Vegetable stew

The recipe for vegetable stew is recommended to mix tomatoes, peppers, and onions in a ratio of 4:3:1. The cook has already prepared both onions and peppers. In contrast, the onions were five kilograms less than the peppers. How many kilograms of tomatoes will he need according to this recipe?

Correct answer:

r =  10 kg

Step-by-step explanation:

$$\begin{gathered} r:p:c=4:3:1 \\ r:p=4:3 \\ p=3c \\ c=p-5 \\ r=\frac{4}{3}\cdot p \\ p=3\cdot c \\ c=p-5 \\ r=4/3\cdot p \\ p=3\cdot c \\ c-p=-5 \\ 4p-3r=0 \\ 3c-p=0 \\ Pivot:Row1\leftrightarrow Row3 \\ 3c-p=0 \\ 4p-3r=0 \\ c-p=-5 \\ Row3-\frac{1}{3}\cdot Row1\to Row3 \\ 3c-p=0 \\ 4p-3r=0 \\ -0.667p=-5 \\ Row3-\frac{-0.66666667}{4}\cdot Row2\to Row3 \\ 3c-p=0 \\ 4p-3r=0 \\ -0.5r=-5 \\ r=\frac{-5}{-0.5}=10 \\ p=\frac{0+3r}{4}=\frac{0+3\cdot 10}{4}=7.5 \\ c=\frac{0+p}{3}=\frac{0+7.5}{3}=2.5 \\ c=\frac{5}{2}=2.5 \\ p=\frac{15}{2}=7.5 \\ r=10=10\text{kg} \end{gathered}$$

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