Solving Equations, Inequalities, And Expressions A Comprehensive Guide

Let's delve into solving equations for the variable 'n'. Understanding how to isolate 'n' is a fundamental skill in algebra. We'll analyze the given equations and break down the steps to find the solution. It's crucial to remember that the goal is to get 'n' by itself on one side of the equation. This involves using inverse operations to undo the operations that are being performed on 'n'. The key to success lies in maintaining balance – whatever operation you perform on one side of the equation, you must perform the same operation on the other side to keep the equation true. Mastering these techniques will not only help solve simple equations but also build a solid foundation for tackling more complex algebraic problems.

B. n - 100 = 107

In this equation, our aim is to isolate 'n'. The equation states that 'n' minus 100 equals 107. To get 'n' alone, we need to undo the subtraction of 100. The inverse operation of subtraction is addition. Therefore, we add 100 to both sides of the equation. This maintains the balance of the equation and allows us to isolate 'n'. When we add 100 to the left side (n - 100 + 100), the -100 and +100 cancel each other out, leaving us with just 'n'. On the right side, we add 100 to 107, resulting in 207. Thus, the equation becomes n = 207. This tells us that the value of 'n' that satisfies the equation is 207. To verify our solution, we can substitute 207 back into the original equation: 207 - 100 = 107, which is true. This confirms that our solution is correct. This method of isolating the variable by using inverse operations is a cornerstone of solving algebraic equations. Understanding and applying this principle allows us to systematically solve for unknowns in various mathematical contexts.

C. 100 + 107 = n

This equation presents a straightforward scenario where 'n' is already isolated on one side. The equation states that the sum of 100 and 107 equals 'n'. To find the value of 'n', we simply need to perform the addition. Adding 100 and 107 gives us 207. Therefore, n = 207. This equation highlights a basic yet fundamental concept in algebra: the direct calculation of a variable when it's already isolated. There are no inverse operations required here, making it a relatively simple equation to solve. However, it's essential to recognize this type of equation, as it reinforces the understanding that the value of a variable can be directly determined when it's expressed as the result of a given arithmetic operation. This clarity in recognizing different equation structures is a crucial step in mastering algebraic problem-solving. The simplicity of this equation also serves as a reminder that sometimes, the solution is readily available with a direct calculation.

D. n + 100 = 107

In this equation, we again need to isolate 'n'. The equation states that 'n' plus 100 equals 107. To isolate 'n', we must undo the addition of 100. The inverse operation of addition is subtraction. Therefore, we subtract 100 from both sides of the equation. This maintains the balance of the equation and allows us to isolate 'n'. When we subtract 100 from the left side (n + 100 - 100), the +100 and -100 cancel each other out, leaving us with just 'n'. On the right side, we subtract 100 from 107, resulting in 7. Thus, the equation becomes n = 7. This tells us that the value of 'n' that satisfies the equation is 7. To verify our solution, we can substitute 7 back into the original equation: 7 + 100 = 107, which is true. This confirms that our solution is correct. This method of isolating the variable by using inverse operations is a critical skill in algebra. The ability to systematically solve for unknowns in equations is a cornerstone of mathematical proficiency. Recognizing the inverse relationship between operations is key to efficiently and accurately solving for variables.

Moving on to inequalities, it's essential to understand that inequalities express a range of possible values for a variable, rather than a single value as in equations. To determine whether a given value is a solution to an inequality, we substitute the value for the variable and check if the resulting statement is true. Understanding inequalities is crucial because they are used extensively in various mathematical fields and real-world applications, such as optimization problems, constraint modeling, and data analysis. Being able to interpret and solve inequalities broadens our problem-solving capabilities and allows us to model and analyze situations where values are not fixed but rather fall within certain limits. This ability to work with ranges of values is a fundamental skill in mathematical reasoning.

3. y < -4 if y = 7

Here, we are given the inequality y < -4 and asked to determine if y = 7 is a solution. To do this, we substitute 7 for y in the inequality, which gives us 7 < -4. Now we need to evaluate whether this statement is true. On the number line, 7 is to the right of -4, meaning 7 is greater than -4. Therefore, the statement 7 < -4 is false. This indicates that y = 7 is not a solution to the inequality y < -4. Understanding the number line and the relationships between numbers is essential when dealing with inequalities. A number is less than another if it lies to the left on the number line. Recognizing this relationship allows us to quickly determine the truth value of an inequality when substituting a value for a variable. This process of substitution and evaluation is the foundation for solving more complex inequalities and understanding their solutions.

Evaluating expressions involves substituting given values for variables and performing the indicated operations. This skill is a fundamental aspect of algebra, serving as the building block for simplifying complex equations and functions. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is crucial in ensuring accurate evaluation. Mastering the evaluation of expressions is essential because it not only reinforces basic arithmetic skills but also develops a deeper understanding of how variables and operations interact in mathematical statements. This understanding is vital for success in more advanced mathematical concepts, where expressions are frequently manipulated and evaluated in various contexts.

4. 45 - 5n, Let n = 3

In this expression, we need to substitute the value of 'n' with 3 and then perform the operations. The expression is 45 - 5n. Substituting n = 3, we get 45 - 5(3). According to the order of operations (PEMDAS), we perform multiplication before subtraction. So, we multiply 5 by 3, which gives us 15. Now the expression becomes 45 - 15. Subtracting 15 from 45, we get 30. Therefore, the value of the expression 45 - 5n when n = 3 is 30. This example highlights the importance of adhering to the order of operations. Failure to do so can lead to incorrect results. The step-by-step process of substitution and then performing the operations in the correct order is a key skill in evaluating algebraic expressions. This method ensures accuracy and builds a solid foundation for more complex algebraic manipulations.

5. (n + 6.8) * 10, Let n = 3

Here, we have an expression involving parentheses, addition, and multiplication. We are given that n = 3, so we substitute 3 for 'n' in the expression (n + 6.8) * 10. This gives us (3 + 6.8) * 10. According to the order of operations (PEMDAS), we first perform the operation inside the parentheses. Adding 3 and 6.8 gives us 9.8. Now the expression becomes 9.8 * 10. Multiplying 9.8 by 10, we get 98. Therefore, the value of the expression (n + 6.8) * 10 when n = 3 is 98. This example further emphasizes the importance of following the order of operations. The parentheses act as a grouping symbol, indicating that the addition within them must be performed before the multiplication. Accurate application of PEMDAS is crucial for obtaining the correct result. This process of substitution and step-by-step evaluation reinforces the understanding of how different operations interact within an expression.

6. (25 - n) + 11, Let n = 3

In this expression, we have parentheses, subtraction, and addition. We substitute n = 3 into the expression (25 - n) + 11, which gives us (25 - 3) + 11. Following the order of operations (PEMDAS), we first perform the operation inside the parentheses. Subtracting 3 from 25 gives us 22. Now the expression becomes 22 + 11. Adding 22 and 11, we get 33. Therefore, the value of the expression (25 - n) + 11 when n = 3 is 33. This example reinforces the concept of prioritizing operations within parentheses. The parentheses group the subtraction operation, indicating that it must be performed before the addition. The step-by-step approach, adhering to PEMDAS, ensures accurate evaluation. Consistency in applying the order of operations is a key to success in algebraic manipulations.

7. n^2 + 6, Let n = 3

This expression involves an exponent and addition. We substitute n = 3 into the expression n^2 + 6, which gives us 3^2 + 6. According to the order of operations (PEMDAS), we evaluate exponents before addition. 3^2 means 3 squared, which is 3 multiplied by itself (3 * 3), resulting in 9. Now the expression becomes 9 + 6. Adding 9 and 6, we get 15. Therefore, the value of the expression n^2 + 6 when n = 3 is 15. This example highlights the importance of understanding exponents and their place in the order of operations. Exponents indicate repeated multiplication, and they must be evaluated before addition or subtraction. Mastery of exponents is essential for working with polynomial expressions and more advanced mathematical concepts.

The discussion category for these problems is mathematics. These problems cover fundamental concepts in algebra, including solving equations, inequalities, and evaluating expressions. These are core topics in mathematics education, serving as the foundation for more advanced studies in algebra, calculus, and other branches of mathematics. A strong understanding of these concepts is essential for students pursuing STEM fields and for anyone seeking to develop analytical and problem-solving skills. Discussing these concepts in a collaborative environment can deepen understanding and help students develop their mathematical reasoning abilities.