Question 647353
A bag contain 3 types of coins namely $1, $2, $5. There are 30 coins 
amounting to $100 in total. There are twice as many $2 coins as $1 
coins.  Find the number of coins in each category using Cramer's rule.
<pre>
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:

{{{system(1x+1y+1z=red(30),
1x+2y+5z=red(100),
-2x+1y+0z=red(0))}}}

Cramer's rule:
 
There are 4 columns,
 
1. The column of x-coefficients {{{matrix(3,1,1,1,-2)}}}
 
2. The column of y-coefficients {{{matrix(3,1,1,2,1)}}}
 
3. The column of z-coefficients {{{matrix(3,1,1,5,0)}}} 
 
4. The column of constants:     {{{red(matrix(3,1,30,100,0))}}}
 
There are four determinants:
 
1. The determinant {{{D}}} consists of just the three columns
of x, y, and z coefficients. in that order, but does not
contain the column of constants.
 
{{{D=abs(matrix(3,3,1,1,1,1,2,5,-2,1,0))}}}. 
 
It has value {{{D=-10}}}.  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 {{{D[x]}}} is like the determinant {{{D}}}
except that the column of x-coefficients is replaced by the
column of constants.  {{{D[x]}}} does not contain the column 
of x-coefficients.
 
{{{D[x]=abs(matrix(3,3,red(30),1,1,red(100),2,5,red(0),1,0))}}}.
 
It has value {{{D[x]=-50}}}.
 
3. The determinant {{{D[y]}}} is like the determinant {{{D}}}
except that the column of y-coefficients is replaced by the
column of constants.  {{{D[y]}}} does not contain the column 
of y-coefficients.
 
{{{D[y]=abs(matrix(3,3,1,red(30),1,1,red(100),5,-2,red(0),0))}}}.
 
It has value {{{D[y]=-100}}}.
 
4. The determinant {{{D[z]}}} is like the determinant {{{D}}}
except that the column of z-coefficients is replaced by the
column of constants.  {{{D[z]}}} does not contain the column 
of z-coefficients.
 
{{{D[z]=abs(matrix(3,3,1,1,red(30),1,2,red(100),-2,1,red(0)))}}}.
 
It has value {{{D[x]=-150}}}.
 
Now the formulas for x, y and z are
 
{{{x=D[x]/D=(-50)/(-10)=5}}}
{{{y=D[y]/D=(-100)/(-10)=10}}}
{{{x=D[z]/D=(-150)/(-10)=15}}}

Answer: 5 $1 coins, 10 $2 coins, 15, $5 coins`
 
Edwin</pre>