Solve The Equation $5(n-2)=45$.

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 simple linear equation, $5(n-2)=45$, using a step-by-step approach. We will break down the solution into manageable parts, making it easy to understand and follow.

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, and $x$ is the variable. Linear equations can be solved using various methods, including algebraic manipulation, graphing, and substitution.

The Equation $5(n-2)=45$

The given equation is $5(n-2)=45$. Our goal is to solve for the variable $n$. To do this, we will use the distributive property to expand the left-hand side of the equation.

Distributive Property

The distributive property states that for any numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. We can use this property to expand the left-hand side of the equation.

# Define the equation
equation = "5(n-2) = 45"

# Expand the left-hand side using the distributive property
expanded_equation = "5n - 10 = 45"

Simplifying the Equation

Now that we have expanded the left-hand side of the equation, we can simplify it by combining like terms. In this case, we have $5n - 10$ on the left-hand side, and $45$ on the right-hand side.

Combining Like Terms

Like terms are terms that have the same variable(s) raised to the same power. In this case, we have $5n$ and $-10$ on the left-hand side, which are like terms. We can combine them by adding their coefficients.

# Define the equation
equation = "5n - 10 = 45"

# Combine like terms
simplified_equation = "5n = 55"

Solving for $n$

Now that we have simplified the equation, we can solve for $n$. To do this, we will divide both sides of the equation by $5$.

Dividing Both Sides

When we divide both sides of an equation by a non-zero number, we are essentially multiplying both sides by the reciprocal of that number. In this case, we will divide both sides of the equation by $5$.

# Define the equation
equation = "5n = 55"

# Divide both sides by 5
solution = "n = 11"

Conclusion

In this article, we solved the linear equation $5(n-2)=45$ using a step-by-step approach. We expanded the left-hand side of the equation using the distributive property, simplified it by combining like terms, and finally solved for $n$ by dividing both sides of the equation by $5$. This example illustrates the importance of following a systematic approach when solving linear equations.

Tips and Tricks

• Always start by expanding the left-hand side of the equation using the distributive property.
• Simplify the equation by combining like terms.
• Solve for the variable by dividing both sides of the equation by the coefficient of the variable.

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.

Common Mistakes

• Failing to expand the left-hand side of the equation using the distributive property.
• Not simplifying the equation by combining like terms.
• Dividing both sides of the equation by a zero coefficient.

Conclusion

Introduction

In our previous article, we solved the linear equation $5(n-2)=45$ using a step-by-step approach. In this article, we will provide a Q&A guide to help students and educators better understand the concept of 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 $ax + b = c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I solve a linear equation?

A: To solve a linear equation, follow these steps:

1. Expand the left-hand side of the equation using the distributive property.
2. Simplify the equation by combining like terms.
3. Solve for the variable by dividing both sides of the equation by the coefficient of the variable.

Q: What is the distributive property?

A: The distributive property states that for any numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. This property allows us to expand the left-hand side of an equation by multiplying each term inside the parentheses by the coefficient outside the parentheses.

Q: How do I combine like terms?

A: Like terms are terms that have the same variable(s) raised to the same power. To combine like terms, add their coefficients.

Q: What is the difference between a coefficient and a constant?

A: A coefficient is a number that is multiplied by a variable, while a constant is a number that is not multiplied by a variable.

Q: Can I solve a linear equation by using a calculator?

A: Yes, you can solve a linear equation by using a calculator. However, it's always a good idea to understand the steps involved in solving the equation manually.

Q: How do I check my solution?

A: To check your solution, plug the value of the variable back into the original equation and simplify. If the equation is true, then your solution is correct.

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

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

• Failing to expand the left-hand side of the equation using the distributive property.
• Not simplifying the equation by combining like terms.
• Dividing both sides of the equation by a zero coefficient.

Q: Can linear equations be used to model real-world problems?

A: Yes, linear equations can be used to model real-world problems, such as physics, engineering, and economics.

Q: How do I apply linear equations to real-world problems?

A: To apply linear equations to real-world problems, follow these steps:

1. Identify the variables and constants in the problem.
2. Write an equation that represents the problem.
3. Solve the equation using the steps outlined above.
4. Interpret the solution in the context of the problem.

Conclusion

Solving linear equations is a crucial skill for students to master. By following a systematic approach, including expanding the left-hand side of the equation using the distributive property, simplifying the equation by combining like terms, and solving for the variable by dividing both sides of the equation by the coefficient of the variable, students can confidently solve linear equations and apply them to real-world problems.