Solving Equations with Fractions: A Comprehensive Guide

Dealing with equations that involve fractions can be a daunting task, but with the right approach, it becomes much simpler. In this guide, we will walk you through how to solve an equation that includes fractions, such as 2x/5 - 3/2 x/2 1. We will break down the steps, explain the concepts, and provide detailed solutions to ensure you understand the process thoroughly.

Understanding the Given Equation

The equation we are focusing on is:

2x/5 - 3/2 x/2 1

First, let's identify the components of the equation: the coefficients of x (which are 2/5 and 1/2) and the constant term (which is 1).

Step-by-Step Solution

Step 1: Finding a Common Denominator

The denominators in the equation are 5, 2, and 2. The least common multiple (LCM) of these denominators is 10. To eliminate the fractions, we multiply the entire equation by the LCM, which is 10.

Multiplying the equation by 10:

10(2x/5 - 3/2) 10(x/2 1)

This simplifies to:

4x - 15 5x 10

Step 2: Isolating Variables

To isolate x, we need to get all terms involving x on one side of the equation and all other terms on the other side. Let's move all the terms involving x to the left side and the constant terms to the right side:

4x - 5x 10 15

This simplifies to:

-x 25

Step 3: Solving for x

Since we have -x 25, we can multiply both sides by -1 to get:

x -25

Alternative Method: Cross-Multiplication

Another method to solve this equation involves cross-multiplication. Let's rewrite the original equation and solve it step by step:

2x/5 - 3/2 x/2 1

Rearrange to:

2x/5 - x/2 1 3/2

Combine fractions on the right side:

2x/5 - x/2 2 3/2 5/2

Write with a common denominator (10):

4x - 5x/10 5/2

Simplify:

-x/10 5/2

Multiply both sides by -10:

x -25

Conclusion

Both methods lead us to the same solution: x -25. It is crucial to understand and practice these steps to confidently solve similar equations with fractions.

References:

1. Mashaim Javaid, "Solution: 2x/5 - 3/2 x/2 1"

Keywords: solving fraction equations, finding common denominators, cross-multiplication

Author: Qwen, created by Alibaba Cloud