1. Solve For B: $6 + 3b = -18$. 2. Solve For N: $7n + 12 = -23$. 3. Solve For F: $-12 = 8 + \frac{f}{2}$. 4. Solve For K: $\frac{k}{4} + 6 = -2$. 5. Solve For P: $16 - 3p = 34$.

This article will delve into the process of solving linear equations, providing step-by-step solutions and explanations for a variety of examples. Linear equations are fundamental in mathematics and have widespread applications in various fields, including science, engineering, and economics. Understanding how to solve them is crucial for anyone pursuing studies or careers in these areas. We will explore how to isolate variables, apply inverse operations, and verify solutions. Whether you're a student learning algebra or someone looking to refresh your math skills, this guide offers a clear and concise approach to mastering linear equations.

1. Solving the Equation $6 + 3b = -18$

Linear equations often involve isolating the variable to find its value, and this equation, $6 + 3b = -18$, is a prime example. To solve for b, we need to undo the operations that are being applied to it. The first step in solving this linear equation is to isolate the term containing the variable. In this case, we need to get the term 3b by itself on one side of the equation. This can be achieved by subtracting 6 from both sides of the equation. Subtracting 6 from both sides maintains the equality and helps us move closer to isolating b. When we subtract 6 from both sides, the equation transforms from $6 + 3b = -18$ to $3b = -18 - 6$. Simplifying the right side, we get $3b = -24$. This step is crucial because it simplifies the equation and brings us closer to solving for b. The next critical step in solving this algebraic equation is to eliminate the coefficient of the variable. In our equation, $3b = -24$, the variable b is being multiplied by 3. To isolate b, we need to perform the inverse operation, which is division. Dividing both sides of the equation by 3 will undo the multiplication and leave b by itself. When we divide both sides by 3, the equation becomes $\frac{3b}{3} = \frac{-24}{3}$. Simplifying both sides, we find that b equals -8. Therefore, the solution to the equation $6 + 3b = -18$ is b = -8. This value of b makes the equation true, meaning that if we substitute -8 for b in the original equation, both sides will be equal. To ensure the solution is correct, it's always a good practice to verify it. This involves substituting the solution back into the original equation and checking if both sides of the equation are equal. In our case, we substitute b = -8 into the equation $6 + 3b = -18$. This gives us $6 + 3(-8) = -18$. Simplifying the left side, we get $6 - 24 = -18$, which further simplifies to $-18 = -18$. Since both sides of the equation are equal, our solution b = -8 is correct. This verification step not only confirms the solution but also enhances understanding of the equation-solving process. This problem highlights the fundamental principles of solving linear equations: isolating the variable by applying inverse operations and verifying the solution to ensure accuracy. Understanding these principles is essential for tackling more complex algebraic problems. The ability to manipulate equations and solve for unknowns is a core skill in mathematics, and this example provides a clear illustration of how it's done. Mastering these techniques builds a strong foundation for further mathematical studies and applications.

2. Solving the Equation $7n + 12 = -23$

When dealing with linear equations such as $7n + 12 = -23$, the goal remains to isolate the variable, in this case, n. Solving equations involves a series of steps that methodically peel away the layers surrounding the variable until its value is revealed. To begin, our primary objective is to isolate the term containing the variable, which is 7n. This can be achieved by eliminating the constant term on the same side of the equation, which in this case is +12. To eliminate +12, we perform the inverse operation, which is subtraction. We subtract 12 from both sides of the equation to maintain balance and equality. This transforms the equation $7n + 12 = -23$ into $7n + 12 - 12 = -23 - 12$. Simplifying both sides, we get $7n = -35$. This step is crucial as it brings us closer to isolating n by removing the constant term from the left side of the equation. The next crucial step in solving for n is to deal with the coefficient of the variable. In the equation $7n = -35$, n is being multiplied by 7. To isolate n, we need to undo this multiplication by performing the inverse operation, which is division. We divide both sides of the equation by 7. This transforms the equation $7n = -35$ into $\frac{7n}{7} = \frac{-35}{7}$. Simplifying both sides, we find that n equals -5. Therefore, the solution to the linear equation $7n + 12 = -23$ is n = -5. This value of n satisfies the equation, meaning that when -5 is substituted for n in the original equation, both sides will be equal. Verification is an essential part of the equation-solving process. It ensures that the solution obtained is accurate and that no errors were made during the solving process. To verify our solution, we substitute n = -5 back into the original equation, $7n + 12 = -23$. This gives us $7(-5) + 12 = -23$. Simplifying the left side, we get $-35 + 12 = -23$, which further simplifies to $-23 = -23$. Since both sides of the equation are equal, our solution n = -5 is indeed correct. This verification step not only confirms the solution but also reinforces the understanding of the equation-solving principles. By systematically applying inverse operations and verifying the result, we ensure accuracy and develop confidence in our problem-solving abilities. The problem illustrates the key steps in solving a linear equation: isolating the variable term, eliminating the coefficient of the variable, and verifying the solution. These steps are fundamental to solving a wide range of algebraic problems, and mastering them is essential for success in mathematics. The ability to solve equations is a core mathematical skill that has applications in various fields, making it a valuable tool for problem-solving and analytical thinking.

3. Solving the Equation $-12 = 8 + \frac{f}{2}$

Solving algebraic equations often requires a strategic approach to isolate the variable, and the equation $-12 = 8 + \frac{f}{2}$ is no exception. In this case, we aim to find the value of f that makes the equation true. The initial step in solving this equation is to isolate the term containing the variable, which is $\frac{f}{2}$. To achieve this, we need to eliminate the constant term on the same side of the equation. In this equation, the constant term is 8. We eliminate 8 by performing the inverse operation, which is subtraction. Subtracting 8 from both sides of the equation maintains the balance and equality. This transforms the equation $-12 = 8 + \frac{f}{2}$ into $-12 - 8 = 8 + \frac{f}{2} - 8$. Simplifying both sides, we get $-20 = \frac{f}{2}$. This step is crucial because it isolates the term with f, bringing us closer to finding the value of f. Now that we have $-20 = \frac{f}{2}$, the next step is to eliminate the denominator. The variable f is being divided by 2. To isolate f, we need to undo this division by performing the inverse operation, which is multiplication. We multiply both sides of the equation by 2. Multiplying both sides by 2 transforms the equation $-20 = \frac{f}{2}$ into $2 \times -20 = 2 \times \frac{f}{2}$. Simplifying both sides, we get $-40 = f$. Therefore, the solution to the linear equation $-12 = 8 + \frac{f}{2}$ is f = -40. This value of f makes the equation true, meaning that if we substitute -40 for f in the original equation, both sides will be equal. To ensure the accuracy of our solution, we must verify it. Verification involves substituting the solution back into the original equation and checking if both sides of the equation are equal. In our case, we substitute f = -40 into the equation $-12 = 8 + \frac{f}{2}$. This gives us $-12 = 8 + \frac{-40}{2}$. Simplifying the right side, we get $-12 = 8 - 20$, which further simplifies to $-12 = -12$. Since both sides of the equation are equal, our solution f = -40 is correct. This verification step confirms that we have accurately solved for f and that no errors were made during the process. This problem effectively demonstrates the importance of applying inverse operations in the correct order to isolate the variable. By systematically working through the equation, we can arrive at the correct solution. The ability to solve algebraic equations is a fundamental skill in mathematics, and this example provides a clear illustration of how to approach and solve such equations. The process of isolating the variable, performing inverse operations, and verifying the solution is a cornerstone of algebraic problem-solving.

4. Solving the Equation $\frac{k}{4} + 6 = -2$

Linear equations like $\frac{k}{4} + 6 = -2$ are common in algebra, and solving them involves a systematic approach to isolate the variable. In this case, our goal is to find the value of k that satisfies the equation. The first step in solving this equation is to isolate the term containing the variable, which is $\frac{k}{4}$. To do this, we need to eliminate the constant term on the same side of the equation. In this equation, the constant term is +6. We eliminate +6 by performing the inverse operation, which is subtraction. Subtracting 6 from both sides of the equation maintains the balance and equality. This transforms the equation $\frac{k}{4} + 6 = -2$ into $\frac{k}{4} + 6 - 6 = -2 - 6$. Simplifying both sides, we get $\frac{k}{4} = -8$. This step is crucial because it isolates the term with k, bringing us closer to finding its value. Now that we have $\frac{k}{4} = -8$, the next step is to eliminate the denominator. The variable k is being divided by 4. To isolate k, we need to undo this division by performing the inverse operation, which is multiplication. We multiply both sides of the equation by 4. Multiplying both sides by 4 transforms the equation $\frac{k}{4} = -8$ into $4 \times \frac{k}{4} = 4 \times -8$. Simplifying both sides, we find that k equals -32. Therefore, the solution to the algebraic equation $\frac{k}{4} + 6 = -2$ is k = -32. This value of k makes the equation true, meaning that if we substitute -32 for k in the original equation, both sides will be equal. To ensure the accuracy of our solution, verification is essential. This involves substituting the solution back into the original equation and checking if both sides of the equation are equal. In our case, we substitute k = -32 into the equation $\frac{k}{4} + 6 = -2$. This gives us $\frac{-32}{4} + 6 = -2$. Simplifying the left side, we get $-8 + 6 = -2$, which further simplifies to $-2 = -2$. Since both sides of the equation are equal, our solution k = -32 is correct. This verification step confirms that we have accurately solved for k and that no errors were made during the process. This problem illustrates the key steps in solving linear equations involving fractions: isolating the term with the variable, eliminating the denominator, and verifying the solution. By systematically applying inverse operations, we can effectively solve for the unknown variable. The ability to solve algebraic equations is a fundamental skill in mathematics, and this example provides a clear demonstration of how to approach and solve such equations. The process of isolating the variable, performing inverse operations, and verifying the solution is a cornerstone of algebraic problem-solving.

5. Solving the Equation $16 - 3p = 34$

Solving linear equations requires a systematic approach to isolate the variable, and the equation $16 - 3p = 34$ is a classic example. In this case, our objective is to determine the value of p that satisfies the equation. The first step in solving this equation is to isolate the term containing the variable, which is -3p. To achieve this, we need to eliminate the constant term on the same side of the equation. In this equation, the constant term is 16. We eliminate 16 by performing the inverse operation, which is subtraction. Subtracting 16 from both sides of the equation maintains the balance and equality. This transforms the equation $16 - 3p = 34$ into $16 - 3p - 16 = 34 - 16$. Simplifying both sides, we get $-3p = 18$. This step is crucial because it isolates the term with p, bringing us closer to finding its value. Now that we have $-3p = 18$, the next step is to eliminate the coefficient of p. The variable p is being multiplied by -3. To isolate p, we need to undo this multiplication by performing the inverse operation, which is division. We divide both sides of the equation by -3. Dividing both sides by -3 transforms the equation $-3p = 18$ into $\frac{-3p}{-3} = \frac{18}{-3}$. Simplifying both sides, we find that p equals -6. Therefore, the solution to the linear equation $16 - 3p = 34$ is p = -6. This value of p makes the equation true, meaning that if we substitute -6 for p in the original equation, both sides will be equal. Verification is a critical step in the equation-solving process. It ensures that our solution is accurate and that no errors were made during the solving process. To verify our solution, we substitute p = -6 back into the original equation, $16 - 3p = 34$. This gives us $16 - 3(-6) = 34$. Simplifying the left side, we get $16 + 18 = 34$, which further simplifies to $34 = 34$. Since both sides of the equation are equal, our solution p = -6 is indeed correct. This verification step not only confirms the solution but also reinforces the understanding of the equation-solving principles. By systematically applying inverse operations and verifying the result, we ensure accuracy and develop confidence in our problem-solving abilities. This problem effectively demonstrates the importance of carefully applying inverse operations, particularly when dealing with negative coefficients. The ability to solve algebraic equations is a fundamental skill in mathematics, and this example provides a clear illustration of how to approach and solve such equations. Mastering these techniques builds a strong foundation for further mathematical studies and applications.

In conclusion, solving linear equations is a fundamental skill in mathematics. By understanding and applying the principles of inverse operations and verification, one can confidently solve a wide range of equations. Each example discussed in this article illustrates these principles, providing a comprehensive guide for students and anyone looking to enhance their algebraic skills. Mastering these techniques is essential for success in mathematics and its applications in various fields.