Introduction to Solving Math Equations

Solving math equations often requires creative and methodical approaches. This article explores a detailed method for finding all solutions in given equations. We will use a step-by-step approach to unravel the mysteries hidden within the equations.

Step-by-Step Guide to Solving Math Equations

Dividing Equations to Simplify Complexity

One common technique in solving math problems involves dividing equations to simplify the complexity. Let's examine two given equations and demonstrate the process:

Original Equations:

(displaystyle x^{2} - 2x y^{2} - 2y 27xy quad text{and} quad x^{2} - 1 - y^{2} - 1 1y)

We divide both sides of the equations by (xy), resulting in:

(displaystyle frac{1}{x} 2frac{1}{y} 2 27 quad text{and} quad frac{1}{x} - 1frac{1}{y} - 1 0)

Now, let's introduce new variables to represent these simplified forms:

(X frac{1}{x} quad text{and} quad Y frac{1}{y})

The simplified equations become:

(X^2 Y^2 27 quad text{and} quad XY 10)

From these, we derive:

(2XY 13 quad text{and} quad XY 10)

Exploring Positive and Negative Solutions

Solutions When (xy geq 0)

When (xy geq 0), the solutions can be found using the following steps:

(displaystyle sqrt{x}frac{1}{sqrt{x}}^2 sqrt{y}frac{1}{sqrt{y}}^2 27)

(displaystyle xfrac{1}{x} yfrac{1}{y} 10)

Introduce variables (displaystyle u sqrt{x}frac{1}{sqrt{x}} quad text{and} quad v sqrt{y}frac{1}{sqrt{y}})

The equations simplify to:

(u^2v^2 27)

(u^2 - 2 v^2 - 2 10)

Further simplification results in:

(u^2v^2 frac{21}{2})

(u^2 - v^2 pm sqrt{(u^2v^2)^2 - 4} pm frac{3}{2})

Solving for (X) and (Y), we get:

(X^2 - 4X 1 0 quadRightarrow quad X 2 pm sqrt{3})

(Y^2 - frac{5}{2}Y 2 0 quadRightarrow quad Y 2 pm frac{1}{2})

The solutions in terms of (x) and (y) are:

(displaystyle x 2 pm sqrt{3} quad text{and} quad y 2 pm frac{1}{2})

Note that there are multiple combinations of solutions:

(displaystyle xy in left{2sqrt{3} cdot frac{2 sqrt{3}}{2 sqrt{3}} - 2 - sqrt{3} cdot frac{2 sqrt{3}}{2 sqrt{3}} - 2 - sqrt{3} cdot frac{1}{2 sqrt{3}} - 2 frac{1}{2}, 2 - sqrt{3} cdot frac{1}{2 - sqrt{3}} - 2 sqrt{3} cdot frac{1}{2 - sqrt{3}} - 2 - sqrt{3} cdot frac{2 sqrt{3}}{2 sqrt{3}} - 2 - sqrt{3} cdot frac{1}{2 sqrt{3}} - 2 frac{1}{2}right})

Solutions When (xy leq 0)

When (xy leq 0), the equations take a different form:

(x^{2} - 2x y^{2} - 2y 27xy)

(x^{2} - 1 - y^{2} - 1 1y)

Introducing similar transformations, we find that the original module is not valid in this scenario. To solve it, we change the signs of (x) and (y).

(displaystyle sqrt{x} - frac{1}{sqrt{x}}^2 sqrt{y} - frac{1}{sqrt{y}}^2 27)

(displaystyle x - frac{1}{x} 10)

Introduce new variables:

(u sqrt{x} - frac{1}{sqrt{x}})

(v sqrt{y} - frac{1}{sqrt{y}})

The equations become:

(u^2v^2 27)

(u^2 - 2 v^2 - 2 10)

Further simplification results in:

(u^2v^2 frac{-21}{2})

No real solutions exist in this case.