Solve For \[$ N \$\].$\[ 3n - 8 = 13 \\]

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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 linear equations of the form ax + b = c, where a, b, and c are constants. We will use the given equation 3n - 8 = 13 as an example to demonstrate the step-by-step process of solving linear equations.

What is a Linear Equation?

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 = c, where a, b, and c are constants. The variable(s) in a linear equation can be a single variable or multiple variables.

The Given Equation

The given equation is 3n - 8 = 13. This equation is a linear equation in one variable, n.

Step 1: Add 8 to Both Sides

To solve the equation, we need to isolate the variable n. The first step is to add 8 to both sides of the equation. This will eliminate the negative term on the left-hand side.

3n - 8 + 8 = 13 + 8

Simplifying the equation, we get:

3n = 21

Step 2: Divide Both Sides by 3

Now that we have isolated the term with the variable n, we need to get rid of the coefficient 3. We can do this by dividing both sides of the equation by 3.

\frac{3n}{3} = \frac{21}{3}

Simplifying the equation, we get:

n = 7

Conclusion

In this article, we solved the linear equation 3n - 8 = 13 using the step-by-step process of adding 8 to both sides and then dividing both sides by 3. We isolated the variable n and found its value to be 7. This demonstrates the importance of following the order of operations and using algebraic properties to solve linear equations.

Tips and Tricks

  • When solving linear equations, always follow the order of operations (PEMDAS).
  • Use algebraic properties, such as adding or subtracting the same value to both sides, to isolate the variable.
  • Check your work by plugging the solution back into the original equation.

Common Mistakes to Avoid

  • Not following the order of operations (PEMDAS).
  • Not using algebraic properties to isolate the variable.
  • Not checking the solution by plugging it back into the original equation.

Real-World Applications

Linear equations have numerous real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects under constant acceleration.
  • Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Linear equations are used to model economic systems and make predictions about future trends.

Conclusion

Introduction

In our previous article, we discussed the step-by-step process of solving linear equations. However, we know that practice makes perfect, and there's no better way to practice than by answering questions. In this article, we'll provide a Q&A guide to help you master the art of solving linear equations.

Q1: What is a linear equation?

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 = c, where a, b, and c are constants.

Q2: How do I solve a linear equation?

To solve a linear equation, follow these steps:

  1. Add or subtract the same value to both sides of the equation to isolate the variable.
  2. Multiply or divide both sides of the equation by the same value to eliminate the coefficient of the variable.
  3. Check your work by plugging the solution back into the original equation.

Q3: What is the order of operations (PEMDAS)?

The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when you have multiple operations in an expression. The acronym PEMDAS stands for:

  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.

Q4: How do I handle fractions in linear equations?

When working with fractions in linear equations, you can multiply both sides of the equation by the denominator to eliminate the fraction. For example, if you have the equation:

23x=4{\frac{2}{3}x = 4}

You can multiply both sides by 3 to get:

2x=12{2x = 12}

Q5: What is the difference between a linear equation and a quadratic equation?

A linear equation is an equation in which the highest power of the variable(s) is 1. A quadratic equation, on the other hand, is an equation in which the highest power of the variable(s) is 2. For example:

Linear equation: 2x + 3 = 5 Quadratic equation: x^2 + 4x + 4 = 0

Q6: How do I check my work when solving a linear equation?

To check your work, plug the solution back into the original equation. If the solution satisfies the equation, then you have found the correct solution. For example, if you have the equation:

2x + 3 = 5

And you find that x = 1 is a solution, you can plug x = 1 back into the equation to check:

2(1) + 3 = 5 2 + 3 = 5 5 = 5

Since the solution satisfies the equation, you have found the correct solution.

Q7: What are some common mistakes to avoid when solving linear equations?

Some common mistakes to avoid when solving linear equations include:

  • Not following the order of operations (PEMDAS).
  • Not using algebraic properties to isolate the variable.
  • Not checking the solution by plugging it back into the original equation.

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By following the step-by-step process outlined in this article and practicing with the Q&A guide, you can become proficient in solving linear equations and apply your skills to real-world problems. Remember to always follow the order of operations (PEMDAS) and use algebraic properties to isolate the variable. With practice and patience, you can become a master of solving linear equations.