Answer to Question #260967 in Algebra for bri

Question

(1) The table below shows the calories, fat, and carbohydrates per ounce for three brands of cereal, and the total amount of calories, fat, and carbohydrates required for a special diet. Calories Fat Carbohydrates Cereal Brand X 50 0 22 Cereal Brand Y 108 0.1 25.5 Cereal Brand Z 127 5.5 18 Total Required 393 5.7 91 (i) Using the information in the table above, write a system of three equations. (ii) Write a system in matrix form Ax b  . (ii) Using the Cramer's Rule or Inverse Method, find the amount of each brand of cereal that will give the level of nutrition required. 

Answer 1

Using cramer's Rule;

$\begin{pmatrix} 50&108& 127\\ 0&0.1 & 5.5\\ 22&25.5&18 \end{pmatrix}\begin{pmatrix} x \\ y \\ z \end{pmatrix}=\begin{pmatrix} 393 \\ 5.7\\ 91 \end{pmatrix}$

Let $D=$ $\begin{pmatrix} 50&108 & 127 \\ 0 & 0.1&5.5\\ 22&25.5&18 \end{pmatrix}$

$D_x=\begin{pmatrix} 393& 108&127\\ 5.7&0.1 & 5.5\\ 91&25.5&18 \end{pmatrix}$

$D_y=\begin{pmatrix} 50&393 & 127 \\ 0 & 5.7&5.5\\ 22&91&18 \end{pmatrix}$

$D_2= \begin{pmatrix} 50&108 & 393 \\ 0&0.1 & 5.7\\ 22&25.5&91 \end{pmatrix}$

Det $D=50\begin{vmatrix} 0.1 & 5.5 \\ 25.5 & 18 \end{vmatrix}-108\begin{vmatrix} 0& 5.5 \\ 22 & 18 \end{vmatrix}+127\begin{vmatrix} 0 & 0.1\\ 22 & 25.5 \end{vmatrix}$

$=50(-138.45)-108(-121)+127(-2.2)$

$=5866.1$

Det $D_x=393\begin{vmatrix} 0.1& 5.5 \\ 25.5& 18 \end{vmatrix}-108\begin{vmatrix} 5.7& 5.5 \\ 91& 18 \end{vmatrix}+127\begin{vmatrix} 5.7 & 0.1\\ 91 & 25.5 \end{vmatrix}$

$=393(-138.45)-108(-397.9)+127(136.25)$

$=5866.1$

Det $D_y=50\begin{vmatrix} 5.7 & 5.5 \\ 91& 18 \end{vmatrix}-393\begin{vmatrix} 0& 5.5 \\ 22& 18 \end{vmatrix}+127\begin{vmatrix} 0& 5.7 \\ 22 & 91 \end{vmatrix}$

$=50(-397.9)-393(-121)+127(-125.4)$

$=11732.2$

Det $D_2=50\begin{vmatrix} 0.1 & 5.7 \\ 25.5 & 91 \end{vmatrix}-108\begin{vmatrix} 0& 5.7 \\ 22 & 91 \end{vmatrix}+393\begin{vmatrix} 0& 0.1\\ 22& 25.5 \end{vmatrix}$

$=50(-136.25)-108(-125.4)+393(-22)$

$=5866.1$

$x=\frac{|D_x|}{|D|}=\frac{5866.1}{5866.1}=1,$

$y=\frac{|D_y|}{|D|}=\frac{11732.2}{5866.1}=2$

$z=\frac{|D_z|}{|D|}=\frac{5866.1}{5866.1}=1$

Amount of each brand of cereal that will give the level of nutrition required

Cereal Brand $X=1$ ounce

Cereal Brand $Y=2$ ounces

Cereal Brand $Z=1$ ounce