पाटीगणित (श्रीधराचार्य - प्राचीन संस्कृत टीका एवं प्रो. कृपाशंकर शुक्ल सविस्तार व्याख्या)

MIXTURES. PURCHASE AND SALE 51
Let the number of pigeons, cranes, swans, and peacocks bought for
100 rûpas, and the prices paid for the respective birds be as follows :
Pigeons Cranes Swans Peacocks
Number 5x 7y 9z 3u } (1)
Price 3x 5y 7z 9u
Then we have
5x+7y+9z+3u = 100 , (2)
3x+5y+7z+9u = 100 . (3)
Multiplying (2) by 3 (which is the price of one peacock) and
subtracting (3) therefrom, we have
12x+16y+20z = 200 . (4)
This is an indeterminate equation having an indefinitely large
number of solutions. But we are in the present case concerned only with
those positive and non-zero solutions of (4), which make
5x , 7y , 9z , and 3u
positive integers.
Proceeding by trial, we obtain the following sixteen such solutions
of (4) :
(1) x=3, y=4, z=5 . (9) x=4, y=7, z=2 .
(2) x=11, y=3, z=1 . (10) x=8, y=4, z=2 .
(3) x=1, y=8, z=3 . (11) x=12, y=1, z=2 .
(4) x=6, y=3, z=4 . (12) x=5, y=5, z=3 .
(5) x=2, y=6, z=4 . (13) x=9, y=2, z=3 .
(6) x=4, y=2, z=6 . (14) x=7, y=1, z=5 .
(7) x=3, y=9, z=1 . (15) x=1, y=3, z=7 .
(8) x=7, y=6, z=1 . (16) x=2, y=1, z=8 .
The corresponding values of u are obtained from (2).
Substituting these values of x, y, z, and u in (1), we obtain sixteen
valid solutions of the problem. (For actual solutions, see the answer).
The above analysis explains the basis of the foregoing rule (Rule
63-64). It may be observed that the multipliers mentioned in the rule
refer to the variables x, y, z, and u.
Indological Truths