Pre-Algebra Examples

$| z | = | z - 1 - i |$

Step 1

Rewrite the equation as $| z - 1 - i | = | z |$ .

$| z - 1 - i | = | z |$

Step 2

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

$z - 1 - i = z$

$z - 1 - i = - z$

$- (z - 1 - i) = z$

$- (z - 1 - i) = - z$

Step 3

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

$z - 1 - i = z$

$z - 1 - i = - z$

Step 4

Solve $z - 1 - i = z$ 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.

$z - 1 - i - z = 0$

Step 4.1.2

Combine the opposite terms in $z - 1 - i - z$ .

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Step 4.1.2.1

Subtract $z$ from $z$ .

$0 - 1 - i = 0$

Step 4.1.2.2

Subtract $1$ from $0$ .

$- 1 - i = 0$

Step 4.2

Since $- 1 - i \neq 0$ , there are no solutions.

No solution

Step 5

Solve $z - 1 - i = - z$ for $z$ .

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Step 5.1

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

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Step 5.1.1

Add $z$ to both sides of the equation.

$z - 1 - i + z = 0$

Step 5.1.2

Add $z$ and $z$ .

$2 z - 1 - i = 0$

Step 5.2

Move all terms not containing $z$ to the right side of the equation.

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Step 5.2.1

Add $1$ to both sides of the equation.

$2 z - i = 1$

Step 5.2.2

Add $i$ to both sides of the equation.

$2 z = 1 + i$

Step 5.3

Divide each term in $2 z = 1 + i$ by $2$ and simplify.

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Step 5.3.1

Divide each term in $2 z = 1 + i$ by $2$ .

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

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.

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

Step 5.3.2.1.2

Divide $z$ by $1$ .

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

Step 6

List all of the solutions.

$z =$