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Algebra is a fundamental part of the GMAT. It involves solving equations and understanding relationships between variables. GMAT algebra often requires you to manipulate equations using basic operations like addition, subtraction, multiplication, and division. In this particular problem, algebra helps us manipulate and combine the given equations. These basic steps help to isolate variables, so we can find their values. Mastering GMAT algebra techniques is crucial because many other sections of the exam build on these skills. For example, you'll often need algebra for solving word problems and data sufficiency questions.

Equation solving methods are essential for the GMAT. In this problem, we use a step-by-step approach: start by labeling and isolating equations, then solve for one variable at a time. Here's a quick breakdown:

1. **Label the Equations**: Identify and organize the equations given in the problem.
2. **Isolate One Variable**: Solve one of the equations for one of the variables. In this example, we solved for \(x\) in terms of \ y \.
3. **Substitute**: Substitute the expression of the isolated variable into the other equation. This substitution allows you to solve for one variable completely.
4. **Solve for Remaining Variable**: Solve the resulting equation for the remaining variable. Ensure to perform cross-multiplications and simplifying steps correctly.
These steps ensure you systematically break down the problem until you find an answer.

Cross-multiplication is a handy technique for solving fractions or ratios. It’s often used in algebra to eliminate fractions and simplify equations. Here's how it applied to our problem:
1. **Set Up the Equations**: From Equation 1: \( \frac{x+y}{x-y} = \frac{1}{2} \), cross-multiply to get \ 2(x+y) = x-y \. This removes the fraction and creates a simpler equation.
2. **Isolate Variables**: Simplify and rearrange terms. For example, from \ 2x + 2y = x - y \, we isolate \ x \ to get \ x = -3y \.
3. **Apply in Other Equations**: Substitute \ x = -3y \ in Equation 2: \( \frac{-3y + y}{y + 1} = 2 \). Cross-multiplying here simplifies the expression: \ -2y = 2(y + 1) \.
Cross-multiplying effectively clears fractions, making equations easier to solve. This technique is particularly useful in GMAT problems involving ratios and proportions.
Understanding when and how to use cross-multiplication will give you an edge in solving complex algebraic equations quickly and accurately.