How calculate intersection directly without Stokes' theorem?

Question

Calculate the line integral directly without Stokes' theorem: \begin{gather*} \oint_\gamma \mathbf{F} \cdot d\mathbf{r} \end{gather*} \begin{gather*} \mathbf{F}(x,y,z)=(2z-3y) {\hat{\mathbf{i}}} + (3x-z){\hat{\mathbf{j}}} + (y-2x){\hat{\mathbf{k}}} \end{gather*} $\gamma$ is the intersection of the sphere \begin{gather*} x^2+y^2+z^2=1 \end{gather*} and the plane \begin{gather*} x+y+z=0 \end{gather*} Solution: First the intersection: \begin{gather*} z=\pm \sqrt{1-x^2-y^2} \\ z=-x-y \end{gather*} Solve with the positive square root: \begin{gather*} \sqrt{1-x^2-y^2} =-x-y \\ 1-x^2-y^2=(-x-y)^2 = x^2+y^2+2xy\\ \iff \\ 2x^2+2y^2+2xy=1 \end{gather*} And here I'm stuck because I can't eliminate x or y in the last equation ...