Solve For $n$.$n + 1 = 4(n - 8$\]Possible Solutions:A. $n = 1$B. $n = 8$C. $n = 11$D. $n = 16$

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific type of linear equation, which is in the form of $n + 1 = 4(n - 8)$. We will break down the solution step by step and provide a clear explanation of each step.

Understanding the Equation

The given equation is $n + 1 = 4(n - 8)$. To solve for $n$, we need to isolate the variable $n$ on one side of the equation. The equation is a linear equation, which means it is in the form of $ax + b = c$, where $a$, $b$, and $c$ are constants.

Step 1: Distribute the 4

The first step in solving the equation is to distribute the 4 to the terms inside the parentheses. This will give us:

$n + 1 = 4n - 32$

Step 2: Add 32 to Both Sides

Next, we need to add 32 to both sides of the equation to get rid of the negative term. This will give us:

$n + 1 + 32 = 4n - 32 + 32$

Simplifying the equation, we get:

$n + 33 = 4n$

Step 3: Subtract $n$ from Both Sides

Now, we need to subtract $n$ from both sides of the equation to isolate the term with $n$. This will give us:

$n + 33 - n = 4n - n$

Simplifying the equation, we get:

$33 = 3n$

Step 4: Divide Both Sides by 3

Finally, we need to divide both sides of the equation by 3 to solve for $n$. This will give us:

$\frac{33}{3} = \frac{3n}{3}$

Simplifying the equation, we get:

$11 = n$

Conclusion

In conclusion, the solution to the equation $n + 1 = 4(n - 8)$ is $n = 11$. This is the only possible solution to the equation.

Possible Solutions

The possible solutions to the equation are:

• $n = 1$
• $n = 8$
• $n = 11$
• $n = 16$

However, only one of these solutions is correct. To determine which solution is correct, we need to plug each solution back into the original equation and check if it is true.

Checking the Solutions

Let's plug each solution back into the original equation and check if it is true.

• For $n = 1$, we get: $1 + 1 = 4(1 - 8)$, which is not true.
• For $n = 8$, we get: $8 + 1 = 4(8 - 8)$, which is not true.
• For $n = 11$, we get: $11 + 1 = 4(11 - 8)$, which is true.
• For $n = 16$, we get: $16 + 1 = 4(16 - 8)$, which is not true.

Therefore, the only possible solution to the equation is $n = 11$.

Final Answer

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Introduction

In our previous article, we solved the linear equation $n + 1 = 4(n - 8)$ and found that the solution is $n = 11$. However, we also provided four possible solutions: $n = 1$, $n = 8$, $n = 11$, and $n = 16$. In this article, we will answer some frequently asked questions about solving linear equations and provide additional examples to help you understand the concept better.

Q&A

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable is 1. In other words, it is an equation that can be written in the form of $ax + b = c$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable on one side of the equation. You can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Q: What is the order of operations when solving a linear equation?

A: When solving a linear equation, you need to follow the order of operations, which is:

1. Parentheses: Evaluate any expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I check if a solution is correct?

A: To check if a solution is correct, you need to plug the solution back into the original equation and check if it is true. If the solution satisfies the equation, then it is a valid solution.

Q: What are some common mistakes to avoid when solving linear equations?

A: Some common mistakes to avoid when solving linear equations include:

• Not following the order of operations
• Not isolating the variable on one side of the equation
• Not checking if the solution is correct
• Not simplifying the equation before solving it

Q: Can you provide more examples of solving linear equations?

A: Yes, here are a few more examples:

• Solve the equation $2x + 5 = 11$.
• Solve the equation $x - 3 = 7$.
• Solve the equation $4x + 2 = 14$.

Example 1: Solve the equation $2x + 5 = 11$

To solve this equation, we need to isolate the variable $x$ on one side of the equation. We can do this by subtracting 5 from both sides of the equation:

$2x + 5 - 5 = 11 - 5$

Simplifying the equation, we get:

$2x = 6$

Next, we need to divide both sides of the equation by 2 to solve for $x$:

$\frac{2x}{2} = \frac{6}{2}$

Simplifying the equation, we get:

$x = 3$

Therefore, the solution to the equation $2x + 5 = 11$ is $x = 3$.

Example 2: Solve the equation $x - 3 = 7$

To solve this equation, we need to isolate the variable $x$ on one side of the equation. We can do this by adding 3 to both sides of the equation:

$x - 3 + 3 = 7 + 3$

Simplifying the equation, we get:

$x = 10$

Therefore, the solution to the equation $x - 3 = 7$ is $x = 10$.

Example 3: Solve the equation $4x + 2 = 14$

To solve this equation, we need to isolate the variable $x$ on one side of the equation. We can do this by subtracting 2 from both sides of the equation:

$4x + 2 - 2 = 14 - 2$

Simplifying the equation, we get:

$4x = 12$

Next, we need to divide both sides of the equation by 4 to solve for $x$:

$\frac{4x}{4} = \frac{12}{4}$

Simplifying the equation, we get:

$x = 3$

Therefore, the solution to the equation $4x + 2 = 14$ is $x = 3$.

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By following the order of operations and isolating the variable on one side of the equation, you can solve linear equations with ease. Remember to check if the solution is correct by plugging it back into the original equation. With practice and patience, you will become proficient in solving linear equations in no time.