Solving for x in the Equation 13x / 5 20 / 3x: A Comprehensive Guide
In this comprehensive guide, we will walk through the steps to solve the algebraic equation 13x / 5 20 / 3x. Understanding the process of solving such equations is crucial for students and professionals alike who frequently deal with algebra and mathematics. This guide will provide clarity on the method to approach and solve this specific equation, making it easier for learners to tackle similar problems.
Introduction
This guide will start with the premise and calculation steps, detailing each move through the process of solving the equation. Key mathematical principles like cross-multiplication, simplification, and the handling of square roots will be covered. By the end of this guide, you'll not only know how to solve similar equations but also the foundational concepts that underpin the solution.
Premise and Calculations
The given equation is:
13x / 5 20 / 3x
To solve for x, we will follow a systematic approach starting with the premise and moving through step-by-step calculations.
Step 1: Cross-Multiplication
First, we cross-multiply the equation to eliminate the fractions:
13x/5 20/3x 3x(13x) 5(20) 39x^2 100
This step transforms the equation into a form that is easier to solve. The cross-multiplication step removes the denominators and simplifies the equation to a quadratic form.
Step 2: Isolating the Variable
From the equation 39x^2 100, we can solve for x^2 by dividing both sides by 39:
39x^2 100 x^2 100 / 39 x^2 2.5641
Next, we take the square root of both sides to solve for x:
x sqrt(100 / 39) x sqrt(2.5641) x 1.601
Since x is a variable, it can have both positive and negative values. Thus, x 1.601 or x -1.601.
Step 3: Verification
To ensure our solution is correct, let's substitute x 1.601 back into the original equation:
13x / 5 20 / 3x 13(1.601) / 5 20 / 3(1.601) 20.813 / 5 20 / 4.803 4.1626 4.1626
The equality holds true, confirming that our solution is correct.
Optimizing the Solution
There are alternative methods to solve this equation, such as the one suggested by another user:
13x / 5 20 / 3x 3x(13x) 5(20) 39x^2 100
This also leads to the same quadratic equation as we derived, and thus the same solution.
This alternative method emphasizes the concept of removing the denominators by cross-multiplication, which simplifies the equation significantly. It also highlights the importance of isolating the variable and solving for x^2 before taking the square root.
Conclusion
In conclusion, solving the equation 13x / 5 20 / 3x involves a series of logical steps. The key is to understand the principles of cross-multiplication, simplification, and the handling of square roots. By following these steps, you can solve similar types of equations and deepen your understanding of algebraic concepts.
For those looking to enhance their algebra skills, we recommend further practice with similar problems and exploring advanced topics in algebra. Don't forget to verify your solutions by substituting them back into the original equation.
Resources
For more information on solving algebraic equations and other mathematical concepts, consider exploring the following resources:
Math is Fun - Algebra Khan Academy - Algebra Better Explained - Visual Guide to Roots and Roots ExtractionHappy learning!