Lopanasthapanabhyam:
By Alternate Elimination and Retention
Lopanasthapanabhyam is the corollary to
Vyashtisamanstih, the eleventh
sutra of
Vedic mathematics.
This corollary is used primarily to
factorise algebraic equations into
quadratic equations by alternately setting the value of two of the
variables at zero.
Example: Factorise the following equation:
3x
2 + 2y
2 + 6z
2 + 7xy + 7yz + 11xz
Step One: With lopanasthapanabhyam ("By Elimination and Retention"), we would start by eliminating one of the variables while retaining the others. Let's simplify the problem by saying z = zero. In other words, we eliminate z while retaining x and y
If z = 0 the equation becomes: 3x
2 + 2y
2 + 7xy
This can be rewritten as the quadratic equation:
(3x + y)(x + 2y)
Step Two: Now factorise the original equation, but set the value of one of the other variables to zero instead. Let's eliminate y while retaining x and z
If y = 0 the equation becomes: 3x
2 + 6z
2 + 11xz
This can be rewritten as the quadratic equation:
(3x + 2z)(x + 3z)
Step Three: Fill in the gaps by joining the two bolded equations in the previous steps:
(3x + y)(x + 2y) combined with
(3x + 2z)(x + 3z) becomes:
(3x + y + 2z)(x + 2y + 3z)
"By Elimination and Retention" can also be used to find the
highest common factor of
algebraic expressions. To learn more about this upsutra, check out the resources below:
RESOURCES:
Vedic Mathematics by Sri Bharati Krisna Tirthaji
http://www.vedamu.org/Mathematics/course.asp
http://www.sanalnair.org/articles/vedmath/intro.htm
http://www.vedicganita.org/ganitsutras.htm
http://hinduism.about.com/library/weekly/aa062901a.htm
http://www.vedicmaths.org/
http://www.hinduism.co.za/vedic.htm
Mathemagics by Arthur Benjamin and Michael B. Shermer
http://en.wikipedia.org/wiki/Vedic_math
http://www.tifr.res.in/~vahia/dani-vmsm.pdf
http://www.sacw.net/DC/CommunalismCollection/ArticlesArchive/NoVedic.html