1. Solve: $6n - 2 = N + 13$A. $n = -3$ B. $n = -2$ C. $n = 2$ D. $n = 3$

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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 a first-degree equation in one variable. We will use the equation 6n−2=n+136n - 2 = n + 13 as an example to demonstrate the step-by-step process of solving linear equations.

What are Linear Equations?

A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form ax+b=cax + b = c, where aa, bb, and cc are constants, and xx is the variable. Linear equations can be solved using various methods, including algebraic manipulation, graphing, and substitution.

The Equation 6n−2=n+136n - 2 = n + 13

The given equation is 6n−2=n+136n - 2 = n + 13. Our goal is to solve for the variable nn. To do this, we will use the method of algebraic manipulation.

Step 1: Add 2 to Both Sides

The first step in solving the equation is to add 2 to both sides of the equation. This will eliminate the negative term on the left-hand side.

6n - 2 + 2 = n + 13 + 2

Simplifying the equation, we get:

6n = n + 15

Step 2: Subtract nn from Both Sides

Next, we will subtract nn from both sides of the equation. This will isolate the term with the variable on the left-hand side.

6n - n = n + 15 - n

Simplifying the equation, we get:

5n = 15

Step 3: Divide Both Sides by 5

Finally, we will divide both sides of the equation by 5. This will solve for the variable nn.

5n / 5 = 15 / 5

Simplifying the equation, we get:

n = 3

Conclusion

In this article, we solved the linear equation 6n−2=n+136n - 2 = n + 13 using the method of algebraic manipulation. We added 2 to both sides of the equation, subtracted nn from both sides, and finally divided both sides by 5 to solve for the variable nn. The solution to the equation is n=3n = 3.

Answer Key

The correct answer is:

A. n=−3n = -3

B. n=−2n = -2

C. n=2n = 2

D. n=3n = 3

The correct answer is D. n=3n = 3.

Tips and Tricks

  • When solving linear equations, always follow the order of operations (PEMDAS).
  • Use algebraic manipulation to isolate the variable on one side of the equation.
  • Check your solution by plugging it back into the original equation.

Practice Problems

Try solving the following linear equations:

  1. 2x+3=5x−22x + 3 = 5x - 2
  2. x−2=3x+1x - 2 = 3x + 1
  3. 4x+2=2x−34x + 2 = 2x - 3

References

About the Author

Introduction

In our previous article, we solved the linear equation 6n−2=n+136n - 2 = n + 13 using the method of algebraic manipulation. In this article, we will answer some frequently asked questions about solving linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form ax+b=cax + b = c, where aa, bb, and cc are constants, and xx is the variable.

Q: How do I solve a linear equation?

A: To solve a linear equation, you can use the method of algebraic manipulation. This involves adding, subtracting, multiplying, or dividing both sides of the equation by the same value to isolate the variable on one side of the equation.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when solving an equation. The order of operations is:

  1. Parentheses: Evaluate 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 my solution?

A: To check your solution, plug it back into the original equation and see if it is true. If the solution is true, then you have solved the equation correctly.

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 your solution
  • Not using algebraic manipulation to solve the equation

Q: Can I use a calculator to solve linear equations?

A: Yes, you can use a calculator to solve linear equations. However, it is always a good idea to check your solution by plugging it back into the original equation to make sure it is true.

Q: How do I solve a linear equation with fractions?

A: To solve a linear equation with fractions, you can multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions.

Q: Can I solve a linear equation with decimals?

A: Yes, you can solve a linear equation with decimals. However, it is always a good idea to round your answer to the nearest hundredth or thousandth to avoid any errors.

Q: How do I solve a linear equation with variables on both sides?

A: To solve a linear equation with variables on both sides, you can add or subtract the same value to both sides of the equation to eliminate the variable on one side.

Q: Can I use algebraic manipulation to solve a linear equation with absolute values?

A: Yes, you can use algebraic manipulation to solve a linear equation with absolute values. However, you will need to consider both the positive and negative cases of the absolute value.

Conclusion

In this article, we answered some frequently asked questions about solving linear equations. We covered topics such as the order of operations, checking solutions, and common mistakes to avoid. We also discussed how to solve linear equations with fractions, decimals, and variables on both sides.

Practice Problems

Try solving the following linear equations:

  1. 2x+3=5x−22x + 3 = 5x - 2
  2. x−2=3x+1x - 2 = 3x + 1
  3. 4x+2=2x−34x + 2 = 2x - 3

References

About the Author

[Your Name] is a mathematics educator with a passion for teaching and learning. With a strong background in mathematics and education, [Your Name] has developed a unique approach to teaching linear equations that is engaging, interactive, and easy to understand.