Introduction to the Equation

The equation y x - x/R may seem straightforward at first glance, but unraveling its complexities reveals a rich tapestry of algebraic principles. This article will guide you through the process of solving this equation for x, providing a detailed explanation of each step and the underlying mathematical concepts.

Understanding the Equation

Let us begin by examining the given equation:

Equation

y x - x/R

This equation is intriguing because it involves both a linear term x and a term that incorporates the variable x divided by a constant R. To solve for x, we need to isolate it on one side of the equation.

Step 1: Convert the Equation to a Quadratic Form

In order to solve for x, we need to convert the equation into a quadratic form. Let's start by clearing the fraction by multiplying every term by R (assuming R ≠ 0):

Equation Conversion

Ry Rx - x

Next, we will rearrange this equation to bring all terms involving x to one side:

Ry x Rx

Now, let's move the term Ry to the right side:

Quadratic Form

x - Rx Ry 0

This can be simplified further:

Simplified Quadratic Form

Ry - x^2/R x 0

Let's multiply every term by R again to clear the fraction:

Clearing the Fraction

Ry x - x^2 0

We can rearrange this equation to form a standard quadratic equation:

Standard Quadratic Equation

x^2 - Rx Ry 0

Step 2: Applying the Quadratic Formula

Now that we have the equation in a standard quadratic form, we can apply the quadratic formula to solve for x. The quadratic formula is given by:

Quadratic Formula

x [-b ± sqrt(b^2 - 4ac)] / (2a)

In our equation x^2 - Rx Ry 0, we identify the coefficients:

a 1 b -R c Ry

Substituting these values into the quadratic formula:

Substituting Values

x [R ± sqrt((R)^2 - 4 · 1 · Ry)] / (2 · 1)

Simplifying the expression:

x [R ± sqrt(R^2 - 4Ry)] / 2

Step 3: Simplifying the Solution

The expression [R ± sqrt(R^2 - 4Ry)] / 2 represents the two possible solutions for x. These solutions can be represented as:

Simplified Solutions

x (R sqrt(R^2 - 4Ry)) / 2 x (R - sqrt(R^2 - 4Ry)) / 2

These solutions provide us with the values of x for any given y and constant R. It is important to note that these solutions are valid only if the discriminant (R^2 - 4Ry) is non-negative. If the discriminant is negative, there are no real solutions for x.

Visualization and Further Insights

The equation y x - x/R can be visualized as a set of parabolas. For various values of R, the parabolas will be positioned differently but will always intersect the line y x.

When R 1, the equation simplifies to:

Parabola at R 1

y x - 1/x

When R 2, the equation becomes:

Parabola at R 2

y x - x/2

And so on for any other values of R.

Each of these parabolas will touch the origin (0,0) and cross the line y x in a manner consistent with the solutions derived for x.

Conclusion

In conclusion, solving the equation y x - x/R involves first converting it to a quadratic form and then applying the quadratic formula. The solutions are:

Solution

x (R sqrt(R^2 - 4Ry)) / 2 x (R - sqrt(R^2 - 4Ry)) / 2

These solutions provide a comprehensive understanding of how to solve the given equation, and the visualization of the parabolas offers additional insights into the behavior of the equation for different values of R.