Solving Complex Fractional Equations: A Step-by-Step Guide

In this article, we will walk through the process of solving a system of complex fractional equations. We will explore the steps and methods to solve equations of the form ( frac{2}{x} - frac{1}{y} 3 ) and ( frac{4}{x} frac{3}{y} 16 ). This guide will be beneficial for students and professionals alike who need to understand and apply algebraic methods in solving such equations.

Introduction to Fractional Equations

Fractional equations are a type of algebraic equation where variables appear in the denominator. These equations can be quite challenging to solve, but with a systematic approach, they can be tackled effectively. In this article, we will present a detailed method to solve a specific pair of fractional equations, highlighting the key steps and mathematical techniques involved.

Problem Statement

The problem at hand is to solve the following system of equations:

( frac{2}{x} - frac{1}{y} 3 ) ( frac{4}{x} frac{3}{y} 16 )

Method 1: Substitution and Elimination

We will solve this problem using a combination of substitution and elimination techniques.

Step 1: Introduce New Variables

To simplify the equations, let's introduce new variables: and . The equations then become:

(2u - v 3) (4u 3v 16)

Step 2: Solve the Linear System

We can solve this system using the method of elimination. First, multiply the first equation by 3:

(6u - 3v 9)

Now, add this to the second equation:

(4u 3v 16)

Adding these two equations eliminates (v), giving us:

10u 25

Therefore,

Step 3: Solve for (v)

Substitute (u frac{5}{2}) into the first equation: (2u - v 3).

(2left(frac{5}{2}right) - v 3)

(5 - v 3)

(-v -2)

(v 2)

Step 4: Find (x) and (y)

Now, take the reciprocals of (u) and (v):

Therefore, the solution to the original system of equations is (x frac{2}{5}) and (y frac{1}{2}).

Method 2: Direct Elimination

Another approach is to multiply the first equation by 3 directly and then add the equations to eliminate (v).

Step 1: Multiply by 3

Multiply both sides of the first equation by 3:

3()

Let’s call this new equation (1):

Fractional equation (1): ( frac{6}{x} - frac{3}{y} 9 )

Step 2: Add the Equations

Now add this to the second equation:

Fractional equation (2): ( frac{4}{x} frac{3}{y} 16 )

Combining like terms, we get:

( frac{x}{10} frac{1}{25} )

( x frac{10}{25} frac{2}{5} )

Step 3: Substitute (x) into the First Equation

Substitute ( x frac{2}{5} ) into the first original equation:

( frac{2}{frac{2}{5}} - frac{1}{y} 3 )

( 5 - frac{1}{y} 3 )

( -frac{1}{y} 3 - 5 )

( -frac{1}{y} -2 )

( frac{1}{y} 2 )

( y frac{1}{2} )

Thus, the solution to the original problem is ( x frac{2}{5} ) and ( y frac{1}{2} ).

Conclusion

This article has presented a detailed solution to a complex system of fractional equations. By introducing new variables, using elimination and substitution methods, and solving the system step-by-step, we have arrived at the solution ( x frac{2}{5} ) and ( y frac{1}{2} ).

(Note:) The methods discussed can be applied to a variety of similar problems, making these techniques valuable for students and professionals in mathematics and related fields.