"By addition and by subtraction."
x+y=10, x-y=4 → Add: 2x=14, x=7. Subtract: 2y=6, y=3.
Solving simultaneous equations by adding and subtracting them to eliminate variables.
Add or subtract two equations to eliminate one variable and solve for the other. The simplest method for symmetric equation pairs.
This sutra provides the simplest approach to simultaneous equations when the coefficients allow easy elimination. Given x + y = S and x − y = D, adding gives 2x = S + D and subtracting gives 2y = S − D. This extends to any pair where adding or subtracting eliminates a variable. The sutra also applies to mental arithmetic: to add 38 + 5, think of it as adding 40 and subtracting 2 (By Addition and By Subtraction). In the Vedic system, this principle of complementary operations — doing one thing and then adjusting — appears throughout many calculations and is a fundamental mental math strategy.
The addition-subtraction method for solving equations appears in the Bakhshali Manuscript (3rd–7th century CE), one of the oldest Indian mathematical manuscripts. The Chinese "fangcheng" method (from The Nine Chapters on the Mathematical Art, ~200 BCE) also uses this principle. In the Vedic framework, this sutra represents the duality principle — every mathematical operation has a complementary reverse. Bharati Krishna Tirtha emphasized that this sutra embodies the Hindu philosophical concept of "dvandva" (pairs of opposites) which together reveal truth. The technique was formalized in European mathematics as Gaussian elimination in the 19th century.
This is the fundamental principle of linear combination in linear algebra. Adding or subtracting equations to eliminate variables is the basis of Gaussian elimination, the most important algorithm in numerical linear algebra. The computational complexity is O(n³) for n equations, but this sutra identifies O(1) special cases where the answer is immediate. In physics, this principle is used in signal processing (adding/subtracting signals for constructive/destructive interference) and in quantum mechanics (symmetric and antisymmetric wavefunctions).