Solve frac{z_{1}}{z_{2}}=z_{1}*z_2^-1

$$\frac { z _ { 1 } } { z _ { 2 } } = z _ { 1 } \cdot z _ { 2 } ^ { - 1 }$$

Solve for z_1 (complex solution)

$$z_{1}\in \mathrm{C}$$

$$z_{2}\neq 0$$

Solution Steps

Multiply both sides of the equation by z_{2}.

$$z_{1}=z_{1}z_{2}^{-1}z_{2}$$

Subtract z_{1}z_{2}^{-1}z_{2} from both sides.

$$z_{1}-z_{1}z_{2}^{-1}z_{2}=0$$

Reorder the terms.

$$z_{1}-\frac{1}{z_{2}}z_{1}z_{2}=0$$

Multiply both sides of the equation by z_{2}.

$$z_{2}z_{1}-\frac{1}{z_{2}}z_{1}z_{2}z_{2}=0$$

Multiply z_{2} and z_{2} to get z_{2}^{2}.

$$z_{2}z_{1}-\frac{1}{z_{2}}z_{1}z_{2}^{2}=0$$

Express \frac{1}{z_{2}}z_{1} as a single fraction.

$$z_{2}z_{1}-\frac{z_{1}}{z_{2}}z_{2}^{2}=0$$

Express \frac{z_{1}}{z_{2}}z_{2}^{2} as a single fraction.

$$z_{2}z_{1}-\frac{z_{1}z_{2}^{2}}{z_{2}}=0$$

Cancel out z_{2} in both numerator and denominator.

$$z_{2}z_{1}-z_{1}z_{2}=0$$

Combine z_{2}z_{1} and -z_{1}z_{2} to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any z_{1}.

$$z_{1}\in \mathrm{C}$$

Solve for z_1

$$z_{1}\in \mathrm{R}$$

$$z_{2}\neq 0$$

Solution Steps

Multiply both sides of the equation by z_{2}.

$$z_{1}=z_{1}z_{2}^{-1}z_{2}$$

Subtract z_{1}z_{2}^{-1}z_{2} from both sides.

$$z_{1}-z_{1}z_{2}^{-1}z_{2}=0$$

Reorder the terms.

$$z_{1}-\frac{1}{z_{2}}z_{1}z_{2}=0$$

Multiply both sides of the equation by z_{2}.

$$z_{2}z_{1}-\frac{1}{z_{2}}z_{1}z_{2}z_{2}=0$$

Multiply z_{2} and z_{2} to get z_{2}^{2}.

$$z_{2}z_{1}-\frac{1}{z_{2}}z_{1}z_{2}^{2}=0$$

Express \frac{1}{z_{2}}z_{1} as a single fraction.

$$z_{2}z_{1}-\frac{z_{1}}{z_{2}}z_{2}^{2}=0$$

Express \frac{z_{1}}{z_{2}}z_{2}^{2} as a single fraction.

$$z_{2}z_{1}-\frac{z_{1}z_{2}^{2}}{z_{2}}=0$$

Cancel out z_{2} in both numerator and denominator.

$$z_{2}z_{1}-z_{1}z_{2}=0$$

Combine z_{2}z_{1} and -z_{1}z_{2} to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any z_{1}.

$$z_{1}\in \mathrm{R}$$

Solve for z_2

$$z_{2}\neq 0$$

Solution Steps

Variable z_{2} cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by z_{2}.

$$z_{1}=z_{1}z_{2}^{-1}z_{2}$$

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

$$z_{1}z_{2}^{-1}z_{2}=z_{1}$$

Reorder the terms.

$$\frac{1}{z_{2}}z_{1}z_{2}=z_{1}$$

Variable z_{2} cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by z_{2}.

$$1z_{1}z_{2}=z_{1}z_{2}$$

Subtract z_{1}z_{2} from both sides.

$$1z_{1}z_{2}-z_{1}z_{2}=0$$

Combine 1z_{1}z_{2} and -z_{1}z_{2} to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any z_{2}.

$$z_{2}\in \mathrm{R}$$

Variable z_{2} cannot be equal to 0.

$$z_{2}\in \mathrm{R}\setminus 0$$