Solve z=1/2sqrt{2}+J1/2sqrt{2}

$$z = \frac { 1 } { 2 } \sqrt { 2 } + J \frac { 1 } { 2 } \sqrt { 2 }$$

Solve for J

$$J=\sqrt{2}z-1$$

Steps for Solving Linear Equation

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

$$\frac{1}{2}\sqrt{2}+J\times \frac{1}{2}\sqrt{2}=z$$

Subtract \frac{1}{2}\sqrt{2} from both sides.

$$J\times \frac{1}{2}\sqrt{2}=z-\frac{1}{2}\sqrt{2}$$

The equation is in standard form.

$$\frac{\sqrt{2}}{2}J=z-\frac{\sqrt{2}}{2}$$

Divide both sides by \frac{1}{2}\sqrt{2}.

$$\frac{2\times \frac{\sqrt{2}}{2}J}{\sqrt{2}}=\frac{2\left(z-\frac{\sqrt{2}}{2}\right)}{\sqrt{2}}$$

Dividing by \frac{1}{2}\sqrt{2} undoes the multiplication by \frac{1}{2}\sqrt{2}.

$$J=\frac{2\left(z-\frac{\sqrt{2}}{2}\right)}{\sqrt{2}}$$

Divide -\frac{\sqrt{2}}{2}+z by \frac{1}{2}\sqrt{2}.

$$J=\sqrt{2}z-1$$

Solve for z

$$z=\frac{\sqrt{2}\left(J+1\right)}{2}$$