2x+y=1
=>y=1-2x
\(x^3+y^3=6xy-8\)
=>\(x^3+\left(1-2x\right)^3=6x\left(1-2x\right)-8\)
=>\(x^3+1-6x+12x^2-8x^3=6x-12x^2-8\)
=>\(-7x^3+12x^2-6x+1+12x^2-6x+8=0\)
=>\(-7x^3+24x^2-12x+9=0\)
=>\(-7x^3+21x^2+3x^2-9x-3x+9=0\)
=>\(-7x^2\left(x-3\right)+3x\left(x-3\right)-3\left(x-3\right)=0\)
=>\(\left(x-3\right)\left(-7x^2+3x-3\right)=0\)
=>x-3=0
=>x=3
\(y=1-2x=1-2\cdot3=1-6=-5\)