Step 1 :Setting the two expressions for $y$ equal to each other, it follows that $2x^{\frac{6}{5}} = 6x^{\frac{3}{5}} - 4$.
Step 2 :Re-arranging, $2x^{\frac{6}{5}} - 6x^{\frac{3}{5}} + 4 = 0$.
Step 3 :Let $t = x^{\frac{3}{5}}$, then the equation becomes $2t^2 - 6t + 4 = 0$.
Step 4 :For there to be exactly one solution for $t$, then the discriminant of the given quadratic must be equal to zero.
Step 5 :Thus, $(-6)^2 - 4 \cdot 2 \cdot 4 = 36 - 32 = 4$.
Step 6 :Since the discriminant is not equal to zero, there is no $t$ that satisfies the equation, which means there is no $x$ that satisfies the original system of equations.
Step 7 :So, there is \(\boxed{\text{no solution}}\).