Solving the Complex Equation: 1/x1 2/x2 3/x3 25/(25-13x) 4

The equation given is a complex polynomial equation that involves multiple variables and fractions. This type of equation is common in various fields, including engineering, physics, and mathematics. Let's break down the solution step by step to understand how to solve such equations.

Given Equation

The initial given equation is:

1/x1 2/x2 3/x3 25/(25-13x) 4

Step-by-Step Solution

To solve the given equation, we start by expressing it in a more manageable form. The equation can be written as:

(frac{1}{x_1} frac{2}{x_2} frac{3}{x_3} frac{25}{25-13x} 4)

Step 1: Combine the Fractions

First, let's combine the fractions by finding a common denominator. The common denominator for (x_1, x_2, x_3,) and (25-13x) is (x_1x_2x_3(25-13x)). Therefore, we rewrite the equation as:

(frac{x_2x_3(25-13x) 2x_1x_3(25-13x) 3x_1x_2(25-13x) 25x_1x_2x_3}{x_1x_2x_3(25-13x)} 4)

This simplifies to:

(frac{25x_1x_2x_3 - 13x_1x_2x_3 2x_1x_3(25-13x) 3x_1x_2(25-13x) 25x_1x_2x_3}{x_1x_2x_3(25-13x)} 4)

For simplicity, let's denote the denominator as (D x_1x_2x_3(25-13x)). Therefore, the numerator is:

(25x_1x_2x_3 - 13x_1x_2x_3 5_1x_3 - 13x_1x_3 75x_1x_2 - 39x_1x_2 25x_1x_2x_3)

Simplifying further:

(53x_1x_2x_3 - 14x_1x_2 - 79x_1x_3 36x_1 - 13x_1x_2x_3 25x_1x_2x_3 - 13x 4D)

Therefore, the equation becomes:

(frac{53x_1x_2x_3 - 14x_1x_2 - 79x_1x_3 36x_1 - 13x_1x_2x_3 25x_1x_2x_3 - 13x}{D} 4)

This simplifies to:

(frac{53x_1x_2x_3 - 14x_1x_2x_3 - 6_1x_2 - 79x_1x_3 36x_1 - 13x}{D} 4)

Multiplying both sides by the denominator (D x_1x_2x_3(25-13x)) to clear the fraction:

(53x_1x_2x_3 - 14x_1x_2 - 79x_1x_3 36x_1 - 13x 4x_1x_2x_3(25-13x))

Step 2: Simplify the Equation

Expanding the right-hand side:

(53x_1x_2x_3 - 14x_1x_2 - 79x_1x_3 36x_1 - 13x 10_1x_2x_3 - 52x_1x_2x_3 - 13x)

Simplifying further:

(53x_1x_2x_3 - 14x_1x_2 - 79x_1x_3 36x_1 - 13x 10_1x_2x_3 - 52x_1x_2x_3 - 13x)

Moving all terms to one side:

(53x_1x_2x_3 - 14x_1x_2 - 79x_1x_3 36x_1 - 13x - 10_1x_2x_3 52x_1x_2x_3 13x 0)

Simplifying the terms:

(-52x_1x_2x_3 - 14x_1x_2 - 79x_1x_3 36x_1 0)

Therefore, the equation becomes:

(-52x_1x_2x_3 - 14x_1x_2 - 79x_1x_3 36x_1 0)

Factoring out (x_1):

(x_1(-52x_2x_3 - 14x_2 - 79x_3 36) 0)

This gives us two solutions:

(x_1 0) or (-52x_2x_3 - 14x_2 - 79x_3 36 0)

For (x_1 0), this is a valid solution.

Step 3: Solve the Remaining Quadratic

For the remaining quadratic equation (-52x_2x_3 - 14x_2 - 79x_3 36 0), we can solve for (x_2) and (x_3) using the quadratic formula. Rewriting it in standard form:

(-79x_3 - 14x_2 36 - 52x_2x_3 0)

(52x_2x_3 79x_3 14x_2 - 36 0)

This is a quadratic equation in (x_2) and (x_3). Using the quadratic formula:

(x_2 frac{-b pm sqrt{b^2 - 4ac}}{2a})

where (a 52x_3, b 14,) and (c -36 - 79x_3)

(x_2 frac{-14 pm sqrt{14^2 - 4(52x_3)(-36 - 79x_3)}}{2(52x_3)})

(x_2 frac{-14 pm sqrt{196 4(52x_3)(36 79x_3)}}{104x_3})

(x_2 frac{-14 pm sqrt{196 7728 16572x_3^2}}{104x_3})

(x_2 frac{-14 pm sqrt{7924 16572x_3^2}}{104x_3})

For (x_2 1):

(53x_1x_2x_3 - 14x_1x_2 - 79x_1x_3 36x_1 - 13x 0)

Substituting (x_2 1) and (x_3 1):

(53 - 14 - 79 36 - 13x 0)

(-15x 0)

(x 0)

Therefore, the complete solution for (x_2) and (x_3) involves solving the quadratic equation with the discriminant:

(x_2 frac{-14 pm sqrt{7924 16572x_3^2}}{104x_3})

(x_3 frac{-211 pm sqrt{3545}}{104})

The final solutions are:

(x_1 0, 1, frac{-211 sqrt{3545}}{104}, frac{-211 - sqrt{3545}}{104})

This concludes the step-by-step solution of the given complex equation.