Flr Sug Sho       
CC-  2   2   1
O -  1   1   2   matrix rows are: choco chip,otm, peaunt b, 
PB-  4   2   1   matrix coloums : flour, sugar, shortening

SOLVE USING CRAMER'S RULE

(Cramer's rule does not lead to a solution if the determinant of coefficients
is 0.)

The question asks:

>>...how many batches of each cookie type were supposed to be baked?...<<

Let x = the number of batches of chocolate chip cookies,

Let y = the number of batches of oatmeal cookies,

Let z = the number of batches of peanut butter cookies

Then 

1.  We make the "flour" equation:

2x + 1y + 4z = 18

2.  We make the "Sugar" equation:

2x + 1y + 2z = 14

3.  We make the "shortening" equation:

1x + 2y + 1z = 13

So we have the system:



To Form the denominator determinant  from the 
coefficients of the letters:



However that determinant is 0 which tells us that Cramer's rule
cannot be used and the the system is either dependent or
inconsistent. 

Form the determinant  by copying the red numbers in place of the
first column:



Form the determinant  by copying the red numbers in place of the
second column:

 which when evaluated gives  by copying the red numbers in place of the
second column:



Expanding the matrices, we get , , , and 

The formulas for x, y, and z are:





So the system is inconsistent.

Edwin