Solving Equations with Fractions: A Comprehensive Guide

When dealing with complex algebraic equations that involve fractions, it can be challenging to isolate the variable and find the solution. In this guide, we will explore various methods to solve equations with fractions, focusing on a specific example and explaining the mathematical steps in detail.

Introduction to Fractional Equations

Fractional equations are equations that involve fractions with variables in the numerator or the denominator. Solving these equations requires a clear understanding of algebraic manipulation and the ability to find common denominators.

Example Problem: Solving the Equation

Consider the equation:

5/(5-x) - 3/(5-x) 2

This equation can be simplified and solved step-by-step. Let's break it down:

Step 1: Combine the Fractions

The equation can be rewritten as:

5/(5-x) - 3/(5-x) 2

Since the denominators are the same, we can combine the fractions:

(5 - 3)/(5-x) 2

(2)/(5-x) 2

Step 2: Clear the Denominator

To clear the denominator, multiply both sides of the equation by (5-x):

2 2(5-x)

2 10 - 2x

Step 3: Isolate the Variable

Now, isolate the variable x by moving all terms involving x to one side of the equation:

2x 10 - 2

2x 8

Divide both sides by 2:

x 4

Step 4: Verify the Solution

Substitute x 4 back into the original equation to verify:

5/(5-4) - 3/(5-4) 2

5/1 - 3/1 2

5 - 3 2

2 2

The equation is satisfied, confirming that x 1 is the correct solution.

Alternative Methods to Solve Fractional Equations

Solving equations with fractions can also be approached using alternative methods. Here are a couple of strategies:

Method 1: Using Cross Multiplication

Consider the equation:

3/(x-5) - 2 5/(5-x)

This equation can be simplified using cross multiplication:

3/(x-5) - 2 5/(5-x)

(5-x)*3 - 2*(x-5) 5*(5-x)

15 - 3x - 2x 10 25 - 5x

25 - 5x 25 - 5x

This equation is always true, which means the solution is valid for all x except x 5 (since the denominator cannot be zero).

Method 2: Common Denominator Technique

Consider another equation:

-3/(5-x) - 2 5/(5-x)

Combine the fractions and simplify:

-3 - 2(5-x) 5

-3 - 10 2x 5

2x - 13 5

2x 18

x 9

However, if we verify this solution, we find that it does not satisfy the original equation due to the denominator constraint.

Conclusion

Solving fractional equations is a crucial skill in algebra. By following systematic steps and utilizing various algebraic techniques, you can simplify and find the solution accurately. Whether using combining fractions, cross multiplication, or finding a common denominator, the key is to methodically manipulate the equation until the variable is isolated.

Remember, always verify your solution by substituting it back into the original equation. This not only ensures your solution is correct but also deepens your understanding of the problem.