91Ó°ÊÓ

Algebra is like the language of mathematics. It allows us to find unknown values using known numbers and symbols. In this problem, we're asked if two variables, \(x\) and \(y\), are the same. Understanding algebraic expressions and how to manipulate them is key. For instance, statement (1) involves manipulating fractions and sums, which are fundamental operations in algebra. When working with expressions like \((x+y)\left(\frac{1}{x}+\frac{1}{y}\right)\), it's crucial to break them down step-by-step:

• Combine like terms: This means grouping similar terms to simplify. In this case, \(\frac{1}{x}+\frac{1}{y}\) simplifies to \(\frac{x+y}{xy}\).
• Apply distributive properties: Distribute \((x+y)\) across the simplified fraction, further simplifying it to \(\frac{(x+y)^2}{xy}\).

Breaking the problem into smaller parts helps simplify the expression and uncover relationships between variables. Such skills are essential when tackling more complex algebra problems.

At the heart of finding out if \(x = y\) is solving equations correctly. The goal is to isolate variables and simplify equations to reveal clear solutions. Statement (2) directly states that \(x - 100 = y - 100\), simplifying to \(x = y\). This is a perfectly solved equation that explicitly answers the problem by itself. Equation solving often involves several steps:

• Look for direct simplifications: Like in statement (2), where subtracting similar numbers on both sides reveals our answer immediately.
• Use substitutions: When statement (1) is considered, after finding \(x = y\), substituting back into the equation helps verify the truth of this statement.

Where possible, simplify equations first by combining similar terms or isolating variables. This makes it easier to see if and when solutions match or differ.

Logical reasoning is about understanding relationships and using statements to form conclusions. In this GMAT problem, reasoning is as important as math skills. The problem presents two statements, and using logic helps determine which one confirms the solution best:

• Deductive logic in math: Statement (2) directly confirms \(x = y\). We use logical deduction to validate the truth of the condition.
• Verification: Even though statement (1) is more complex, using the solution from statement (2), you can confirm correctness through substitution back into the equation.

Logical reasoning makes it easier to decide the value of information each statement provides. By combining logic with mathematical methods, you validate the findings comprehensively. Use logical reasoning to assess whether assumptions fit, ensuring solutions are both mathematically and logically sound.