Question
Sat April 10, 2010 By:

Integral solutions

Expert Reply
Fri April 23, 2010

Dear Student,

I believe you are asking about the number of non-negative integral solutions of the given equation, otherwise there can be infinite number of solutions if we allow x and y to be negative.

Assuming that we seek for non-negative solutions, consider two cases :

Case 1: if y is even => y = 2k + 1 ( where k is some non-negative integer).

Putting y = 2k +1 in given eqn., we get

2x + 3(2k +1) = 2001

=> 2x + 6k = 2001 -3 = 1998

=> 2x = 1998 - 6k

=> x = 999 - 3k

It is evident that for all values of k ∈ {0,1,2,3,4,...,333}, x will be a non-negative integer.

Hence, the ordered pair (x,y) ≡ (999 - 3k , 2k +1) , where k ∈ {0,1,2,3,4,...,333}.

i.e., there are 334 such ordered pairs or non-negative integral solutions of the equation.

Therefore, option (b) is correct.

 

Regards,

Topperlearning.

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