Question

a patient started a new diet that requires each meal to have 460 calories, 6 grams of fiber, and 11 g of fat. the following table shows the fiber, fat and calories content of three foods: fruit, cottage cheese, toast.
food\tfruit\tcottage cheese\ttoast
fiber (g)\t2\t0\t2
fat (g)\t0\t5\t1
calories\t60\t120\t100
how many servings of each food should the patient eat to follow this diet?
fruit serving(s) cottage cheese serving(s) toast serving(s)

Explanation:

Step1: Set up equations

Let $x$ be the number of servings of fruit, $y$ be the number of servings of cottage - cheese, and $z$ be the number of servings of toast.
For fiber: $2x+0y + 2z=6$ (Equation 1), which simplifies to $x + z=3$ or $x=3 - z$.
For fat: $0x+5y + z=11$ (Equation 2), so $5y=11 - z$ or $y=\frac{11 - z}{5}$.
For calories: $60x+120y+100z = 460$ (Equation 3).

Step2: Substitute $x$ and $y$ into the calorie - equation

Substitute $x = 3 - z$ and $y=\frac{11 - z}{5}$ into Equation 3:
\[

$$\begin{align*} 60(3 - z)+120\times\frac{11 - z}{5}+100z&=460\\ 180-60z + 24(11 - z)+100z&=460\\ 180-60z+264 - 24z+100z&=460\\ (-60z-24z + 100z)+(180 + 264)&=460\\ 16z+444&=460\\ 16z&=460 - 444\\ 16z&=16\\ z&=1 \end{align*}$$

\]

Step3: Find $x$ and $y$

Since $z = 1$, then $x=3 - z=3 - 1 = 2$.
And $y=\frac{11 - z}{5}=\frac{11 - 1}{5}=2$.

Answer:

Fruit: 2 servings, Cottage cheese: 2 servings, Toast: 1 serving