91Ó°ÊÓ

We have been given that Nutrition a dietitian at Palos Community Hospital wants a patient to have a meal that has 78 grams (g) of protein, 59 g of carbohydrates, and 75 milligrams (mg) of vitamin A. The hospital food service tells the dietitian that the dinner for today is salmon steak, baked eggs, and acorn squash. Each serving of salmon steak has 30 g of protein, 20 g of carbohydrates, and 2 mg of vitamin A. Each serving of baked eggs contains 15 g of protein, 2 g of carbohydrates, and 20 mg of vitamin A. Each serving of acorn squash contains 3 g of protein, 25 g of carbohydrates, and 32 mg of vitamin A.

We have to find how many servings of each food should the dietitian provide for the patient.

$$\begin{gathered} \begin{array}{r}30x+15y+3z=78\\ 20x+2y+25z=59\\ 2x+20y+32z=75\end{array} \\ \mathrm{The}\mathrm{corresponding}\mathrm{augmented}\mathrm{matrix}\mathrm{is}: \\ \left[\begin{array}{cccc}30& 15& 3& 78\\ 20& 2& 25& 59\\ 2& 20& 32& 75\end{array}\right] \end{gathered}$$

$$\begin{gathered} R_3↔R_1 \\ \left[\begin{array}{cccc}2& 20& 32& 75\\ 20& 2& 25& 59\\ 30& 15& 3& 78\end{array}\right] \\ {R}_2→R_2−10R_1; \\ {R}_3→R_3−15R_1 \\ \left[\begin{array}{cccc}2& 20& 32& 75\\ 0& −198& −295& −691\\ 0& −285& −477& −1047\end{array}\right] \end{gathered}$$

$$\begin{gathered} \begin{array}{r}{R}_3→R_3−\frac{285}{198}{R}_2\end{array} \\ :\left[\begin{array}{cccc}2& 20& 32& 75\\ 0& −198& −295& −691\\ 0& 0& −\frac{3457}{66}& −\frac{3457}{66}\end{array}\right] \\ {R}_3→−\frac{66}{3457}{R}_3 \\ {R}_3→R_3−\frac{285}{198}{R}_2: \\ \left[\begin{array}{cccc}2& 20& 32& 75\\ 0& −198& −295& −691\\ 0& 0& 1& 1\end{array}\right] \end{gathered}$$

$$\begin{gathered} 2x+20y+32z=75\left(1\right) \\ −198y−295z=−691\left(2\right) \\ z=1\left(3\right) \\ From\left(2\right):−198y−295=−691 \\ −198y=−396 \\ y=2 \\ From\left(1\right):2x+20â‹\dots 2+32â‹\dots 1=75 \\ 2x+72=75 \\ 2x=3 \\ x=1.5 \end{gathered}$$

The dietician should give the patient 1.5 salmon steak, 2 baked eggs and 1 acorn squash.