If \(y=x / 3+3 / x\) and \(3 x=1\), then \(y\) is: A. \(10 / 9\) B. \(18 / 3\) C. \(82 / 9\) D. \(8 / 9\) E. None of the above

Algebraic equations form the backbone of numerous mathematical problems, including those found on the GMAT. In our given exercise, we have the equations:

\(y = \frac{x}{3} + \frac{3}{x}\)
and
\(3x = 1\).

These equations help us relate different variables and find unknown values. By breaking these equations down, we can solve for one variable to understand its relationship with others.

To begin with, solving for x in the equation \(3x = 1\) yields \(x = \frac{1}{3}.\) Using algebraic principles like isolating variables and performing basic arithmetic operations such as addition, subtraction, multiplication, and division allows for finding the exact values of unknowns.

Grasping algebraic equations is crucial, as this knowledge will enable solving more complex problems effectively.

The substitution method is a highly valued technique for solving algebraic equations. This method involves solving one equation for one variable and then substituting that solution into another equation.

In our problem, we first found the value of x by solving the equation \(3x = 1\). This gave us \(x = \frac{1}{3}.\)

Next, we substituted \(x = \frac{1}{3}\) into the original equation \(y = \frac{x}{3} + \frac{3}{x}.\) Substitution transforms our equation to:

\(y = \frac{\frac{1}{3}}{3} + \frac{3}{\frac{1}{3}} \).
This method simplifies our problem by turning it into a single-variable equation, making it much easier to solve.

The substitution was essential to break down the problem into manageable parts, demonstrating how versatile this technique is in solving various algebra problems.

Providing solutions step-by-step is incredibly beneficial for students, as it allows for a better understanding of the process involved in solving complex problems. Here, let's break down the steps used to solve our problem:

• **Step 1**: Identify the given equations and solve for one variable. Given \(3x = 1\), solving for x gives \(x = \frac{1}{3}.\)


• **Step 2**: Substitute the value of x into the second equation. Substituting \(x = \frac{1}{3}\) into \(y = \frac{x}{3} + \frac{3}{x}\) gives us \(y = \frac{\frac{1}{3}}{3} + \frac{3}{\frac{1}{3}}.\)


• **Step 3**: Simplify the resulting equation step-by-step. The fraction \( \frac{\frac{1}{3}}{3} \) simplifies to \( \frac{1}{9}\) and \( \frac{3}{\frac{1}{3}} \) simplifies to 9, thus the equation becomes \( y = \frac{1}{9} + 9.\)


• **Step 4**: Combine the terms and calculate the final value. Add the fractions: \( y = \frac{1}{9} + \frac{81}{9} = \frac{82}{9}.\)

Step-by-step solutions enable a deeper understanding of the reasoning and processes involved in solving math problems. Reviewing each step ensures that no detail is overlooked, making it easier to grasp even the most challenging concepts.