Simplify The Expression:${\frac{5xy}{20}(4xy - 3y)}$

Introduction

Algebraic manipulation is a crucial skill in mathematics, and simplifying expressions is an essential part of it. In this article, we will focus on simplifying the given expression: ${\frac{5xy}{20}(4xy - 3y)}$. We will break down the process into manageable steps, making it easier to understand and follow along.

Understanding the Expression

Before we start simplifying the expression, let's take a closer look at it. The given expression is a product of two terms: $\frac{5xy}{20}$ and $(4xy - 3y)$. To simplify the expression, we need to apply the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$.

Step 1: Apply the Distributive Property

To simplify the expression, we need to apply the distributive property to the product of the two terms. This means that we need to multiply each term in the first factor by each term in the second factor.

$${\frac{5xy}{20}(4xy - 3y) = \frac{5xy}{20} \cdot 4xy + \frac{5xy}{20} \cdot (-3y)}$$

Step 2: Simplify the First Term

Now that we have applied the distributive property, let's simplify the first term: $\frac{5xy}{20} \cdot 4xy$. To simplify this term, we can start by multiplying the numerators and denominators separately.

$${\frac{5xy}{20} \cdot 4xy = \frac{5 \cdot 4 \cdot x \cdot y}{20 \cdot x \cdot y}}$$

Step 3: Cancel Out Common Factors

Now that we have multiplied the numerators and denominators, let's cancel out any common factors. In this case, we can cancel out the $x$ and $y$ terms in the numerator and denominator.

$${\frac{5 \cdot 4 \cdot x \cdot y}{20 \cdot x \cdot y} = \frac{5 \cdot 4}{20}}$$

Step 4: Simplify the Second Term

Now that we have simplified the first term, let's simplify the second term: $\frac{5xy}{20} \cdot (-3y)$. To simplify this term, we can start by multiplying the numerators and denominators separately.

$${\frac{5xy}{20} \cdot (-3y) = \frac{5 \cdot (-3) \cdot x \cdot y}{20 \cdot x \cdot y}}$$

Step 5: Cancel Out Common Factors

Now that we have multiplied the numerators and denominators, let's cancel out any common factors. In this case, we can cancel out the $x$ and $y$ terms in the numerator and denominator.

$${\frac{5 \cdot (-3) \cdot x \cdot y}{20 \cdot x \cdot y} = \frac{5 \cdot (-3)}{20}}$$

Step 6: Combine the Terms

Now that we have simplified both terms, let's combine them. To do this, we can add the two simplified terms together.

$${\frac{5 \cdot 4}{20} + \frac{5 \cdot (-3)}{20} = \frac{20 - 15}{20}}$$

Step 7: Simplify the Final Expression

Now that we have combined the terms, let's simplify the final expression. To do this, we can simplify the fraction by dividing the numerator and denominator by their greatest common divisor.

$${\frac{20 - 15}{20} = \frac{5}{20}}$$

Step 8: Simplify the Fraction

Now that we have simplified the fraction, let's simplify it further by dividing the numerator and denominator by their greatest common divisor.

$${\frac{5}{20} = \frac{1}{4}}$$

The final answer is $\boxed{\frac{1}{4}}$.

Conclusion

Simplifying the expression ${\frac{5xy}{20}(4xy - 3y)}$ requires careful application of the distributive property, simplification of terms, and cancellation of common factors. By following the steps outlined in this article, we can simplify the expression and arrive at the final answer of $\boxed{\frac{1}{4}}$. This process demonstrates the importance of algebraic manipulation in mathematics and provides a clear understanding of how to simplify complex expressions.

Frequently Asked Questions

• Q: What is the distributive property? A: The distributive property is a mathematical property that states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$.
• Q: How do I simplify a fraction? A: To simplify a fraction, you can divide the numerator and denominator by their greatest common divisor.
• Q: What is the greatest common divisor? A: The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder.

Additional Resources

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

References

• "Algebra" by Michael Artin
• "Calculus" by Michael Spivak
• "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
  

Introduction

In our previous article, we explored the process of simplifying the expression ${\frac{5xy}{20}(4xy - 3y)}$. We broke down the process into manageable steps, making it easier to understand and follow along. In this article, we will continue to provide a Q&A guide to algebraic manipulation, covering common questions and topics related to simplifying expressions.

Q&A: Algebraic Manipulation

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. This property allows us to expand expressions by multiplying each term in the first factor by each term in the second factor.

Q: How do I simplify a fraction?

A: To simplify a fraction, you can divide the numerator and denominator by their greatest common divisor (GCD). The GCD is the largest number that divides two or more numbers without leaving a remainder.

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. For example, the GCD of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 without leaving a remainder.

Q: How do I cancel out common factors?

A: To cancel out common factors, you need to identify the common factors in the numerator and denominator and then divide both by the common factor. For example, if you have the expression $\frac{6x}{12x}$, you can cancel out the common factor of $x$ by dividing both the numerator and denominator by $x$.

Q: What is the difference between simplifying and solving an equation?

A: Simplifying an expression involves reducing it to its simplest form, while solving an equation involves finding the value of the variable that makes the equation true. For example, the expression $\frac{6x}{12x}$ can be simplified by canceling out the common factor of $x$, but it is not a solution to an equation.

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

A: You should simplify an expression when it is necessary to make the expression easier to work with or to make it more understandable. For example, if you have a complex expression with many terms, simplifying it can make it easier to solve or manipulate.

Q: Can I simplify an expression with variables?

A: Yes, you can simplify an expression with variables. In fact, simplifying expressions with variables is a crucial skill in algebra and mathematics. You can use the same techniques for simplifying expressions with variables as you would for simplifying expressions with constants.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you need to apply the rules of exponents, such as the product rule and the power rule. For example, if you have the expression $x^2 \cdot x^3$, you can simplify it by applying the product rule, which states that $x^a \cdot x^b = x^{a+b}$.

Conclusion

Simplifying expressions is an essential skill in mathematics, and it requires careful application of the distributive property, simplification of terms, and cancellation of common factors. By following the steps outlined in this article and the previous article, you can simplify complex expressions and arrive at the final answer. Remember to always check your work and to use the correct techniques for simplifying expressions.

Frequently Asked Questions

• Q: What is the distributive property? A: The distributive property is a mathematical property that states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$.
• Q: How do I simplify a fraction? A: To simplify a fraction, you can divide the numerator and denominator by their greatest common divisor (GCD).
• Q: What is the greatest common divisor (GCD)? A: The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder.

Additional Resources

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

References

• "Algebra" by Michael Artin
• "Calculus" by Michael Spivak
• "Mathematics for Computer Science" by Eric Lehman and Tom Leighton