Technique explanation
The Urdhva-Tiryagbhyam sutra ("Vertically and crosswise") can be applied to solve any system of two linear equations of the form a₁x + b₁y = c₁ and a₂x + b₂y = c₂. This method uses a deterministic cross-product formula to find x and y directly.
The formula for x and y is derived from the cross-multiplication of coefficients and constants:
- x = (c₁b₂ - c₂b₁) / (a₁b₂ - a₂b₁)
- y = (a₁c₂ - a₂c₁) / (a₁b₂ - a₂b₁)
Method Condition: Reliable for any non-parallel linear system (where the denominator is non-zero).
Standard fallback: Cramer's Rule or the method of elimination.
Worked examples
Example 1: Positive Coefficients
Solve: 3x + 4y = 10 and 5x + 2y = 12
- Step 1: Calculate denominator (D): (3 2) - (5 4) = 6 - 20 = -14
- Step 2: Calculate x: (10 2 - 12 4) / -14 = (20 - 48) / -14 = -28 / -14 = 2
- Step 3: Calculate y: (3 12 - 5 10) / -14 = (36 - 50) / -14 = -14 / -14 = 1
Check: Verify with the second equation: 5(2) + 2(1) = 10 + 2 = 12. The result matches the constant term.
Example 2: Mixed Signs
Solve: 2x - 3y = 1 and 3x + 2y = 8
- Step 1: Denominator (D): (2 2) - (3 -3) = 4 + 9 = 13
- Step 2: x: (1 2 - 8 -3) / 13 = (2 + 24) / 13 = 26 / 13 = 2
- Step 3: y: (2 8 - 3 1) / 13 = (16 - 3) / 13 = 13 / 13 = 1
Check: Verify with the second equation: 3(2) + 2(1) = 6 + 2 = 8. The result matches the constant term.
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