If x 0 and 3x + 4|y| = 33, then how many integer pairs of (x, y) are there?

Question

A bit confused in this question, please help me in this ?

Simran Mehta · Accepted Answer

First, we need to find integer values of x such that 3x &le; 33 and the remaining part is divisible by 4|y|.
Let&rsquo;s iterate x from 1 upwards:For x = 1: 3(1) + 4|y| = 33 -> 3 + 4|y| = 33 -> 4|y| = 30 -> |y| = 30/4 (not an integer)For x = 2: 3(2) + 4|y| = 33 -> 6 + 4|y| = 33 -> 4|y| = 27 -> |y| = 27/4 (not an integer)For x = 3: 3(3) + 4|y| = 33 -> 9 + 4|y| = 33 -> 4|y| = 24 -> |y| = 24/4 -> |y| = 6For x = 4: 3(4) + 4|y| = 33 -> 12 + 4|y| = 33 -> 4|y| = 21 -> |y| = 21/4 (not an integer)For x = 5: 3(5) + 4|y| = 33 -> 15 + 4|y| = 33 -> 4|y| = 18 -> |y| = 18/4 (not an integer)For x = 6: 3(6) + 4|y| = 33 -> 18 + 4|y| = 33 -> 4|y| = 15 -> |y| = 15/4 (not an integer)For x = 7: 3(7) + 4|y| = 33 -> 21 + 4|y| = 33 -> 4|y| = 12 -> |y| = 12/4 -> |y| = 3For x = 8: 3(8) + 4|y| = 33 -> 24 + 4|y| = 33 -> 4|y| = 9 -> |y| = 9/4 (not an integer)For x = 9: 3(9) + 4|y| = 33 -> 27 + 4|y| = 33 -> 4|y| = 6 -> |y| = 6/4 (not an integer)For x = 10: 3(10) + 4|y| = 33 -> 30 + 4|y| = 33 -> 4|y| = 3 -> |y| = 3/4 (not an integer)For x = 11: 3(11) + 4|y| = 33 -> 33 + 4|y| = 33 -> 4|y| = 0 -> |y| = 0/4 -> |y| = 0
Thus, valid (x, y) pairs are:(x, |y|) = (3, 6), (7, 3), and (11, 0)
Since |y| can be positive or negative:- (3, 6), (3, -6)- (7, 3), (7, -3)- (11, 0)
There are 5 valid integer pairs (x, y).

If x > 0 and 3x + 4|y| = 33, then how many integer pairs of (x, y) are there?