Simple linear equations in one variable | Part-3

Solving Linear Equations in One Variable with Rational Coefficients Using Vedic Sutras

Linear equations with one variable are fundamental in algebra. When these equations involve rational coefficients—fractions or ratios—they can initially seem complex. However, by reducing the equation to a standard linear form and applying efficient techniques from Vedic mathematics, these problems become simpler and quicker to solve.

Understanding the Problem

A linear equation in one variable typically looks like ax + b = 0, where a and b are numbers and x is the variable to solve for. When a and/or b are rational numbers (fractions), the first step is to simplify the equation into this standard form.

Step 1: Reduce to Standard Linear Form

Consider an example:

(3/4)x - 5/6 = 7/12

To solve this, first eliminate fractions by finding the least common denominator (LCD) of all denominators: 4, 6, and 12. The LCD is 12.

Multiply every term by 12 to clear the denominators:

  12 × (3/4)x - 12 × (5/6) = 12 × (7/12)
  

Which simplifies to:

  9x - 10 = 7
  

This is now a linear equation with integer coefficients, making it easier to handle.

Step 2: Solve the Reduced Equation

We now solve:

  9x - 10 = 7
  9x = 7 + 10
  9x = 17
  x = 17 / 9
  

Step 3: Applying Vedic Sutras for Efficient Calculation

Vedic mathematics offers sutras (simple formulaic approaches) that simplify arithmetic operations and algebraic manipulations. Two useful sutras for linear equations are:

• “Nikhilam Navatashcaramam Dashatah” (All from 9 and the last from 10): Helpful in working with numbers close to powers of 10, useful for quick multiplication and division.
• “Paravartya Yojayet” (Transpose and adjust): Useful for transposing terms across the equation with sign changes, speeding up the process.

Using the Paravartya Yojayet sutra, when moving -10 to the other side, we directly transpose it as +10, reducing the need to write intermediate steps:

  9x = 7 + 10
  

Next, dividing by 9 can be efficiently handled by Vedic multiplication shortcuts or direct fraction simplification, depending on the context.

Summary and Benefits

Multiplying by the least common denominator eliminates the fractions, converting the equation into one with whole number coefficients. Then, by applying Vedic Sutras like Paravartya Yojayet, the transposition of terms becomes faster and less error-prone. This combination makes solving linear equations quicker and mentally manageable.

Practice Problem

Solve the equation: (5/8)x + 3/4 = 7/6 using these steps.

Solution:

• Find LCD of 8, 4, 6 → 24
• Multiply entire equation by 24:

24 × (5/8)x + 24 × (3/4) = 24 × (7/6)

• Simplify:

15x + 18 = 28

• Using the Vedic principle 'Paravartya Yojayet', move 18 across the equation:

 15x = 28 - 18

15x = 10

• Divide both sides by 15:

x = 10 / 15 = 2 / 3

Answer: x = 2/3

Solving linear equations with rational coefficients becomes straightforward by reducing them to linear form and applying Vedic mathematics techniques. These techniques streamline the calculation process while also boosting mental sharpness in algebraic thinking.