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(x2+y2–1)3-x2y3=0

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(x2+y2–1)3-x2y3=0

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(x2+y2–1)3-x2y3=0 equation

The solution set is obviously symmetric with respect to the $y$-axis. Therefore we may assume $x\geq 0$. In the domain $\{(x,y)\in {\mathbb R}^2\ |\ x\geq0\}$ the equation is equivalent with\n$$x^2+ y^2 -1=x^{2/3} y\ ,$$\nwhich can easily be solved for $y$:\n$$y={1\over2}\bigl(x^{2/3}\pm\sqrt{x^{4/3}+4(1-x^2)}\bigr)\ .$$\nNow plot this, taking both branches of the square root into account. You might have to numerically solve the equation $x^{4/3}+4(1-x^2)=0$ in order to get the exact $x$-interval.

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The solution

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(x2 + y2 - 1)*3 - x2*y3 = 0
$$- x_{2} y_{3} + 3 \left(\left(x_{2} + y_{2}\right) - 1\right) = 0$$
Detail solution
Given the linear equation:
(x2+y2-1)*3-x2*y3 = 0

Expand brackets in the left part
x2*3+y2*3-1*3-x2*y3 = 0

Looking for similar summands in the left part:
-3 + 3*x2 + 3*y2 - x2*y3 = 0

Move free summands (without y3)
from left part to right part, we given:
$$- x_{2} y_{3} + 3 x_{2} + 3 y_{2} = 3$$
Move the summands with the other variables
from left part to right part, we given:
$$- x_{2} y_{3} + 3 y_{2} = \left(-3\right) x_{2} + 3$$
Divide both parts of the equation by (3*y2 - x2*y3)/y3
y3 = 3 - 3*x2 / ((3*y2 - x2*y3)/y3)

We get the answer: y3 = 3*(-1 + x2 + y2)/x2
The solution of the parametric equation
Given the equation with a parameter:
$$- x_{2} y_{3} + 3 x_{2} + 3 y_{2} - 3 = 0$$
The coefficient at y3 is equal to
$$- x_{2}$$
then possible cases for x2 :
$$x_{2} < 0$$
$$x_{2} = 0$$
Consider all cases in more detail:
With
$$x_{2} < 0$$
the equation
$$3 y_{2} + y_{3} - 6 = 0$$
its solution
$$y_{3} = 6 - 3 y_{2}$$
With
$$x_{2} = 0$$
the equation
$$3 y_{2} - 3 = 0$$
its solution
The graph
Rapid solution [src]
        /  3*(-1 + re(x2) + re(y2))*im(x2)   3*(im(x2) + im(y2))*re(x2)\   3*(im(x2) + im(y2))*im(x2)   3*(-1 + re(x2) + re(y2))*re(x2)
y31 = I*|- ------------------------------- + --------------------------| + -------------------------- + -------------------------------
        |           2         2                    2         2         |         2         2                     2         2           
        \         im (x2) + re (x2)              im (x2) + re (x2)     /       im (x2) + re (x2)               im (x2) + re (x2)       
$$y_{31} = i \left(\frac{3 \left(\operatorname{im}{\left(x_{2}\right)} + \operatorname{im}{\left(y_{2}\right)}\right) \operatorname{re}{\left(x_{2}\right)}}{\left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}} - \frac{3 \left(\operatorname{re}{\left(x_{2}\right)} + \operatorname{re}{\left(y_{2}\right)} - 1\right) \operatorname{im}{\left(x_{2}\right)}}{\left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}}\right) + \frac{3 \left(\operatorname{im}{\left(x_{2}\right)} + \operatorname{im}{\left(y_{2}\right)}\right) \operatorname{im}{\left(x_{2}\right)}}{\left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}} + \frac{3 \left(\operatorname{re}{\left(x_{2}\right)} + \operatorname{re}{\left(y_{2}\right)} - 1\right) \operatorname{re}{\left(x_{2}\right)}}{\left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}}$$
y31 = i*(3*(im(x2) + im(y2))*re(x2)/(re(x2)^2 + im(x2)^2) - 3*(re(x2) + re(y2) - 1)*im(x2)/(re(x2)^2 + im(x2)^2)) + 3*(im(x2) + im(y2))*im(x2)/(re(x2)^2 + im(x2)^2) + 3*(re(x2) + re(y2) - 1)*re(x2)/(re(x2)^2 + im(x2)^2)
Sum and product of roots [src]
sum
  /  3*(-1 + re(x2) + re(y2))*im(x2)   3*(im(x2) + im(y2))*re(x2)\   3*(im(x2) + im(y2))*im(x2)   3*(-1 + re(x2) + re(y2))*re(x2)
I*|- ------------------------------- + --------------------------| + -------------------------- + -------------------------------
  |           2         2                    2         2         |         2         2                     2         2           
  \         im (x2) + re (x2)              im (x2) + re (x2)     /       im (x2) + re (x2)               im (x2) + re (x2)       
$$i \left(\frac{3 \left(\operatorname{im}{\left(x_{2}\right)} + \operatorname{im}{\left(y_{2}\right)}\right) \operatorname{re}{\left(x_{2}\right)}}{\left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}} - \frac{3 \left(\operatorname{re}{\left(x_{2}\right)} + \operatorname{re}{\left(y_{2}\right)} - 1\right) \operatorname{im}{\left(x_{2}\right)}}{\left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}}\right) + \frac{3 \left(\operatorname{im}{\left(x_{2}\right)} + \operatorname{im}{\left(y_{2}\right)}\right) \operatorname{im}{\left(x_{2}\right)}}{\left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}} + \frac{3 \left(\operatorname{re}{\left(x_{2}\right)} + \operatorname{re}{\left(y_{2}\right)} - 1\right) \operatorname{re}{\left(x_{2}\right)}}{\left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}}$$
=
  /  3*(-1 + re(x2) + re(y2))*im(x2)   3*(im(x2) + im(y2))*re(x2)\   3*(im(x2) + im(y2))*im(x2)   3*(-1 + re(x2) + re(y2))*re(x2)
I*|- ------------------------------- + --------------------------| + -------------------------- + -------------------------------
  |           2         2                    2         2         |         2         2                     2         2           
  \         im (x2) + re (x2)              im (x2) + re (x2)     /       im (x2) + re (x2)               im (x2) + re (x2)       
$$i \left(\frac{3 \left(\operatorname{im}{\left(x_{2}\right)} + \operatorname{im}{\left(y_{2}\right)}\right) \operatorname{re}{\left(x_{2}\right)}}{\left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}} - \frac{3 \left(\operatorname{re}{\left(x_{2}\right)} + \operatorname{re}{\left(y_{2}\right)} - 1\right) \operatorname{im}{\left(x_{2}\right)}}{\left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}}\right) + \frac{3 \left(\operatorname{im}{\left(x_{2}\right)} + \operatorname{im}{\left(y_{2}\right)}\right) \operatorname{im}{\left(x_{2}\right)}}{\left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}} + \frac{3 \left(\operatorname{re}{\left(x_{2}\right)} + \operatorname{re}{\left(y_{2}\right)} - 1\right) \operatorname{re}{\left(x_{2}\right)}}{\left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}}$$
product
  /  3*(-1 + re(x2) + re(y2))*im(x2)   3*(im(x2) + im(y2))*re(x2)\   3*(im(x2) + im(y2))*im(x2)   3*(-1 + re(x2) + re(y2))*re(x2)
I*|- ------------------------------- + --------------------------| + -------------------------- + -------------------------------
  |           2         2                    2         2         |         2         2                     2         2           
  \         im (x2) + re (x2)              im (x2) + re (x2)     /       im (x2) + re (x2)               im (x2) + re (x2)       
$$i \left(\frac{3 \left(\operatorname{im}{\left(x_{2}\right)} + \operatorname{im}{\left(y_{2}\right)}\right) \operatorname{re}{\left(x_{2}\right)}}{\left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}} - \frac{3 \left(\operatorname{re}{\left(x_{2}\right)} + \operatorname{re}{\left(y_{2}\right)} - 1\right) \operatorname{im}{\left(x_{2}\right)}}{\left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}}\right) + \frac{3 \left(\operatorname{im}{\left(x_{2}\right)} + \operatorname{im}{\left(y_{2}\right)}\right) \operatorname{im}{\left(x_{2}\right)}}{\left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}} + \frac{3 \left(\operatorname{re}{\left(x_{2}\right)} + \operatorname{re}{\left(y_{2}\right)} - 1\right) \operatorname{re}{\left(x_{2}\right)}}{\left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}}$$
=
3*(I*((im(x2) + im(y2))*re(x2) - (-1 + re(x2) + re(y2))*im(x2)) + (im(x2) + im(y2))*im(x2) + (-1 + re(x2) + re(y2))*re(x2))
---------------------------------------------------------------------------------------------------------------------------
                                                       2         2                                                         
                                                     im (x2) + re (x2)                                                     
$$\frac{3 \left(i \left(\left(\operatorname{im}{\left(x_{2}\right)} + \operatorname{im}{\left(y_{2}\right)}\right) \operatorname{re}{\left(x_{2}\right)} - \left(\operatorname{re}{\left(x_{2}\right)} + \operatorname{re}{\left(y_{2}\right)} - 1\right) \operatorname{im}{\left(x_{2}\right)}\right) + \left(\operatorname{im}{\left(x_{2}\right)} + \operatorname{im}{\left(y_{2}\right)}\right) \operatorname{im}{\left(x_{2}\right)} + \left(\operatorname{re}{\left(x_{2}\right)} + \operatorname{re}{\left(y_{2}\right)} - 1\right) \operatorname{re}{\left(x_{2}\right)}\right)}{\left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}}$$
3*(i*((im(x2) + im(y2))*re(x2) - (-1 + re(x2) + re(y2))*im(x2)) + (im(x2) + im(y2))*im(x2) + (-1 + re(x2) + re(y2))*re(x2))/(im(x2)^2 + re(x2)^2)
The graph
(x2+y2–1)3-x2y3=0 equation
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