Solving Complex Equations: A Comprehensive Guide

In this guide, we'll walk through the process of solving a complex algebraic equation involving fractions. Let's take the equationfrac{x}{2} - frac{1}{5} frac{x}{3} frac{1}{4}

Step-by-Step Solution of the Equation

Given the equation:

frac{x}{2} - frac{1}{5} frac{x}{3} frac{1}{4}

To solve this equation, we need to find the value of x. Here's the detailed step-by-step process:

Step 1: Find the Least Common Multiple (LCM)

The denominators of the fractions in the equation are 2, 5, 3, and 4. The least common multiple of these numbers is 60. We'll multiply every term in the equation by 60 to eliminate the fractions:

60 * 3 - 12 2 15

Step 2: Simplify the Equation

Now that we've cleared the fractions, we can simplify the equation:

3 - 12 2 15 3 - 2 15 12 1 27 x frac{27}{10} x 2.7

Step 3: Verify the Solution

To verify the solution, substitute x 2.7 back into the original equation:

Left-hand side (LHS): 30(2.7) - 12 81 - 12 69/60 Right-hand side (RHS): 20(2.7) 15 54 15 69/60 LHS RHS, hence the solution is correct.

Understanding the Process

Let's briefly go through each step of the process:

Step 1: Find the LCM: The LCM of the denominators 2, 5, 3, and 4 is 60. Multiplying the entire equation by 60 gets rid of the fractions, making it easier to solve. Step 2: Simplify: After eliminating the fractions, we can combine like terms and isolate the variable on one side of the equation. Step 3: Solve and Verify: Solve for the variable and then substitute the value back into the original equation to check if it satisfies the equation.

Conclusion

Solving complex algebraic equations, especially those involving fractions, can be systematic with the right steps. By finding the LCM, clearing the fractions, and simplifying, we can efficiently find and verify the solution.

For more help with algebraic equations and other math concepts, feel free to explore more tutorials and resources.