2x + 3y + 1 = 0
2x = -3y - 1
x = (-3/2)y - 1/2
.
x^2 + 6xy + 5 = 0
( (-3/2)y - 1/2 )^2 + 6y( (-3/2)y - 1/2 ) + 5 = 0
(9/4)y^2 +(3/4)y + (3/4)y + 1/4 - 9y^2 - 3y + 5 = 0
(9/4 - 9)y^2 + (3/4 + 3/4 - 3)y + 5 = 0
(9/4 - 36/4)y^2 + (3/2 - 6/2)y + 5 = 0
(-27/4)y^2 + (-3/2)y + 5 = 0
(27/4)y^2 + (3/2)y - 5 = 0
27y^2 + 6y - 20 = 0
quadratic formula
y = (-6 +- sqrt(6^2 - 4(27)(-20)) ) / 2(27)
y = (-6 +- sqrt(36 + 1080) ) / 54
y = (-6 +- sqrt(1116))/54
y = (-6 +- sqrt(4*9*31))/54
y = (-6 +- 6sqrt(31))/54
y = (-1 +- sqrt(31))/9
y = (-1 + sqrt(31))/9, (-1 - sqrt(31))/9
.
y = (-1 + sqrt(31))/9
2x + 3y = 1
2x + 3( (-1 + sqrt(31))/9, (-1 - sqrt(31))/9 ) = 1
2x + (-1 + sqrt(31))/9, (-1 - sqrt(31))/3 = 1
2x = 1 - (-1 + sqrt(31))/9, (-1 - sqrt(31))/9
2x = (9 + 1 - sqrt(31))/9
2x = (10 - sqrt(31))/9
x = (10 - sqrt(31))/18
.
y = (-1 - sqrt(31))/9
2x + 3y = 1
2x + 3( (-1 - sqrt(31))/9 ) = 1
2x + (-1 - sqrt(31))/3 = 1
2x = 3/3 + (1 + sqrt(31))/3
2x = (3 + 1 + sqrt(31))/3
2x = (4 + sqrt(31))/3
x = (4 + sqrt(31))/6
Answer: x = (10 - sqrt(31))/18, y = (-1 + sqrt(31))/9 or x = (4 + sqrt(31))/6, y = (-1 - sqrt(31))/9