Find x, y, u, and v, satisfying the system of four equations
x+7y+3v+5u = 16
8x+4y+6v+2u = -16
2x+6y+4v+8u = 16
5x+3y+7v+u = -16

Source: George Polya, How to Solve It: A New Aspect of Mathematical Method, Princeton Mathematics Press, 1973.

Level: Senior

Label the equations as (1), (2), (3), and (4) in order. Adding (1) and (4),
6x+10y+10v+6u = 0,
(5) 6(x+u)+10(y+v) = 0.
Adding (2) and (3),
10x+10y+10v+10u = 0,
(6) 10(x+u)+10(y+v) = 0.
Subtracting (5) from (6) and solving for x+u, we obtain x+u = 0. Substituting x+u = 0 into (5) and solving for y+v, we have y+v = 0. Therefore u = -x and v = -y. Substituting these two expressions for u and v into (1) and (2) and simplying, we derived a simplier system
(7) -x+y = 4;
(8) 3x-y = -8.
Adding (7) and (8) and solving for x yields x = -2. Thus, by (7), y = 4+x = 2. It follows from the earlier results that u = -x = 2 and v = -y = -2. Hence the solution is (x, y, u, v) = (-2, 2, 2, -2).