Question

What is the value of \(x y\) ? (1) \(x^3 y^3=8\) (2) \(5 \mathrm{x}+10 \mathrm{y}=25\) A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. 1 and 2 together are not sufficient

Short answer

Option A is correct: Equation (1) alone is sufficient.

Step-by-step solution

- Analyze Equation (1)

Given equation (1) is \(x^3 y^3 = 8\). This can be rewritten using the property of exponents:\((xy)^3 = 8\). Taking the cube root of both sides, we get:\(xy = \root{3}{8} = 2\). So, equation (1) is sufficient to find the value of \(xy\).

- Analyze Equation (2)

Given equation (2) is \(5x + 10y = 25\). Dividing the entire equation by 5:\(x + 2y = 5\). Without additional information, there are infinite combinations of \(x\) and \(y\) that satisfy this equation, making it not sufficient alone to find the value of \(xy\).

- Combining Both Equations

Since equation (1) already provided a direct value for \(xy\), there is no need to combine the equations. Equation (1) alone is sufficient to determine the value of \(xy\).

Conclusion

From the analysis, equation (1) alone is sufficient to find the value of \(xy\), while equation (2) alone is not sufficient. Thus, the answer is option A.

Key concepts

Algebra

Algebra is a critical branch of mathematics focused on solving equations and understanding variables. In this particular problem, we're dealing with equations involving variables like x and y. The goal is to determine the value of the product xy. One useful algebraic step we took was to manipulate equation (2) by simplifying it. Initially, it was given as 5x + 10y = 25. By dividing all terms by 5, we simplified it to x + 2y = 5. This simplification helps to see relationships more clearly, even though it alone does not solve for xy.

Exponents

Exponents are another fundamental concept in mathematics. They denote repeated multiplication of a base number. In the problem, equation (1) is given as \(x^3 y^3 = 8\). By understanding the properties of exponents, we rewrote this equation as \((xy)^3 = 8\). This transformation made it easier to solve by taking the cube root of both sides, resulting in \(xy = \sqrt[3]{8} = 2\). This shows how powerful understanding and applying exponent rules can be.

Equation Analysis

Analyzing equations involves understanding their structure and implications. For this problem, we first analyzed equation (1) and determined it was adequate to find the value of xy directly. By contrast, equation (2), even after simplification to \(x + 2y = 5\), did not yield a direct value for xy without additional information. This analysis showed that not all equations will directly lead to a solution; some may provide boundaries or possible relationships instead of direct answers.

Data Sufficiency

In GMAT math problems, determining if the provided data is sufficient to answer the question is crucial. The steps showed that equation (1) was sufficient to determine the value of the product xy, while equation (2) was not. Understanding when you have enough information is key in data sufficiency questions. The correct answer was to identify that equation (1) alone suffices, demonstrating effective data analysis skills in problem-solving contexts.