How to Solve Equations with Fractions Having Different Denominators

When facing an equation that involves two fractions with different denominators and a single variable, it can be daunting initially. However, by following a systematic approach, you can effectively solve these equations. This guide aims to provide you with a step-by-step method to handle such equations efficiently.

Understanding the Equation

First, let's identify the form of the equation you want to solve. The general form looks like this:

( frac{a}{b} frac{c}{d} )

Here, 'a', 'b', 'c', and 'd' can be expressions involving the variable. For example, you might encounter an equation like:

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

Find the Common Denominator

The next step is to identify the least common denominator (LCD) of the fractions. The LCD is the smallest multiple of the denominators that can be used to eliminate the fractions. For the example above, the denominators are 3 and 4, and the LCD is 12.

Multiplying Both Sides by the LCD

Multiply each term in the equation by the LCD to eliminate the fractions:

( 12 cdot frac{2x}{3} 12 cdot frac{x - 1}{4} )

This simplifies to:

( 4 cdot 2x 3 cdot (x - 1) )

Further simplification results in:

( 8x 3x - 3 )

Isolating the Variable

Now, isolate the variable 'x' on one side of the equation. You can do this by rearranging the terms:

( 8x - 3x -3 )

This simplifies to:

( 5x -3 )

Solving for the Variable

Finally, solve for 'x' by dividing both sides by 5:

( x frac{-3}{5} )

Verifying the Solution

To ensure your solution is correct, substitute the value of 'x' back into the original equation:

( frac{2 cdot frac{-3}{5}}{3} frac{frac{-3}{5} - 1}{4} )

This simplifies to:

( frac{frac{-6}{5}}{3} frac{frac{-3}{5} - frac{5}{5}}{4} )

Further simplification confirms:

( frac{-2}{5} frac{-2}{5} )

Since both sides are equal, the solution is verified.

By following these steps, you can solve equations involving fractions with different denominators effectively. Understanding the process and applying it systematically will help you handle similar problems with confidence.

Getting the Most Out of This Guide

This guide provides a comprehensive understanding of solving equations with fractions of different denominators. By practicing with different examples, you can enhance your skills and gain a deeper insight into the process. Remember to always check your solutions for accuracy to ensure your answers are correct.