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Linear Diophantine Equations

What amounts of money can you make with only $5 and $7 bills? Can you make $23 with those 2 types of bills?

System of Diophantine Equations

         

If we solve the following system of equations \[ x + y + 2z = 10 \\ 3x - y + z = 5, \] where \(x, y, z\) are non-negative integers, what is the sum of all possible values of \(x?\)

In the post office they sell souvenir stamps of three denominations: 25 cents, 10 cents, and 5 cents. If you want to buy 20 stamps for exactly 2 dollars, how many combinations of the three kinds of stamps are there that you can buy?

Old Spinner's Manufacture employed 20 people at their shop. Their weekly salary was 3 shillings for a man, 2 shillings for a woman, and half a shilling for a child below 15. If the 20 people altogether received exactly 20 shillings a week and there were at least one man and one woman, how many children were employed?

Solve the following system of linear Diophantine equations \[ \left \{ \begin{array}{ll} 3x -4y -z = 5 \qquad & (1) \\ 3y -4z +2w = -5 \qquad & (2) \\ 2x + w = 10 \qquad & (3) \end{array} \right. \] for non-negative integers \( x, y, z, w. \) Find the value of \( x+ y+ z + w. \)

Jane bought 5 watermelons, 3 apples and 20 cherries for 46 dollars. Lizzy bought 2 watermelons, 5 apples and 12 cherries for 27 dollars. If the prices of a watermelon and an apple are both integers, what is the price of an apple?

Give your answer in dollars.

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