Multiply Using The Distributive Property A Comprehensive Guide

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$(5x - 5y - 9)(-xy) = $

Let's delve into the world of algebraic expressions and explore how to effectively multiply them using the distributive property. This fundamental property is a cornerstone of algebra, enabling us to simplify complex expressions and solve equations with ease. In this comprehensive guide, we'll break down the distributive property, illustrate its application with a specific example, and provide you with the knowledge and skills to confidently tackle similar problems.

The distributive property, in its essence, provides a method for multiplying a single term by a group of terms enclosed within parentheses. It states that for any numbers or algebraic expressions a, b, and c, the following holds true:

a( b + c) = a b + a c

In simpler terms, to multiply a term by a sum or difference within parentheses, we distribute the term to each individual term inside the parentheses. This involves multiplying the term by each term inside, maintaining the original signs. The distributive property extends to situations with more than two terms inside the parentheses as well. For instance:

a( b + c + d) = a b + a c + a d

Now, let's apply this powerful property to the problem at hand:

$(5x - 5y - 9)(-xy) = $

Our goal is to multiply the term (-xy) by the expression (5x - 5y - 9). We'll achieve this by distributing (-xy) to each term within the parentheses.

Step 1: Distribute (-xy) to the first term (5x)

(-xy) * (5x) = -5x²y

Here, we multiply the coefficients (-1 and 5), and then multiply the variables. x times x gives us x², and y remains as it is.

Step 2: Distribute (-xy) to the second term (-5y)

(-xy) * (-5y) = 5xy²

In this step, we multiply the coefficients (-1 and -5), which results in a positive 5. Then, we multiply the variables. y times y gives us y², and x remains as it is.

Step 3: Distribute (-xy) to the third term (-9)

(-xy) * (-9) = 9xy

Here, we multiply the coefficients (-1 and -9), which results in a positive 9. The variables x and y remain as they are.

Step 4: Combine the results

Now, we combine the results from the previous steps to obtain the final simplified expression:

-5x²y + 5xy² + 9xy

Therefore, (5x−5y−9)(−xy)=−5x2y+5xy2+9xy(5x - 5y - 9)(-xy) = -5x^2y + 5xy^2 + 9xy

This is our simplified answer. We have successfully multiplied the expression using the distributive property.

Key Concepts and Considerations

  • Sign Awareness: Pay close attention to the signs of the terms when distributing. A negative term multiplied by a negative term results in a positive term, while a negative term multiplied by a positive term results in a negative term.
  • Variable Multiplication: When multiplying variables with exponents, remember to add the exponents if the bases are the same. For example, x * x* = x^(1+1) = x².
  • Combining Like Terms: After distributing, check if there are any like terms that can be combined to further simplify the expression. Like terms are terms that have the same variables raised to the same powers.

Practice Problems

To solidify your understanding of the distributive property, try solving the following practice problems:

  1. (3a + 2b)(a)
  2. (-2p + 7q - 4)(3pq)
  3. (4m - 6n + 1)(-2mn)

Real-World Applications

The distributive property isn't just an abstract mathematical concept; it has practical applications in various real-world scenarios. For instance:

  • Calculating Area: If you need to find the area of a rectangular garden with a length of (x + 5) meters and a width of x meters, you can use the distributive property to multiply x by (x + 5), resulting in the area x² + 5x square meters.
  • Budgeting: Imagine you're planning a party and need to calculate the total cost. If you're buying 10 items that cost (y + $2) each, the distributive property helps you find the total cost: 10(y + $2) = 10y + $20.
  • Discounts and Sales: When calculating sale prices, the distributive property can be used to determine the discounted amount. For example, if an item costs $z and is on sale for 20% off, you can calculate the discount by multiplying 0.20 by $z.

Conclusion

The distributive property is a fundamental concept in algebra that empowers us to simplify expressions and solve equations. By mastering this property, you'll gain a valuable tool for tackling a wide range of mathematical problems. Remember to practice regularly and apply the concepts to real-world scenarios to further enhance your understanding.

By understanding and applying the distributive property, you'll be well-equipped to tackle algebraic expressions and solve equations with confidence. This foundational skill will serve you well in your mathematical journey.

Multiply the expression (5x - 5y - 9) by (-xy) using the distributive property.

Multiply Using the Distributive Property: Step-by-Step Guide with Examples