Verified Answer :

  • Identify the distributive property: \(a(b + c) = ab + ac\).
  • Distribute \(-15\) to both terms inside the parentheses: \(-15 \times 5x = -75x\) and \(-15 \times 6y = -90y\).
  • Combine the results: \(-75x - 90y\).
  • Select the matching option: \(\boxed{-75x - 90y}\).

Explanation

  1. Understanding the Distributive Property

The expression to simplify is \(-15(5x + 6y)\). We are asked to rewrite this expression using the distributive property. The distributive property of multiplication over addition states that for any real numbers \(a\), \(b\), and \(c\), the following equality holds: \[a(b + c) = ab + ac.\] In our expression, we identify the variables as \(a = -15\), \(b = 5x\), and \(c = 6y\).

  1. Applying the Property and Simplifying

To apply the property, we distribute the multiplier \(-15\) to each of the terms inside the parentheses:
\[-15(5x + 6y) = (-15 \times 5x) + (-15 \times 6y).\]
Calculating each product individually:

  • For the first term, we have \(-15 \times 5x = -75x\).
  • For the second term, we have \(-15 \times 6y = -90y\).

Combining these, we get \(-75x + (-90y)\), which simplifies to \(-75x - 90y\).

  1. Final Result and Matching Option

Comparing our result, \(-75x - 90y\), with the given options:

  • \(-75x - 90y\)
  • \(-75x + 90y\)
  • \(-75x + 6y\)
  • \(-5x - 21y\)

Our result matches the first option. Thus, the final expression is \(-75x - 90y\).

Examples

The distributive property is a fundamental tool for simplifying complex expressions in algebra. For instance, if you are calculating the total cost of a shopping trip where you buy 15 units of two different types of items that cost \(5x\) and \(6y\) respectively, the expression \(-15(5x + 6y)\) represents the net change in your budget. By distributing the multiplier, you can quickly find the total change for each item type individually, making accounting for personal finances or inventory management much more efficient.

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