Simple linear equations in one variable | Part -2

>How to Solve Linear Equations with Rational Terms Using Traditional and Vedic Techniques

Linear equations in one variable, especially those with rational coefficients, are a fundamental part of algebra. These equations are also known as first-degree equations because the variable is raised only to the power of 1. Solving such equations involves isolating the variable to find its value.

Sometimes, these equations are not in standard form and must first be simplified or reduced to a linear form. This includes clearing fractions, combining like terms, and transposing variables. In this article, we’ll explore both the traditional algebraic approach and alternative strategies inspired by Vedic mathematics.

1. Understanding the Linear Form

A linear equation in one variable typically has the form:

ax + b = c

In this form, a, b, and c are rational numbers (which may be integers or fractions), and x represents the unknown. The objective is to determine the value of x that makes the equation true.

2. Steps to Solve with Rational Coefficients

Let’s consider an example:

Example: Solve: (3/4)x - (5/2) = 1/2

Step 1: Eliminate denominators by multiplying the entire equation by the least common denominator (LCD). The LCD of 4 and 2 is 4.

Multiplying both sides by 4:

(4 × (3/4)x) - (4 × 5/2) = 4 × (1/2)

→ 3x - 10 = 2

Step 2: Solve the linear equation.

3x - 10 = 2

Add 10 to both sides: 3x = 12

Divide both sides by 3: x = 4

Therefore, the solution is: x = 4

3. Applying Vedic Math Logic

Vedic Mathematics provides intuitive techniques that often help speed up mental calculations. One Vedic sutra that proves especially helpful in solving linear equations is "Sankalana-Vyavakalanabhyam," which translates to "through the process of addition and subtraction."

Let’s revisit the same problem using Vedic insight:

(3/4)x - (5/2) = 1/2

Think: What number, when multiplied by 3/4 and then reduced by 5/2, gives 1/2?

Reverse the operations:

First, undo the subtraction: 1/2 + 5/2 = 6/2 = 3

Now solve: (3/4)x = 3 → Multiply both sides by 4: 3x = 12 → x = 4

In this method, instead of algebraically solving step-by-step, we reverse the operations mentally, making the process more intuitive and faster.

4. Why Learn Both Methods?

Traditional algebra ensures accuracy and builds procedural understanding. Vedic sutras, on the other hand, offer flexibility and faster problem-solving—especially helpful during timed exams or mental math scenarios.

When combined, both approaches help students develop a more complete and strategic mathematical toolkit.

Solving linear equations with rational coefficients often requires a preliminary step of reducing the equation into standard form. Whether you prefer the structured algebraic method or enjoy the rapid mental strategies from Vedic Mathematics, both methods lead to the same solution. Mastering both can improve your confidence, speed, and accuracy in mathematics.