Steps to Solve Complex Algebraic Equations: A Comprehensive Guide

Introduction

Algebra remains a fundamental tool in mathematics, with applications ranging from basic problem-solving to advanced scientific and engineering computations. Solving algebraic equations, especially complex ones, requires a systematic approach. This guide will walk you through the step-by-step process of solving a given algebraic equation, ensuring clarity and efficiency. By following these steps, you can confidently tackle even the most complex equations.

The Problem: Solving the Equation

Consider the equation:

2/2x – 53/3x – 10 2/x – 3

Step 1: Isolate the Common Denominator

To eliminate the denominators, multiply both sides of the equation by the least common multiple (LCM) of the denominators. The LCM of 2, 3, and x is 6x. Let's multiply the equation by 6x:

6x(2/2x – 53/3x – 10) 6x(2/x – 3)

103x – 1 – 3 152x – 5x – 3 102x – 53x – 10

Step 2: Expand the Binomials

Expand the binomials on the left and right sides of the equation:

103x^2 – 19x 30 – 152x^2 – 11x 15 106x^2 – 35x – 50

Step 3: Combine Like Terms

Combine the like terms to simplify the equation:

3^2 – 19300 – 103^2 165x – 225 6^2 – 35500

-5x 25 0

Step 4: Solve for x

Add 5x to both sides of the equation:

-5x 25 0 5x

25 5x

Divide both sides by 5:

x 5

Step 5: Verify the Solution

Substitute x 5 back into the original equation to verify:

10/10 – 5(15/15 – 10) 10/5 – 3

1 – 3 2 – 3

1 – 3 –1

–2 –2 (True!)

General Tips for Solving Algebraic Equations

While solving equations, keep in mind the following tips:

Always handle the equation symmetrically, meaning what you do to one side, do to the other side.

Avoid performing operations that can simplify the problem unnecessarily. For example, dividing by a common factor is often not recommended unless it simplifies the problem significantly.

Always simplify your equations step by step and keep each step clear and understandable.

Check your solution by substituting it back into the original equation.

Conclusion

By following these systematic steps, you can solve even the most complex algebraic equations efficiently and accurately. Whether you are a student, a mathematician, or someone working in a field that requires algebraic problem-solving, these methods will serve you well. Remember, practice is key to mastering this skill.