The price of the bartofón 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.