1. ## Similarity between equations

So I have these two equations:

x^3 + by + c = 0

y^3 + bx + d = 0

where x and y are functions of b, and c and d are constants. The task is to show that y = -x when d = -c. Which should be simple enough,

y^3 + bx + d = 0
y^3 + bx - c = 0
y^3 + bx - c = x^3 + by + c
-y^3 - bx = x^3 + by

but to me it looks like there's no way to go further than this, or am I missing something? Thanks! =))

2. Originally Posted by gralla55
So I have these two equations:

x^3 + by + c = 0

y^3 + bx + d = 0

where x and y are functions of b, and c and d are constants. The task is to show that y = -x when d = -c. Which should be simple enough,

y^3 + bx + d = 0
y^3 + bx - c = 0
y^3 + bx - c = x^3 + by + c
-y^3 - bx = x^3 + by

but to me it looks like there's no way to go further than this, or am I missing something? Thanks! =))
$\displaystyle x^3 + by + c = 0$
$\displaystyle y^3 + bx - c = 0$
-----------------
$\displaystyle (x^3+y^3) + b(x+y) = 0$

$\displaystyle (x+y)(x^2 -xy+y^2) + b(x+y) = 0$

$\displaystyle (x+y)[(x^2 -xy+y^2) + b] = 0$

$\displaystyle x+y = 0$

$\displaystyle y = -x$