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उपसूत्र १२. विलोकनं

(Upasūtra 12. vilokanaṃ) - By mere observation.
 

The Upasūtra: vilokanaṃ (By mere observation) is used to find solutions to equations. It observes that sometimes we form a habit of finding solutions in a conventional method, despite the fact that the solution may be obvious - and may be found by mere observation.
 
As an illustration, let us use this Upasūtra to solve the Simultaneous Quadratic Equation:
x + y = 9 and xy = 14
  Steps x + y = 9 and xy = 14
1. Find a pattern in the given parameters, and arrange them in a similar pattern. In this case,
x + y = 9
xy = 14
2. Observe carefully to assume combinations that fit the pattern. In this case, for xy = 14
14 may be written as 2 × 7, and -2 × -7, other than 1 × 14
So, possible combinations are {x,y} = {2,7}, {-2,-7}, {1,14}, {-1,-14}
3. Test the combinations, to finalize assumptions. In this case, testing the combinations in: x + y = 9
 
{x,y} = {2,7} gives:
2 + 7 = 9
This assumption is finalized.
 
{x,y} = {-2,-7} gives:
-2 - 7 = -9 9, and thus discarded.
 
Similarly, {x,y} = {1,14} and {-1,-14} are discarded.
4. Find other values, if required, using the finalized combinations. Using the finalized assumption {x,y} = {2,7}, we get:
 
Replacing x = 2,
2 + y = 9 gives y = 7
And, replacing x = 7
7 + y = 9 gives y = 2
 
Thus, the solutions are {x,y} = {2,7} or {7,2}
So, for a practitioner of Vedic Mathematics, for something like:
    1   5
x + - = -
    x   2
  5 may be written as 4 + 1, and thus,
5   4+1   4   1       1
- = --- = - + - = 2 + -
2    2    2   2       2
By the above observation, x=1/2 will also satisfy the expression.
 
Thus, for x + 1/x = 5/2, x = 2 or x = 1/2
Again, for something like:
 x    x+2   34
--- + --- = --
x+2    x    15
  15 may be written as 3 × 5, -3 × -5, 1 × 15, -1 × -15
 
Using this assumption:
Replacing x = 3,
3   5   9 + 25   34
- + - = ------ = --
5   3     15     15
This assumption is finalized.
 
Similarly, of the other assumptions, only x = -5 is finalized.
 
Thus, for the above equation, x = 3 or x = -5
 
Note that this being a Quadratic equation, after finalizing two assumptions, we do not need to test the others.
Please note that this Upasūtra encourages us actively work on assumptions, or guesswork, which we sometimes do anyway. This Upasūtra asserts, that when the solutions lie in front of us, why work for it instead of observing carefully. However, if no pattern is visualized, and the solution cannot be presumed - one should not waste time, and proceed by the conventional method - as in the next example, for something like:
5x+9   5x-9      82
---- + ---- = 2 ---
5x-9   5x+9     319
  Note that,
   82   720   841-121
2 --- = --- = -------
  319   319     319
This gives us,
5x+9   5x-9   29   11
---- + ---- = -- - --
5x-9   5x+9   11   29
 
Thus, (5x+9)/(5x-9) = 29/11 or -11/29
 
Again, 29 = 20 + 9 = (5 × 4) + 9
And, 11 = 20 - 9 = (5 × 4) + 9
This observation in (5x+9)/(5x-9) = 29/11 gives us x = 4
 
And, working the other option (in conventional method), gives us x = -81/100
As in the example above, the key to the solution is in representing 319. This might not stike somebody at a particular moment. This Upasūtra does not enforce (as in the other Sūtras and Upasūtras) a rule, to stare at the problem and keep wondering to fit in a pattern - per say. On the contrary, it encourages one to have a good look at the problem to find a pattern - before moving into the conventional methods.
 
The core to this Upasūtra is to save time, if the solution is obvious - and may be found by mere observation. Taking this ahead in resolving partial fractions, for something like:
   2x+7
----------
(x+3)(x+4)
  The mere observation of 2x+7 gives us x+x+3+4.
Thus, 2x+7 may be written as (x+3) + (x+4), which transforms the given expression to:
 
   2x+7      (x+3) + (x+4)
---------- = -------------
(x+3)(x+4)    (x+3) (x+4)
 
   1     1
= --- + ---
  x+3   x+4
When actively practised, a practitioner is able to visualize similar patterns through experience.
 
Lastly, please remember that, as in any other form of mathematics, the mastery of Vedic Mathematics require practice and the judgement of applying the optimal method for a given scenario - a guideline of which, is presented in Applications »
 
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