Solving Equations Involving Fractions: A Comprehensive Guide

Understanding how to solve equations involving fractions is a fundamental skill in algebra. In this article, we will explore the step-by-step process of solving a specific fraction equation, as well as provide a general approach that can be applied to similar problems. Let's break down the equation 54y-13123y-1.

Step-by-Step Solution

Let's start by simplifying the right side of the equation:

12cdot 3y - 1 frac{1}{2} cdot 3y - frac{1}{2} frac{3}{2}y - frac{1}{2}

Now, the equation looks like this:

54y-1332y-12

Eliminating Fractions

To eliminate the fractions, find the least common denominator (LCD) of the fractions in the equation. The LCD of 4, 3, and 2 is 12. Multiply every term by 12:

12left(frac{5}{4}yright) - 12left(frac{1}{3}right) 12left(frac{3}{2}yright) - 12left(frac{1}{2}right)

This simplifies to:

15y - 4 18y - 6

Isolating the Variable

Rearrange the equation to isolate y:

15y - 18y -6 - 4

This simplifies to:

-3y -2

Solving for y

Solve for y by dividing both sides of the equation by -3:

y frac{-2}{-frac{3}{1}} frac{2}{3}

Therefore, the solution to the equation is:

y frac{2}{3}

In decimal form, this is 0.66666 repeating.

Verification

To verify the solution, substitute y frac{2}{3} back into the original equation:

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

Both sides of the equation simplify to 0.6666666666666666, confirming the solution.

Conclusion

In summary, to solve equations involving fractions, we can follow these steps:

Eliminate fractions by finding the least common denominator and multiplying every term. Rearrange the equation to isolate the variable. Solve for the variable by performing the necessary operations. Verify the solution by substituting the value of the variable back into the original equation.

This method can be applied to various types of fraction equations and is a valuable tool in algebra. Understanding these steps will help you solve more complex equations in the future.

Keywords

Equation solving, fractions, algebra

Conclusion

By following the steps outlined in this article, you can effectively solve equations involving fractions. Whether you are a student, a teacher, or a professional, these skills are essential for success in mathematics and related fields. Happy solving!