Question

Find x and y such that :
xy+(x+y)=19 and xy(x+y)=84.

Answer 1

Let a = xy and b = x+y. Then
a + b = 19
ab = 84.

Write b = 19-a. Then substituting into the second equation,
a(19-a) = 84
=> a^2 - 19a + 84 = 0
=> a = 12 or 7

For a = 12, b = 7; and for a = 7, b = 12. We expected that btw, because the equations in a and b are symmetric.

Now look at the equations for x and y, given a = 12, b = 7.

xy = a = 12
x+y = b = 7 => y = 7 - x.

Substitute that equation for y now in the equation xy = 12. Then solve.