Solve For { N $} : : : { 4n + 15 = 101 \}

Mar 10, 2025 by ADMIN 42 views

Solving Linear Equations: A Step-by-Step Guide to Finding the Value of n

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, where the variable n is isolated on one side of the equation. We will use the equation 4n + 15 = 101 as an example to demonstrate the step-by-step process of solving for n.

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at what it means. The equation 4n + 15 = 101 is a linear equation, which means it is an equation in which the highest power of the variable (in this case, n) is 1. The equation is in the form of ax + b = c, where a, b, and c are constants.

In this equation, the variable n is multiplied by 4, and then 15 is added to the result. The equation states that the result of this operation is equal to 101.

Solving for n: A Step-by-Step Guide

To solve for n, we need to isolate the variable n on one side of the equation. We can do this by using the inverse operations of addition and multiplication.

Step 1: Subtract 15 from Both Sides

The first step in solving for n is to subtract 15 from both sides of the equation. This will help us get rid of the constant term on the left-hand side of the equation.

4n + 15 - 15 = 101 - 15

This simplifies to:

4n = 86

Step 2: Divide Both Sides by 4

Now that we have isolated the term with the variable n, we can divide both sides of the equation by 4 to solve for n.

(4n) / 4 = 86 / 4

This simplifies to:

n = 21.5

Conclusion

In this article, we have demonstrated the step-by-step process of solving a linear equation for the variable n. By using the inverse operations of addition and multiplication, we were able to isolate the variable n on one side of the equation and solve for its value.

Tips and Tricks

When solving linear equations, it's essential to follow the order of operations (PEMDAS) to ensure that you are performing the operations in the correct order.
Make sure to isolate the variable on one side of the equation by using inverse operations.
Check your work by plugging the solution back into the original equation to ensure that it is true.

Real-World Applications

Solving linear equations is a crucial skill in many real-world applications, including:

Finance: Solving linear equations can help you calculate interest rates, investment returns, and other financial metrics.
Science: Linear equations are used to model real-world phenomena, such as the motion of objects, the growth of populations, and the behavior of electrical circuits.
Engineering: Solving linear equations is essential in designing and optimizing systems, such as bridges, buildings, and electronic circuits.

Common Mistakes to Avoid

Make sure to follow the order of operations (PEMDAS) to avoid errors.
Be careful when using inverse operations, as they can sometimes lead to incorrect solutions.
Check your work by plugging the solution back into the original equation to ensure that it is true.

Additional Resources

For more information on solving linear equations, check out the following resources:

Khan Academy: Linear Equations
Mathway: Solving Linear Equations
Wolfram Alpha: Linear Equations

By following the step-by-step guide outlined in this article, you should be able to solve linear equations with confidence. Remember to always check your work and use inverse operations to isolate the variable on one side of the equation.

Introduction

In our previous article, we explored the step-by-step process of solving linear equations for the variable n. However, we know that practice makes perfect, and sometimes, it's helpful to have a refresher on the concepts. In this article, we'll address some common questions and concerns that students may have when solving linear equations.

Q&A: Solving Linear Equations

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (in this case, n) is 1. It can be written in the form of ax + b = c, where a, b, and c are constants.

Q: How do I know if an equation is linear?

A: To determine if an equation is linear, look for the highest power of the variable. If it's 1, then the equation is linear. If it's greater than 1, then the equation is not linear.

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

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when solving an equation. The acronym PEMDAS stands for:

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

Q: How do I isolate the variable on one side of the equation?

A: To isolate the variable, use inverse operations to get rid of the constant term on the same side of the equation as the variable. For example, if you have the equation 4n + 15 = 101, you can subtract 15 from both sides to get 4n = 86.

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

A: Some common mistakes to avoid include:

Not following the order of operations (PEMDAS)
Not isolating the variable on one side of the equation
Not checking your work by plugging the solution back into the original equation

Q: How do I check my work?

A: To check your work, plug the solution back into the original equation and make sure it's true. For example, if you solved the equation 4n + 15 = 101 and got n = 21.5, plug n = 21.5 back into the original equation to make sure it's true.

Q: What are some real-world applications of solving linear equations?

A: Solving linear equations has many real-world applications, including:

Finance: Solving linear equations can help you calculate interest rates, investment returns, and other financial metrics.
Science: Linear equations are used to model real-world phenomena, such as the motion of objects, the growth of populations, and the behavior of electrical circuits.
Engineering: Solving linear equations is essential in designing and optimizing systems, such as bridges, buildings, and electronic circuits.

Q: What are some additional resources for learning more about solving linear equations?

A: Some additional resources for learning more about solving linear equations include:

Khan Academy: Linear Equations
Mathway: Solving Linear Equations
Wolfram Alpha: Linear Equations