Prices at the market

Statement

At the fruit market: $3$ kg of apples, $2$ kg of pears and $1$ kg of kiwis cost $13$ euros; $1$ kg of apples, $4$ kg of pears and $2$ kg of kiwis cost $16$ euros; $2$ kg of apples, $1$ kg of pears and $3$ kg of kiwis cost $10$ euros. How much does $1$ kg of each fruit cost?

Solution

Let $a$, $p$, $k$ be the price per kg of apples, pears and kiwis: $\begin{cases} 3a + 2p + k = 13\\ a + 4p + 2k = 16\\ 2a + p + 3k = 10 \end{cases}$ From the first, $k = 13 - 3a - 2p$. Substitute into the second: $a + 4p + 2(13 - 3a - 2p) = 16 \implies a + 4p + 26 - 6a - 4p = 16 \implies -5a = -10 \implies a = 2.$ Then $k = 13 - 6 - 2p = 7 - 2p$. Substitute $a$ and $k$ into the third: $2\cdot 2 + p + 3(7 - 2p) = 10 \implies 4 + p + 21 - 6p = 10 \implies -5p = -15 \implies p = 3.$ Hence $k = 7 - 2\cdot 3 = 1$. Check: $3\cdot2+2\cdot3+1 = 13$, $2+4\cdot3+2 = 16$, $2\cdot2+3+3 = 10$. Correct. $\boxed{\text{apples } 2, \quad \text{pears } 3, \quad \text{kiwis } 1 \ \text{(euro/kg)}}$