A dietician is planning a meal of beef, rice and carrots.  The meal must
contain 96.3g of carbs, 50.0g of protein, and 22.8g of fat. Using the nutrition
information below, use the Gaussian elimination method to determine how many
grams of beef, rice, and carrots, the meal must contain in order to meet the
dietician's requirements.

                 CARBS       PROTEIN      FAT
beef              1             22         12
rice             25              2          0
carrots           8              1          0


I wrote the following equations:

1b + 25r + 8c = 96.3
22b + 2r + 1c = 50
12b = 22.8

I then used b=1.9, but got the wrong answer.


You've done nothing wrong in setting it up.
The solution is  b=1.9g, r=3.2g, c=1.8g

Is that the answer you got?  If that differs
from what the back of the book says is the 
answer, then there is a typo in the book.

Is your error in the Gaussian elimination?
If so post again asking how to do that.

Edwin