Simplify The Expression:${ \frac{x-4}{3} - \frac{x+7}{5} }$

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill that every student and professional should possess. In this article, we will delve into the world of algebraic manipulation and provide a step-by-step guide on how to simplify the given expression: $\frac{x-4}{3} - \frac{x+7}{5}$. We will explore various techniques and strategies to make the expression more manageable and easier to understand.

Understanding the Expression

Before we dive into the simplification process, let's take a closer look at the given expression: $\frac{x-4}{3} - \frac{x+7}{5}$. This expression consists of two fractions, each with a different denominator. The first fraction has a denominator of 3, while the second fraction has a denominator of 5. Our goal is to simplify this expression by combining the two fractions into a single fraction with a common denominator.

Finding the Common Denominator

To simplify the expression, we need to find the least common multiple (LCM) of the two denominators, which are 3 and 5. The LCM of 3 and 5 is 15. Therefore, we will multiply the numerator and denominator of each fraction by the necessary multiples to obtain a common denominator of 15.

Multiplying the Fractions

To multiply the fractions, we will multiply the numerators and denominators separately. For the first fraction, we will multiply the numerator by 5 and the denominator by 5. For the second fraction, we will multiply the numerator by 3 and the denominator by 3.

$\frac{x-4}{3} \cdot \frac{5}{5} = \frac{5(x-4)}{15}$

$\frac{x+7}{5} \cdot \frac{3}{3} = \frac{3(x+7)}{15}$

Combining the Fractions

Now that we have both fractions with a common denominator of 15, we can combine them by adding or subtracting the numerators.

$\frac{5(x-4)}{15} - \frac{3(x+7)}{15} = \frac{5(x-4) - 3(x+7)}{15}$

Simplifying the Numerator

To simplify the numerator, we will distribute the negative sign to the terms inside the parentheses and then combine like terms.

$\frac{5(x-4) - 3(x+7)}{15} = \frac{5x - 20 - 3x - 21}{15}$

$\frac{5x - 20 - 3x - 21}{15} = \frac{2x - 41}{15}$

Conclusion

In conclusion, we have successfully simplified the given expression: $\frac{x-4}{3} - \frac{x+7}{5}$. By finding the common denominator, multiplying the fractions, and combining them, we arrived at the simplified expression: $\frac{2x - 41}{15}$. This expression is now more manageable and easier to understand, making it a valuable tool for students and professionals alike.

Tips and Tricks

• When simplifying expressions, always look for common factors in the numerator and denominator.
• Use the distributive property to simplify complex expressions.
• Combine like terms to make the expression more manageable.
• Use the least common multiple (LCM) to find the common denominator.

Real-World Applications

Simplifying expressions has numerous real-world applications in various fields, including:

• Physics: Simplifying expressions is crucial in physics, where complex equations are used to describe the behavior of physical systems.
• Engineering: Engineers use simplifying expressions to design and optimize complex systems.
• Economics: Economists use simplifying expressions to model and analyze economic systems.

Final Thoughts

Simplifying expressions is a fundamental skill that every student and professional should possess. By following the steps outlined in this article, you can simplify even the most complex expressions and make them more manageable. Remember to always look for common factors, use the distributive property, combine like terms, and use the least common multiple to find the common denominator. With practice and patience, you will become proficient in simplifying expressions and be able to tackle even the most challenging problems.

Introduction

In our previous article, we explored the world of algebraic manipulation and provided a step-by-step guide on how to simplify the given expression: $\frac{x-4}{3} - \frac{x+7}{5}$. We also discussed various techniques and strategies to make the expression more manageable and easier to understand. In this article, we will address some of the most frequently asked questions (FAQs) related to simplifying expressions.

Q&A

Q: What is the purpose of simplifying expressions?

A: The purpose of simplifying expressions is to make them more manageable and easier to understand. Simplifying expressions helps to:

• Reduce the complexity of the expression
• Make it easier to solve
• Identify patterns and relationships
• Make it easier to communicate the solution

Q: What are some common techniques used to simplify expressions?

A: Some common techniques used to simplify expressions include:

• Factoring: breaking down the expression into simpler factors
• Canceling: canceling out common factors in the numerator and denominator
• Distributing: distributing the terms in the expression
• Combining like terms: combining terms with the same variable and coefficient

Q: How do I find the common denominator?

A: To find the common denominator, you need to find the least common multiple (LCM) of the denominators. The LCM is the smallest number that both denominators can divide into evenly.

Q: What is the difference between a numerator and a denominator?

A: The numerator is the top part of the fraction, and the denominator is the bottom part. The numerator is the number being divided, and the denominator is the number by which we are dividing.

Q: Can I simplify an expression with a variable in the denominator?

A: Yes, you can simplify an expression with a variable in the denominator. However, you need to be careful when simplifying expressions with variables in the denominator, as it can lead to undefined expressions.

Q: How do I know when to simplify an expression?

A: You should simplify an expression when:

• The expression is complex and difficult to understand
• The expression is part of a larger equation or inequality
• The expression is used to solve a problem or make a decision

Q: Can I use a calculator to simplify expressions?

A: Yes, you can use a calculator to simplify expressions. However, it's always a good idea to check your work by hand to ensure that the calculator is giving you the correct answer.

Real-World Applications

Simplifying expressions has numerous real-world applications in various fields, including:

• Physics: Simplifying expressions is crucial in physics, where complex equations are used to describe the behavior of physical systems.
• Engineering: Engineers use simplifying expressions to design and optimize complex systems.
• Economics: Economists use simplifying expressions to model and analyze economic systems.

Final Thoughts

Simplifying expressions is a fundamental skill that every student and professional should possess. By following the steps outlined in this article and practicing regularly, you can become proficient in simplifying expressions and be able to tackle even the most challenging problems.

Additional Resources

• Khan Academy: Simplifying Expressions
• Mathway: Simplifying Expressions
• Wolfram Alpha: Simplifying Expressions

Conclusion

In conclusion, simplifying expressions is a crucial skill that every student and professional should possess. By understanding the techniques and strategies outlined in this article, you can simplify even the most complex expressions and make them more manageable. Remember to always look for common factors, use the distributive property, combine like terms, and use the least common multiple to find the common denominator. With practice and patience, you will become proficient in simplifying expressions and be able to tackle even the most challenging problems.