Sweets

Three chocolates and seven cakes cost 85 CZK. Two chocolates and six cakes cost 86 CZK. How much are five chocolates and nine cakes?

I wonder how to get the result, but only by logic, without using a system of equations.

Correct answer:

x =  123 Kč

Step-by-step explanation:

$$\begin{gathered} 3a+7b=85 \\ 6a+2b=86 \\ 3\cdot a+7\cdot b=85 \\ 6\cdot a+2\cdot b=86 \\ 3a+7b=85 \\ 6a+2b=86 \\ Pivot:Row1\leftrightarrow Row2 \\ 6a+2b=86 \\ 3a+7b=85 \\ Row2-\frac{3}{6}\cdot Row1\to Row2 \\ 6a+2b=86 \\ 6b=42 \\ b=\frac{42}{6}=7 \\ a=\frac{86-2b}{6}=\frac{86-2\cdot 7}{6}=12 \\ a=12 \\ b=7 \\ x=5\cdot a+9\cdot b=5\cdot 12+9\cdot 7=123\text{Kc} \end{gathered}$$

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