces the process of combining like terms, including both variable and constant terms. The third problem, 4(a + 2) - 3a, involves the distributive property. First, distribute the 4 across the terms inside the parentheses: 4a + 8. Then, subtract 3a: 4a + 8 - 3a. Finally, combine like terms 4a and -3a to get a, resulting in the simplified expression a + 8. This problem emphasizes the importance of following the order of operations and applying the distributive property correctly. The fourth problem, 7b + 2(b - 1), also involves the distributive property. Distribute the 2 across the terms inside the parentheses: 7b + 2b - 2. Then, combine like terms 7b and 2b to get 9b, resulting in the simplified expression 9b - 2. This problem further reinforces the use of the distributive property and combining like terms. By working through these practice problems and their solutions, you can reinforce your understanding of simplifying expressions and build the skills necessary to tackle more complex algebraic problems. Consistent practice is the key to mastering this fundamental mathematical concept.

By practicing these problems, you can reinforce your understanding and improve your skills in simplifying expressions.