# Thread: solve - Non-linear System of equations

1. ## solve - Non-linear System of equations

solve

2. Originally Posted by dapore
solve

From equation 1:

$x^3 = 3xy^2 + 11$

$x^3 - 11 = 3xy^2$

$\frac{x^3 - 11}{3x} = y^2$

$y = \sqrt{\frac{x^3 - 11}{3x}}$.

Substituting into equation 2:

$y^3 = 3x^2y + 2$

$\left(\sqrt{\frac{x^3 - 11}{3x}}\right)^3 = 3x^2\sqrt{\frac{x^3 - 11}{3x}} + 2$

Now try to solve for $x$.

3. $(1) + i(2)$ and $(1) - i(2)$ we have

$(x-iy)^3 = 11+2i ~~ (x+iy)^3 = 11-2i$

$x-iy = \sqrt[3]{11+2i} ~~ x+iy = \sqrt[3]{11-2i}$

$x = Re(\sqrt[3]{11+2i}) ~~ , ~~ y = - Im(\sqrt[3]{11+2i})$