(n, d, q) = (1, 102, 2)
(n, d, q) = (4, 98, 3)
(n, d, q) = (7, 94, 4)
(n, d, q) = (10, 90, 5)
(n, d, q) = (13, 86, 6)
(n, d, q) = (16, 82, 7)
(n, d, q) = (19, 78, 8)
(n, d, q) = (22, 74, 9)
(n, d, q) = (25, 70, 10)
(n, d, q) = (28, 66, 11)
(n, d, q) = (31, 62, 12)
(n, d, q) = (34, 58, 13)
(n, d, q) = (37, 54, 14)
(n, d, q) = (40, 50, 15)
(n, d, q) = (43, 46, 16)
(n, d, q) = (46, 42, 17)
(n, d, q) = (49, 38, 18)
(n, d, q) = (52, 34, 19)
(n, d, q) = (55, 30, 20)
(n, d, q) = (58, 26, 21)
(n, d, q) = (61, 22, 22)
(n, d, q) = (64, 18, 23)
(n, d, q) = (67, 14, 24)
(n, d, q) = (70, 10, 25)
(n, d, q) = (73, 6, 26)
(n, d, q) = (76, 2, 27)

A piggy bank contains only nickels, dimes, and quarters. There is a total of 105 coins and the total value of those coins is $10.75.

Find a general solution of the number of each coin.
Use the general solutions from part 1 to list 3 specific solutions (for n, d, & q).

n=nickel
d=dime
q=a quarter

Since number of nickels, dimes, and quarters are n, d, and q, we get the following COIN-COUNT equation:
  n + d + q = 105 --- eq (i), and total value-equation, .05n + .1d + .25q = 10.75, or 20(.05n + .1d + .25q) = 20(10.75), or
n + 2d + 5q = 215 --- eq (ii)
     d + 4q = 110 ----- Subtracting eq (i) from eq (ii) 
         4q = 110 - d 
          ----- Solving for q ----- eq (iii)

The SMALLEST number of dimes (d) that can be substituted         The next "d" integer-value that'll create the next      
into the above equation, and which also yields an                smaller integer in numerator that's a MULTIPLE of 4 is 6,
INTEGER-number of quarters is 2, since 2 leads to a              since 6 leads to a numerator of 104, a multiple of 4, and                       
numerator of 108, a MULTIPLE of 4 (the denominator).             which also yields an INTEGER-VALUE for q.
      We then get:                                            We then get: 
Therefore, for the above, we get: Number of dimes: 2               Therefore, for the above, we get: Number of dimes: 6                              
Number of quarters: 27                                             Number of quarters: 26
Number of nickels, 105 - (2 + 27) = 105 - 29 = 76                  Number of nickels, 105 - (6 + 26) = 105 - 32 = 73
This gives us, (n, d, q) = (76, 2, 27)                             This gives us, (n, d, q) = (73, 6, 26)

                                          So far, we have: (n, d, q) = (76, 2, 27)
                                                           (n, d, q) = (73, 6, 26)

Looking at the above SEQUENCE, it's quite obvious that the nickels decrease by 3, the
                                                           dimes INCREASE by 4, and the
                                                           quarters DECREASE by 1. 

                               Thus, we get the following: (n, d, q) = (76,  2, 27)
                                                           (n, d, q) = (73,  6, 26)
                                                           (n, d, q) = (70, 10, 25)
                                                           (n, d, q) = (67, 14, 24)
                                                           (n, d, q) = (64, 18, 23)

You don't need that many because you were asked for just 3, but you should get the "picture." CAPISCHE?

t=4:   q=4; d=110-4(4) =94; n=3(4)-5 = 7   (n,d,q) = (7,94,4)
t=15: q=15; d=110-4(15)=50; n=3(15)-5=40   (n,d,q) = (40,50,15)
t=22: q=22; d=110-4(22)=22; n=3(22)-5=61   (n,d,q) = (61,22,22)