Basic Math Examples

$| z - 4 i | = | z |$

Step 1

Rewrite the equation as $| z | = | z - 4 i |$ .

$| z | = | z - 4 i |$

Step 2

Rewrite the absolute value equation as four equations without absolute value bars.

$z = z - 4 i$

$z = - (z - 4 i)$

$- z = z - 4 i$

$- z = - (z - 4 i)$

Step 3

After simplifying, there are only two unique equations to be solved.

$z = z - 4 i$

$z = - (z - 4 i)$

Step 4

Solve $z = z - 4 i$ for $z$ .

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Step 4.1

Move all terms containing $z$ to the left side of the equation.

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Step 4.1.1

Subtract $z$ from both sides of the equation.

$z - z = - 4 i$

Step 4.1.2

Subtract $z$ from $z$ .

$0 = - 4 i$

Step 4.2

Since $0 \neq - 4 i$ , there are no solutions.

No solution

Step 5

Solve $z = - (z - 4 i)$ for $z$ .

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Step 5.1

Simplify $- (z - 4 i)$ .

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Step 5.1.1

Apply the distributive property.

$z = - z - (- 4 i)$

Step 5.1.2

Multiply $- 4$ by $- 1$ .

$z = - z + 4 i$

Step 5.2

Move all terms containing $z$ to the left side of the equation.

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Step 5.2.1

Add $z$ to both sides of the equation.

$z + z = 4 i$

Step 5.2.2

Add $z$ and $z$ .

$2 z = 4 i$

Step 5.3

Divide each term in $2 z = 4 i$ by $2$ and simplify.

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Step 5.3.1

Divide each term in $2 z = 4 i$ by $2$ .

$\frac{2 z}{2} = \frac{4 i}{2}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $2$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{2 z}{2} = \frac{4 i}{2}$

Step 5.3.2.1.2

Divide $z$ by $1$ .

$z = \frac{4 i}{2}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Cancel the common factor of $4$ and $2$ .

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Step 5.3.3.1.1

Factor $2$ out of $4 i$ .

$z = \frac{2 (2 i)}{2}$

Step 5.3.3.1.2

Cancel the common factors.

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Step 5.3.3.1.2.1

Factor $2$ out of $2$ .

$z = \frac{2 (2 i)}{2 (1)}$

Step 5.3.3.1.2.2

Cancel the common factor.

$z = \frac{2 (2 i)}{2 \cdot 1}$

Step 5.3.3.1.2.3

Rewrite the expression.

$z = \frac{2 i}{1}$

Step 5.3.3.1.2.4

Divide $2 i$ by $1$ .

$z = 2 i$

Step 6

List all of the solutions.

$z =$