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I have an invoice that tells me that eight astrodeniums, a bartofón, three cartunes and three dosefríos were bought for my company a month and a half ago and we were charged a total of 350 euros.

Just one month ago, we have another invoice from the same supplier worth 250 euros for buying five astrodennes, two bartofones, two cartunes and a dosefrio.

Last week, I received a third invoice of three astrodenia, three bartofones, a cartun and two dosefríos. Total, 220 euros.

We have lost the company's price list, but the price of the bartophones can be calculated.

How much does each bartofón cost?

#### Solution

The first invoice is translated into the equation we will call F1: 8A + B + 3C + 3D = 350. The second invoice, the equation F2: 5A + 2B + 2C + D = 250. And the last invoice, gives us F3: 3A + 3B + C + 2D = 220.

We can multiply the second invoice by 3 and we will get 15A + 6B + 6C + 3D = 750, if we neatly subtract the elements of the first, we will get 7A + 5B + 3C = 400, in which the unknown D is missing.

On the other hand, if we multiply the second by 2, we have to 10A + 4B + 4C + 2D = 500, and if we subtract the third now, we get that 7A + B + 3C = 280, a different equation that also lacks the unknown D.

Subtracting again, neatly the two equations obtained in which D is missing (since we observe an obvious similarity between the coefficients of the two unknowns A and C), we obtain that 4B = 120, that is, that B = 30. Each bartofón is worth 30 euros.