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A company produces three combinations of mixed vegetables that sell in 1Kg package. Italian style combines 0.3kg of zucchini, 0.3kg of broccoli and 0.4kg of carrots. French style combines 0.6kg of broccoli and 0.4kg of carrots. Oriental style combines 0.2kg of zucchini, 0.5kg of broccoli and 0.3kg of carrots. The company has a stock of 16,200 kg of zucchini, 41400 kg of broccoli and 29,400 kg of carrots. How many packages of each style should it prepare with the existing supplies?

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If $x$ is the number of italian style packages, $y$ the number of french style, and $z$ the number of oriental style, then the amount of zucchini is $0.3x + 0y + 0.2z$. The amount of broccoli is $0.3x + 0.6y + 0.5z$; and the amount of carrots is $0.4x + 0.4y + 0.3z$.

You want the quantity of zucchini to equal 16,200, broccoli to equal 41,400, and carrots to equal 29,400, so your system of equations is: \begin{align*} 0.3x & + && 0y & + && 0.2x & = && 16200\\ 0.3x & + && 0.6y & + && 0.5z & = && 41400\\ 0.4x & + && 0.4y & + && 0.3z & = && 29400. \end{align*}

I expect you know how to solve a system of linear equations using matrices.

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