Solve |frac{z_{1}}{z_{2}}|=frac{|z_{1}|}{|z_{2}|}

$$| \frac { z _ { 1 } } { z _ { 2 } } | = \frac { | z _ { 1 } | } { | z _ { 2 } | }$$

Solve for z_1

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

$$z_{2}\neq 0$$

Steps Using Definition of Absolute Value

Combine like terms and use the properties of equality to get the variable on one side of the equal sign and the numbers on the other side. Remember to follow the order of operations.

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

Subtract \frac{1}{|z_{2}|}|z_{1}| from both sides of the equation.

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

The absolute value of a product is the product of the absolute values.

$$|\frac{1}{z_{2}}||z_{1}|+\left(-\frac{1}{|z_{2}|}\right)|1||z_{1}|=0$$

Factor out |z_{1}|.

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

Add \frac{1}{|z_{2}|} to -\frac{1}{|z_{2}|}.

$$\text{true}$$

Divide both sides by 0.

$$|z_{1}|=\text{Indeterminate}$$

Use the definition of absolute value.

$$z_{1}=\text{Indeterminate} z_{1}=\text{Indeterminate}$$

Solve for z_2

$$z_{2}\neq 0$$