How large was the audience of each of the four companies at the end of that year?

Question

At the end of a certain year, the four companies with the largest number of home Internet users in the United States were Microsoft, Time Warner, Yahoo, and Google, with a combined audience of 284 million users. Taking x to be the Microsoft audience in millions, y the Time Warner audience in millions, z the Yahoo audience in millions, and u the Google audience in millions, it was observed that
z − u = 3(x − y) + 14
x + y = 46 + z + u
and x − y + z − u = 42.
How large was the audience of each of the four companies at the end of that year?

Answer 1

You have to take the given equations and solve for one of the variables in terms of another variable. (x=y+7) then substitute that new equation (y+7) for x in one of the given equations. Eventually you have all of the variables in terms of one other variable; x = u+44, y = u+37 & z = u + 35. Next you can solve for u using equation [4] and substituting x,y,& z with the formulas above in terms of u. Once you know u is 42 you can go back through those three equations again and solve for x, y & z. Numbers in bold and brackets are referencing those equation numbers.

[1] z − u = 3(x − y) + 14

[2] x + y = 46 + z + u

[3] x − y + z − u = 42.

[4] x+y+z+u = 284 (From the text of the question)

[3] x-y + [1] = 42

x-y + 3(x-y) +14 = 42

x - y + 3x - 3y = 28

4x - 4y = 28

x-y = 7

[5] x = y+7 or y = x-7

[3] y+7-y +z-u = 42

7+z-u = 42

z-u = 35

[6] z = u + 35

[2] y+7+y = 46 +[6] + u

2y+7 = 46 +u +35 +u

2y +7 = 2u + 81

2y = 2u + 74

[7] y = u + 37

[5] = [7]

x-7 = u+37

[8] x = u +44

[4] u+44 +u+37 + u +35 +u = 284

4u + 116 = 284

4u = 168

u= 42

z = u + 35

z= 77

x = u + 44

x = 86

y = u + 37

y = 79

check: 86+79+77+42 = 284