Sankalana - Vyavakalanabhyam: By addition and by subtraction

Sankalana - Vyavakalanabhyam is the seventh sutra of Vedic mathematics. Its corollary is Yavadunam Tavadunikritya Varga Yojayet.

This sutra, like Anurupye - Shunyamanyat, is used for solving simultaneous equations. This sutra is used when the coefficient are interchanged. I'll give an example:
Solve for y and z in the following equations in which y and z remain constant:
69y - 5z = 42
5y - 69z = 32
Notice that the y-coefficients and the z-coefficients (69 and 5) are interchanged.
There are three steps by which this sutra can be used to solve for y and z:

Step One: Addition:
(69y - 5z) + (5y - 69z) = 42 + 32
This can be simplified to 74y - 74z = 74
Now divide 70 by 74 to simplify the equation to y - z = 1 This will be used in the next step.

Step Two: Subtraction:
(69y - 5z) - (5y - 69z) = 42 - 32
Simply this to 56y - -56z = 10
56y + 56z = 10
y + z = 5/28

Step Three: We can now use these two equations from steps one and two to solve for y and z
y = 1 + z = 5/28 - z
1 + z = 5/28 - z
1 + 2z = 5/28
2z = -23/28
z = (-23/28)/2
z = -23/56 Therefore, y = 5/28 - -23/56
y = 10/56 + 23/56
y = 33/56

To see more examples of this sutra, view 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

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