91Ó°ÊÓ

Algebraic equations are mathematical statements where two expressions are set equal to each other. In the given exercise, we have two algebraic equations: \(\frac{x}{y} = 2\) and \(\frac{y}{x} = \frac{1}{2}\). These equations describe a specific relationship between the variables x and y. Understanding these relationships is crucial as they allow us to manipulate and solve for the variables.

When confronted with algebraic equations, keep these key points in mind:

• Identify the given variables and their relationships.
• Look for ways to isolate one of the variables.
• Use algebraic properties like addition, subtraction, multiplication, and division to manipulate the equations.

In our problem, multiplying both sides of \(\frac{x}{y} = 2\) by y helps us isolate x, yielding \(x = 2y\). This isolated equation is crucial for the next step.

The substitution method is a technique used to solve systems of equations where one equation is solved for one variable and then substituted into another equation. It's especially useful when the equations are intertwined, as in our exercise.

From the equation \(x = 2y\), we know x in terms of y. We can substitute this expression for x into the second equation \(\frac{y}{x} = \frac{1}{2}\). This substitution changes our second equation to \(\frac{y}{2y} = \frac{1}{2}\), which simplifies to \(\frac{1}{2} = \frac{1}{2}\).

Substitution method steps:

• First, isolate one of the variables in one of the equations.
• Next, substitute the isolated variable’s expression into the other equation.
• Finally, simplify and solve.

This simplification indicates that our chosen substitution is accurate, as both sides of the equation match.

Value determination is the process of finding the actual values of variables that satisfy given equations. After performing substitution and simplification, the objective is to check if a specific value of the variable can be determined.

In our exercise, after substituting \(x = 2y\) into \(\frac{y}{x} = \frac{1}{2}\), we got an identity \(\frac{1}{2} = \frac{1}{2}\). This shows that there is no contradiction and the relationship holds true for any positive y.

The following steps are used in value determination:

• Simplify equations until you get a relationship or a contradiction.
• Check if the simplified equation allows you to determine a specific value for the variable.
• If no specific value is required (as in our case), acknowledge that the variable can take any permissible value that satisfies the equation.

Here, since our equations didn't restrict y to a specific number, we conclude that y can be any positive number. Hence, in the context of the GMAT problem, correct understanding of the problem leads us to see that y cannot be a specific set of values from the given options, indicating the answer is: Any positive number.