Let x = the number of $1 coins
Let y = the number of $2 coins
Let z = the number of $5 coins

 x +  y +  z =  30     <-- 30 coins
1x + 2y + 5z = 100     <-- $100 total
           y = 2x      <-- twice as many $2 coins as $1 coins

Rearrange the 3rd equation, so that like letters line up in the
system:


  x +  y +  z =  30   
  x + 2y + 5z = 100     
-2x +  y      =   0

To solve using Cramer's rule:
 
Write in all the 1 coefficients and 0 for
the coefficient of the missing variable z in the
third equation:



Cramer's rule:
 
There are 4 columns,
 
1. The column of x-coefficients 
 
2. The column of y-coefficients 
 
3. The column of z-coefficients  
 
4. The column of constants:     
 
There are four determinants:
 
1. The determinant  consists of just the three columns
of x, y, and z coefficients. in that order, but does not
contain the column of constants.
 
. 
 
It has value .  I'm assuming you know how to find the
value of a 3x3 determinant, for that's a subject all by itself.
If you don't know how, post again asking how. 
 
2. The determinant  is like the determinant 
except that the column of x-coefficients is replaced by the
column of constants.   does not contain the column 
of x-coefficients.
 
.
 
It has value .
 
3. The determinant  is like the determinant 
except that the column of y-coefficients is replaced by the
column of constants.   does not contain the column 
of y-coefficients.
 
.
 
It has value .
 
4. The determinant  is like the determinant 
except that the column of z-coefficients is replaced by the
column of constants.   does not contain the column 
of z-coefficients.
 
.
 
It has value .
 
Now the formulas for x, y and z are
 




Answer: 5 $1 coins, 10 $2 coins, 15, $5 coins`
 
Edwin