A distribution company ships apples and oranges in boxes, Apples are shipped in boxes that weigh 35 pounds each. Oranges are shipped in boxes that weigh 55 pounds each. A shipment going to a market weighs 5,000 pounds total and contains 100 boxes. How many boxes of apples are in this shipment?

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We wish to set up equations for the number of boxes and the total weight of the boxes. If $A$ and $O$ represent the number of apples and oranges, we get:

Weight in pounds:

$$\underbrace{35A}_\text{35 pounds per box of Apple} + \underbrace{55O}_\text{55 pounds per box of orange} = \underbrace{5{,}000}_\text{total pounds of all boxes}$$

Boxes:

$$A + O = 100$$

We can solve this problem using elimination. Multiplying both sides of the second equation by 55 in order to eliminate 55O,

$$A + O = 100$$ $$55A+55O=5{,}500$$

Bringing down the weight equation and subtracting:

$$\begin{array} { c c c c} &55A & 55O & 5{,}500\\ -&35A & 55O & 5{,}000 \\ \hline & 20A & 0 & 500 \end{array}$$

Solving for $A$:

$$20A=500$$ $$A=\boxed{25}$$