Notice that the second equation,
4x ² + 3xy + y ² = 2
can be written as
x (4x + 3y) + y ² = 2
and the first equation says 4x + 3y = 1, so this reduces to
x + y ² = 2
Solve the first equation for x :
4x + 3y = 1
4x = 1 - 3y
x = (1 - 3y)/4
Substitute this into the reduced second equation to get a quadratic equation in y, which happens to be easily factorized and solved:
(1 - 3y)/4 + y ² = 2
1 - 3y + 4y ² = 8
4y ² - 3y - 7 = 0
(4y - 7) (y + 1) = 0
4y - 7 = 0 or y + 1 = 0
y = 7/4 or y = -1
Solve for x :
x = (1 - 3 (7/4))/4 or x = (1 - 3 (-1))/4
x = -17/16 or x = 1
So the two solutions are (x, y) = (-17/16, 7/4) and (x, y) = (1, -1).
