Understanding and Solving for x: The Equation Y x / (x 1)

Introduction to the Problem

The equation Y x/(x 1) presents an interesting algebraic challenge. Often, such problems require one to solve for a variable in terms of another. This article will walk you through the steps to solve for x in terms of y.

Step-by-Step Solution

Let's start with the given equation:

Y x / (x 1)

Our goal is to express x in terms of y. Here are the steps to achieve this:

Step 1: Multiply Both Sides by (x 1)

First, we multiply both sides of the equation by x 1 to eliminate the denominator:

Y(x 1) x

Step 2: Distribute Y on the Left Hand Side

Next, distribute Y on the left-hand side:

Yx Y x

Step 3: Subtract x from Both Sides

Now, we subtract x from both sides to collect like terms:

Yx - x Y 0

Simplifying gives us:

Yx - x -Y

Step 4: Factor Out x

We can now factor out x from the left-hand side:

(Y - 1)x -Y

Step 5: Solve for x

Finally, we divide both sides by Y - 1 to solve for x in terms of y:

x -Y / (Y - 1)

Alternative Simplification

A simpler form of the solution can be obtained by multiplying numerator and denominator by -1:

x -1 / (-1) × -Y / (Y - 1)

Which simplifies to:

x Y / (1 - Y)

Conclusion

In conclusion, the equation Y x / (x 1) can be solved for x in terms of y as x Y / (1 - Y). This form of the equation is particularly useful in various mathematical applications, such as calculus and optimization problems.

More Information

For further reading on related concepts, consider exploring algebraic manipulations, solving equations for variables, and understanding the behavior of rational functions. These concepts are fundamental in many areas of mathematics and have practical applications in fields such as engineering, physics, and computer science.