At the end of a certain year, the four organizations ranking 5 through 8 in home

Question

At the end of a certain year, the four organizations ranking 5 through 8 in home Internet users in the United States were eBay, the U.S. Government, Amazon, and Lycos, with a combined audience of 112 million users. Taking x to be the eBay audience in millions, y the U.S. Government audience in millions, z the Amazon audience in millions, and u the Lycos audience in millions, it was observed that y - z = z - u x - y = 3(y - u) + 7 and x - y + z - u = 28. How large was the audience of each of the four companies at the end of that year? eBay U.S. Government Amazon Lycos

Explanation / Answer

Let x,y,z and u be the audiences for ebay, U.S.govt, amazon and Lycos.

Then the conditions given can be written as

x+y+z+u = 112 ... i

x-4y+3u = 7 ... ii

x-y+z-u = 28 ... iii

   y -2z+u =0 ... iv

There are 4 equations in 4 variables and there exists a unique solution.

Solving by matrix method let the equation be Ax = B

Then inverse of A =

Using X = inverse of A * B we get

{ x = 49, y = 24, z = 21, u = 18 }.

A1 A2 A3 A4 1 0 0 1 1 2 3/14 -1/7 -1/14 1/7 3 1/2 0 -1/2 -1 4 2/7 1/7 -3/7 -1/7

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