I came across an interesting system of equations the other day:

a + b = 14,    a^3 + b^3 = 812, solve for a, b
Use substitution to get:
a=14 - b
(14 - b)^3 + b^3 =812
Solve for b:
(14 - b)^3 + b^3 = 812

Expand out terms of the left hand side:
42 b^2 - 588 b + 2744 = 812

Divide both sides by 42:
b^2 - 14 b + 196/3 = 58/3

Subtract 196/3 from both sides:
b^2 - 14 b = -46

Add 49 to both sides:
b^2 - 14 b + 49 = 3

Write the left hand side as a square:
(b - 7)^2 = 3

Take the square root of both sides:
b - 7 = sqrt(3) or b - 7 = -sqrt(3)

Add 7 to both sides:
b = 7 + sqrt(3) or b - 7 = -sqrt(3)

Add 7 to both sides:

b = 7 + sqrt(3) or b = 7 - sqrt(3)
Sub this value of b into: a + b = 14 and should get:

 

a = 7 - sqrt(3)  and  b = 7 + sqrt(3)
a = 7 + sqrt(3)  and b = 7 - sqrt(3)