Solve z=2+it/2-ita

$$z = \frac { 2 + i t } { 2 - i t } a$$

Solve for a

$$\left\{\begin{matrix}a=\frac{z\left(2-it\right)}{it+2}\text{, }&t\neq 2i\text{ and }t\neq -2i\\a\in \mathrm{C}\text{, }&t=2i\text{ and }z=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by -it+2.

$$z\left(-it+2\right)=\left(2+it\right)a$$

Use the distributive property to multiply z by -it+2.

$$-izt+2z=\left(2+it\right)a$$

Use the distributive property to multiply 2+it by a.

$$-izt+2z=2a+ita$$

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

$$2a+ita=-izt+2z$$

Combine all terms containing a.

$$\left(2+it\right)a=-izt+2z$$

The equation is in standard form.

$$\left(it+2\right)a=2z-itz$$

Divide both sides by 2+it.

$$\frac{\left(it+2\right)a}{it+2}=\frac{z\left(2-it\right)}{it+2}$$

Dividing by 2+it undoes the multiplication by 2+it.

$$a=\frac{z\left(2-it\right)}{it+2}$$

Solve for t

$$\left\{\begin{matrix}t=\frac{2i\left(a-z\right)}{z+a}\text{, }&a\neq 0\text{ and }z\neq -a\\t\neq -2i\text{, }&z=0\text{ and }a=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable t cannot be equal to -2i since division by zero is not defined. Multiply both sides of the equation by -it+2.

$$z\left(-it+2\right)=\left(2+it\right)a$$

Use the distributive property to multiply z by -it+2.

$$-izt+2z=\left(2+it\right)a$$

Use the distributive property to multiply 2+it by a.

$$-izt+2z=2a+ita$$

Subtract ita from both sides.

$$-izt+2z-ita=2a$$

Subtract 2z from both sides.

$$-izt-ita=2a-2z$$

Combine all terms containing t.

$$\left(-iz-ia\right)t=2a-2z$$

Divide both sides by -iz-ia.

$$\frac{\left(-iz-ia\right)t}{-iz-ia}=\frac{2a-2z}{-iz-ia}$$

Dividing by -iz-ia undoes the multiplication by -iz-ia.

$$t=\frac{2a-2z}{-iz-ia}$$

Divide 2a-2z by -iz-ia.

$$t=\frac{2i\left(a-z\right)}{z+a}$$

Variable t cannot be equal to -2i.

$$t=\frac{2i\left(a-z\right)}{z+a}\text{, }t\neq -2i$$