By mere Observation / Vilokanam

Generally we come across problems which can be solved by mere observation.But we follow the conventional procedure to get our results.But the hint behind the Sutra enables us to observe the problem completely and find the pattern and finally solve the problem by just observation.


Example 1:
Solve : x + (1/x) =3/2


Conventional method :


x +1/x =5/2


x² +1        5
------- = -----
x               2
2 x²+2 =5x
2x²-5x+2=0
2x²-4x-x+2=0
2x(x-2)-(x-2)=0
(x-2)(2x-1)=0
or x=2 ,(1/2)


Now by Vilokanam or by mere observation :


x + 1      5
    --- = ----
     x       2    can be viewed as ,


x + 1              1
     ----= 2 +-----
      x              2   where we get our results straight away as x= 2 ,(1/2)




Example 2 : 

x + 5      x + 6      113
____ + _____ = ___
x + 6      x + 5      56

Now  by Vilokanam or by mere observation :


113      49 + 64          7           8
___    = _______ = ___ +     ___
56            7 x 8          8           7

x + 5         7                x+5      8
____    = __          or ____ = __
x + 6        8                 x+6      7  by splitting the R.H.S and L.H.S on both sides

8x + 40 = 7x + 42  and 7x + 35 = 8x + 48
Solving for x,
x = 42 - 40 = 2 -x = 48 – 35 = 13
x = 2 or x = -13.

Example 3:

Simultaneous Quadratic Equations:

Solve:    x + y = 9 and xy = 14.

By mere observation,

xy = 14 gives x = 2, y = 7 or x = 7, y = 2

These two sets satisfy x + y = 9 since 2 + 7 = 9 or 7 + 2 = 9. 

Hence the solution.

Similarly, 

Solve : 5x – y = 7 and xy = 6.

By mere observation,

xy = 6 gives x = 2, y = 3 or x = 3, y = 2

These two sets satisfy 5x - y = 7 since 5(2) -3 = 7 but not 5(3)- 2 = 12.

Hence the solution is x=2 and y =3

Try them ,you will be surprised of Vedic Maths ! 

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