Question 901591
4*q=n
25q=12*p
This hints that p=25 and q=12
then that n=48
48+12+25+d=91
d=6

p+5*n+10*d+25*q=625 
12p-25q=0
n-4q=0
p+n+d+q=91


1,5,10,25,625
12,0,0,-25,0
1,1,1,1,91
0,1,0,-4,0

divide row 1 by 1
1,5,10,25,625
12,0,0,-25,0
1,1,1,1,91
0,1,0,-4,0

add  down (-12) *row 1 to row 2
1,5,10,25,625
0,-60,-120,-325,-7500
1,1,1,1,91
0,1,0,-4,0

add  down (-1) *row 1 to row 3
1,5,10,25,625
0,-60,-120,-325,-7500
0,-4,-9,-24,-534
0,1,0,-4,0

add  down (0) *row 1 to row 4
1,5,10,25,625
0,-60,-120,-325,-7500
0,-4,-9,-24,-534
0,1,0,-4,0

divide row 2 by -60
1,5,10,25,625
0,1,2,-325/-60,125
0,-4,-9,-24,-534
0,1,0,-4,0

add  down (4) *row 2 to row 3
1,5,10,25,625
0,1,2,65/12,125
0,0,-1,-28/12,-34
0,1,0,-4,0

add  down (-1) *row 2 to row 4
1,5,10,25,625
0,1,2,65/12,125
0,0,-1,-7/3,-34
0,0,-2,-113/12,-125

divide row 3 by -1
1,5,10,25,625
0,1,2,65/12,125
0,0,1,-7/-3,34
0,0,-2,-113/12,-125

add  down (2) *row 3 to row 4
1,5,10,25,625
0,1,2,65/12,125
0,0,1,7/3,34
0,0,0,-171/36,-57

divide row 4 by -19/4
1,5,10,25,625
0,1,2,65/12,125
0,0,1,7/3,34
0,0,0,1,12

We now have the value for the last variable.
We will work our way up and get the other solutions.

add up  (-7/3) *row 4 to row 3
1,5,10,25,625
0,1,2,65/12,125
0,0,1,0,6
0,0,0,1,12

add up  (-65/12) *row 4 to row 2
1,5,10,25,625
0,1,2,0,60
0,0,1,0,6
0,0,0,1,12

add up  (-25) *row 4 to row 1
1,5,10,0,325
0,1,2,0,60
0,0,1,0,6
0,0,0,1,12

add up  (-2) *row 3 to row 2
1,5,10,0,325
0,1,0,0,48
0,0,1,0,6
0,0,0,1,12

add up  (-10) *row 3 to row 1
1,5,0,0,265
0,1,0,0,48
0,0,1,0,6
0,0,0,1,12

add up  (-5) *row 2 to row 1
1,0,0,0,25
0,1,0,0,48
0,0,1,0,6
0,0,0,1,12

final
1,0,0,0,25
0,1,0,0,48
0,0,1,0,6
0,0,0,1,12

"25","48","6","12"
(25,48,6,12)