Equivalent Expressions Solving (7x + 3y)(8x + 5y)

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In the realm of algebra, simplifying expressions is a fundamental skill. This article delves into the process of finding an equivalent expression for the given product (7x + 3y)(8x + 5y). We will systematically explore the expansion and simplification steps, guiding you through the process and ultimately revealing the correct answer.

The Challenge: Finding the Equivalent Expression

Our starting point is the expression (7x + 3y)(8x + 5y). The goal is to determine which of the provided options (A, B, C, or D) is mathematically equivalent to this expression. To achieve this, we'll employ the distributive property, a cornerstone of algebraic manipulation.

Method 1: Step-by-Step Expansion

Keywords: Distributive property, FOIL method, expanding expressions, simplifying polynomials

To find the equivalent expression, we must expand the product using the distributive property. This method involves multiplying each term in the first binomial by each term in the second binomial. A helpful mnemonic for this process is the FOIL method, which stands for First, Outer, Inner, Last. By applying the distributive property carefully and combining like terms, we can transform the initial expression into a simpler, equivalent form. This step-by-step approach ensures accuracy and clarity in the simplification process, making it easier to identify the correct equivalent expression among the given options.

Applying the Distributive Property (FOIL Method)

  • First: Multiply the first terms of each binomial: 7x * 8x = 56x²
  • Outer: Multiply the outer terms: 7x * 5y = 35xy
  • Inner: Multiply the inner terms: 3y * 8x = 24xy
  • Last: Multiply the last terms: 3y * 5y = 15y²

This gives us the expanded expression: 56x² + 35xy + 24xy + 15y². Now, we need to combine the like terms to simplify further.

Combining Like Terms

  • We identify the like terms, which in this case are the 'xy' terms: 35xy and 24xy.
  • Adding these terms together, we get: 35xy + 24xy = 59xy

This simplification leads to the final equivalent expression: 56x² + 59xy + 15y².

Identifying the Correct Option

Keywords: Equivalent expression, polynomial simplification, combining like terms, algebraic manipulation

Having expanded and simplified the given expression, we now have 56x² + 59xy + 15y². We compare this result with the provided options to determine which one matches exactly. Option C, 56x² + 59xy + 15y², perfectly aligns with our derived expression. Therefore, Option C is the correct answer. By meticulously applying the distributive property and combining like terms, we have successfully identified the equivalent expression. This underscores the importance of careful algebraic manipulation in solving mathematical problems.

Why Other Options Are Incorrect

Keywords: Incorrect solutions, error analysis, polynomial expansion mistakes, common algebraic errors

It's important to understand why the other options are incorrect. Let's briefly analyze each one to pinpoint the potential errors in their derivation.

  • Option A: 7x² + 23xy + 5y² – This option is incorrect because it doesn't correctly multiply the coefficients and variables during the expansion process. For instance, the x² term should be 7x * 8x = 56x², not 7x². Similarly, the xy term and y² term calculations are flawed.
  • Option B: 7x² + 24xy + 8y² – This option suffers from similar errors in multiplication. The coefficients and variables are not properly handled during expansion. The x² term, xy term, and y² term are all incorrect when compared to the correct expansion.
  • Option D: 56x² + 35xy + 15y² – This option correctly calculates the x² and y² terms but makes a mistake in combining the xy terms. It only considers the 35xy term from 7x * 5y but misses the 24xy term from 3y * 8x. The correct xy term should be the sum of both, which is 59xy.

By understanding the mistakes that lead to these incorrect options, we reinforce our understanding of the correct algebraic procedures. This kind of error analysis is crucial for avoiding similar mistakes in the future.

Conclusion

Keywords: Simplifying expressions, algebraic skills, problem-solving strategies, mathematical proficiency

In summary, the expression (7x + 3y)(8x + 5y) is equivalent to 56x² + 59xy + 15y², which corresponds to option C. This solution was obtained by systematically applying the distributive property (FOIL method) and combining like terms. This exercise highlights the importance of mastering basic algebraic techniques for simplifying expressions. Developing strong algebraic skills is crucial for success in more advanced mathematical topics. By practicing and understanding these fundamental concepts, we enhance our problem-solving strategies and improve our overall mathematical proficiency.

Understanding how to expand and simplify expressions like this is a crucial skill in algebra. It forms the foundation for solving more complex equations and problems. By mastering these techniques, you'll be well-equipped to tackle various mathematical challenges.

Method 2: Substitution (Verification)

Keywords: Substitution method, verifying solutions, alternative methods, checking answers

Another way to approach this problem, especially for verification, is the substitution method. This involves substituting numerical values for the variables (x and y) in the original expression and the potential equivalent expressions. If an option is truly equivalent, it should yield the same result as the original expression for any chosen values of x and y. This method serves as a valuable check, ensuring the accuracy of the simplification process. However, it's important to choose values that are easy to calculate and avoid special cases (like 0 or 1) that might mask errors.

Applying the Substitution Method

Let's choose x = 1 and y = 1 for simplicity.

  • Original Expression: (7x + 3y)(8x + 5y) = (7(1) + 3(1))(8(1) + 5(1)) = (10)(13) = 130

Now, we substitute these values into each of the options:

  • Option A: 7x² + 23xy + 5y² = 7(1)² + 23(1)(1) + 5(1)² = 7 + 23 + 5 = 35
  • Option B: 7x² + 24xy + 8y² = 7(1)² + 24(1)(1) + 8(1)² = 7 + 24 + 8 = 39
  • Option C: 56x² + 59xy + 15y² = 56(1)² + 59(1)(1) + 15(1)² = 56 + 59 + 15 = 130
  • Option D: 56x² + 35xy + 15y² = 56(1)² + 35(1)(1) + 15(1)² = 56 + 35 + 15 = 106

Only Option C yields the same result (130) as the original expression. This verifies that Option C is indeed the equivalent expression.

Advantages and Limitations

The substitution method provides a quick way to check answers and confirm the equivalence of expressions. However, it's not a foolproof method for deriving the solution from scratch. It's primarily used for verifying solutions obtained through other methods, like the distributive property. Additionally, choosing specific values might not reveal subtle errors that could exist for other values. Therefore, while substitution is a valuable tool, it's best used in conjunction with other algebraic techniques.

Conclusion: Solidifying Understanding

Keywords: Comprehensive understanding, algebraic techniques, problem-solving approach, mathematical skills

In conclusion, both the step-by-step expansion method and the substitution method confirm that Option C, 56x² + 59xy + 15y², is the equivalent expression for (7x + 3y)(8x + 5y). Understanding both methods provides a comprehensive understanding of the problem and reinforces our algebraic techniques. By mastering these techniques, we develop a strong problem-solving approach that can be applied to a wide range of mathematical challenges. This exercise emphasizes the importance of not only finding the solution but also understanding the underlying principles and being able to verify the answer using different methods. Consistent practice and a solid grasp of fundamental concepts are key to building strong mathematical skills and achieving success in algebra and beyond.

By mastering these techniques, you'll be well-equipped to tackle various mathematical challenges and develop a deeper appreciation for the elegance and power of algebra.