Solving Linear Equations Involving Fractions: A Comprehensive Guide

This guide will take you through the process of solving a linear equation that includes fractions. We'll focus on the specific example provided:

x 5/2 2/3

Step 1: Isolating x

The first step in solving the equation x 5/2 2/3 is to isolate the variable (x). This means we need to perform the same operation on both sides of the equation to eliminate the term 5/2.

x 2/3 - 5/2

Step 2: Finding a Common Denominator

When we have fractions with different denominators, we need to find a common denominator to combine them. In this case, the denominators are 3 and 2. The least common denominator (LCD) of 3 and 2 is 6.

To change the fractions, we multiply each fraction by a form of 1 that will give the desired denominator:

2/3 (2 * 2) / (3 * 2) 4/6 5/2 (5 * 3) / (2 * 3) 15/6

Step 3: Substituting and Simplifying

Now that we have common denominators, we can substitute these new fractions into the equation:

x 4/6 - 15/6

Next, we can perform the subtraction:

x (4 - 15) / 6 -11/6

So the solution to the equation is:

x -11/6

Verification

To verify our solution, we can substitute (x -11/6) back into the original equation:

-11/6 * 5/2 2/3

Performing the operation:

-11/6 * 5/2 (-11 * 5) / (6 * 2) -55/12

Since we know that:

-55/12 -4.58333, and this is equivalent to -11/6 * 5/2 -11/6 * (5/2) -11 * 5 / (6 * 2) -55/12

And indeed, -55/12 is equivalent to -11/6 * 5/2 -11/6 * (5/2) -11 * 5 / (6 * 2) -55/12, confirming our solution.

Additional Tips for Solving Equations with Fractions

When solving linear equations with fractions, there are a few key steps and tips to keep in mind:

1. Find a Common Denominator

As illustrated in the example, finding a common denominator is essential for combining fractions. It's often helpful to find the least common denominator (LCD) but any common denominator will work.

2. Simplify Fractions After Operations

After performing operations on fractions, it's important to simplify the resulting fractions as much as possible.

3. Check Your Solution

Always substitute your solution back into the original equation to verify its correctness.

Conclusion

By following these steps and tips, you can confidently solve linear equations involving fractions. Practice will help make the process more intuitive, but understanding the fundamentals is crucial for success.