Solve z=arcsinx/3+sqrt{x-y}

$$z = \arcsin \frac { x } { 3 } + \sqrt { x - y }$$

Solve for y

$$y=-\left(-\arcsin(\frac{x}{3})+z\right)^{2}+x$$

$$-\arcsin(\frac{x}{3})+z\geq 0\text{ and }|x|\leq 3$$

Steps for Solving Radical Equations

Swap sides so that all variable terms are on the left hand side.

$$\arcsin(\frac{x}{3})+\sqrt{x-y}=z$$

Subtract \arcsin(\frac{x}{3}) from both sides.

$$\sqrt{x-y}=z-\arcsin(\frac{x}{3})$$

Square both sides of the equation.

$$-y+x=\left(-\arcsin(\frac{x}{3})+z\right)^{2}$$

Subtract x from both sides of the equation.

$$-y+x-x=\left(-\arcsin(\frac{x}{3})+z\right)^{2}-x$$

Subtracting x from itself leaves 0.

$$-y=\left(-\arcsin(\frac{x}{3})+z\right)^{2}-x$$

Divide both sides by -1.

$$\frac{-y}{-1}=\frac{\left(-\arcsin(\frac{x}{3})+z\right)^{2}-x}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$y=\frac{\left(-\arcsin(\frac{x}{3})+z\right)^{2}-x}{-1}$$

Divide \left(z-\arcsin(\frac{x}{3})\right)^{2}-x by -1.

$$y=-\left(-\arcsin(\frac{x}{3})+z\right)^{2}+x$$

Solve for y (complex solution)

$$y=-\left(-\arcsin(\frac{x}{3})+z\right)^{2}+x$$

$$z=\arcsin(\frac{x}{3})\text{ or }arg(-\arcsin(\frac{x}{3})+z)<\pi$$