Equivalent Expression For 5x + 10y - 15

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In this article, we'll delve into the process of identifying the expression equivalent to the algebraic expression 5x + 10y - 15. This involves understanding the principles of factoring and the distributive property, which are fundamental concepts in algebra. We'll break down the given expression, explore the potential equivalent expressions, and use step-by-step explanations to arrive at the correct answer. By the end of this guide, you'll not only be able to solve this specific problem but also gain a solid foundation for tackling similar algebraic challenges. So, let's embark on this mathematical journey together and unravel the mysteries of equivalent expressions.

Understanding the Problem

To find the equivalent expression, we need to factor out the greatest common factor (GCF) from the terms of the expression 5x + 10y - 15. The greatest common factor is the largest number that divides evenly into all the coefficients (the numbers in front of the variables) and the constant term. In this case, the coefficients are 5 and 10, and the constant term is -15. Let's break down each term to identify the GCF.

  • The first term, 5x, has a coefficient of 5.
  • The second term, 10y, has a coefficient of 10.
  • The third term, -15, is a constant term.

The factors of 5 are 1 and 5. The factors of 10 are 1, 2, 5, and 10. The factors of 15 are 1, 3, 5, and 15. By comparing these factors, we can see that the greatest common factor of 5, 10, and 15 is 5. This means we can factor out 5 from the entire expression. Factoring is essentially the reverse of distribution; it involves identifying a common factor and extracting it from each term. This process simplifies the expression while maintaining its value, making it easier to work with in various mathematical operations. Understanding factoring is crucial for solving equations, simplifying expressions, and tackling more advanced algebraic concepts.

Factoring the Expression

Now that we've identified the GCF as 5, we can factor it out of the expression 5x + 10y - 15. This involves dividing each term by the GCF and writing the expression as a product of the GCF and the resulting terms in parentheses. Here’s how we do it:

  1. Divide each term by 5:
    • 5x / 5 = x
    • 10y / 5 = 2y
    • -15 / 5 = -3
  2. Write the expression with the GCF (5) outside the parentheses and the results of the division inside the parentheses:
    • 5(x + 2y - 3)

So, factoring out the greatest common factor from the original expression transforms it into a more compact and simplified form. The expression 5(x + 2y - 3) is equivalent to the original 5x + 10y - 15, but it represents the expression in a factored form. Factoring not only simplifies expressions but also provides valuable insights into the underlying structure of the algebraic relationship. It's a key skill in algebra that unlocks the ability to solve complex problems more efficiently. This factored form is easier to manipulate in many algebraic operations, making it a useful tool in various mathematical contexts.

Evaluating the Answer Choices

Now, let's compare the factored expression we found, 5(x + 2y - 3), with the answer choices provided:

A. 5(x - 2y - 3) B. 5(x + 5y - 10) C. 5(x + 2y - 3) D. 5(x + 2y - 15)

By direct comparison, we can see that answer choice C, 5(x + 2y - 3), matches our factored expression perfectly. The other options differ in the signs or coefficients within the parentheses, indicating that they are not equivalent to the original expression. To confirm our answer, we can use the distributive property to expand answer choice C and see if it matches the original expression. The distributive property states that a(b + c) = ab + ac. Applying this to our expression:

  • 5(x + 2y - 3) = 5 * x + 5 * 2y + 5 * (-3) = 5x + 10y - 15

As we can see, expanding the expression 5(x + 2y - 3) gives us the original expression 5x + 10y - 15, which further confirms that answer choice C is indeed the correct equivalent expression. This process of comparing and verifying the factored expression against the given choices is a crucial step in ensuring the accuracy of the solution. It reinforces the understanding of equivalence in algebraic expressions and builds confidence in problem-solving skills.

The Correct Answer

Based on our analysis, the expression equivalent to 5x + 10y - 15 is 5(x + 2y - 3). Therefore, the correct answer is C. This conclusion is supported by our step-by-step process of identifying the greatest common factor, factoring it out, and comparing the resulting expression with the provided answer choices. By applying the distributive property, we further validated our answer, ensuring its accuracy. Understanding these algebraic principles is essential for solving a wide range of mathematical problems, and this example demonstrates the importance of a systematic approach to problem-solving. In mathematics, precision and accuracy are paramount, and by carefully following each step, we can confidently arrive at the correct solution.

Why Other Options are Incorrect

It's also crucial to understand why the other options are incorrect. This helps solidify our understanding of the concepts and prevents similar mistakes in the future. Let's examine each incorrect option:

  • A. 5(x - 2y - 3): If we distribute the 5, we get 5x - 10y - 15, which differs from the original expression in the sign of the 10y term. This highlights the importance of paying attention to the signs when factoring and distributing.
  • B. 5(x + 5y - 10): Distributing the 5 here gives us 5x + 25y - 50, which is significantly different from the original expression. The coefficients of the y term and the constant term do not match, indicating that this expression is not equivalent.
  • D. 5(x + 2y - 15): Distributing the 5 results in 5x + 10y - 75. While the x and y terms match the original expression, the constant term is incorrect. This underscores the necessity of ensuring that all terms match when determining equivalence.

By analyzing these incorrect options, we can see how subtle differences in signs and coefficients can lead to incorrect expressions. This reinforces the importance of careful and methodical application of algebraic principles when simplifying and factoring expressions. Recognizing these potential pitfalls is a valuable step in mastering algebraic problem-solving.

Key Takeaways

This problem illustrates several key algebraic concepts:

  • Greatest Common Factor (GCF): Identifying the GCF is the first step in factoring expressions. It simplifies the expression while preserving its value.
  • Factoring: Factoring is the reverse of distribution and involves extracting the GCF from each term.
  • Distributive Property: The distributive property is used to expand expressions and verify the equivalence of factored forms.
  • Equivalence: Equivalent expressions represent the same mathematical relationship, even if they appear different.

Mastering these concepts is crucial for success in algebra and beyond. By understanding the principles behind factoring and the distributive property, you can confidently tackle a wide range of algebraic problems. This problem-solving approach not only helps in finding the correct answer but also builds a strong foundation for more advanced mathematical studies.

Conclusion

In summary, we've successfully identified that the expression equivalent to 5x + 10y - 15 is 5(x + 2y - 3). This was achieved by finding the greatest common factor, factoring it out of the expression, and verifying the answer using the distributive property. Understanding these concepts is fundamental to mastering algebra. By practicing these techniques, you'll enhance your problem-solving skills and build a solid foundation for future mathematical endeavors. Remember, mathematics is a journey of discovery, and each problem solved is a step forward in your understanding.