Finite Math Examples

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

Step 1

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

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

Step 2

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

$1 - i z = 1 + i z$

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

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

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

Step 3

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

$1 - i z = 1 + i z$

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

Step 4

Solve $1 - i z = 1 + i z$ 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 $i z$ from both sides of the equation.

$1 - i z - i z = 1$

Step 4.1.2

Subtract $i z$ from $- i z$ .

$1 - 2 i z = 1$

Step 4.2

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

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Step 4.2.1

Subtract $1$ from both sides of the equation.

$- 2 i z = 1 - 1$

Step 4.2.2

Subtract $1$ from $1$ .

$- 2 i z = 0$

Step 4.3

Divide each term in $- 2 i z = 0$ by $- 2 i$ and simplify.

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Step 4.3.1

Divide each term in $- 2 i z = 0$ by $- 2 i$ .

$\frac{- 2 i z}{- 2 i} = \frac{0}{- 2 i}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $- 2$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{- 2 i z}{- 2 i} = \frac{0}{- 2 i}$

Step 4.3.2.1.2

Rewrite the expression.

$\frac{i z}{i} = \frac{0}{- 2 i}$

Step 4.3.2.2

Cancel the common factor of $i$ .

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Step 4.3.2.2.1

Cancel the common factor.

$\frac{i z}{i} = \frac{0}{- 2 i}$

Step 4.3.2.2.2

Divide $z$ by $1$ .

$z = \frac{0}{- 2 i}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Cancel the common factor of $0$ and $- 2$ .

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Step 4.3.3.1.1

Factor $2$ out of $0$ .

$z = \frac{2 (0)}{- 2 i}$

Step 4.3.3.1.2

Cancel the common factors.

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Step 4.3.3.1.2.1

Factor $2$ out of $- 2 i$ .

$z = \frac{2 (0)}{2 (- i)}$

Step 4.3.3.1.2.2

Cancel the common factor.

$z = \frac{2 \cdot 0}{2 (- i)}$

Step 4.3.3.1.2.3

Rewrite the expression.

$z = \frac{0}{- i}$

Step 4.3.3.2

Multiply the numerator and denominator of $\frac{0}{- i}$ by the conjugate of $- i$ to make the denominator real.

$z = \frac{0}{- i} \cdot \frac{i}{i}$

Step 4.3.3.3

Multiply.

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Step 4.3.3.3.1

Combine.

$z = \frac{0 i}{- i i}$

Step 4.3.3.3.2

Multiply $0$ by $i$ .

$z = \frac{0}{- i i}$

Step 4.3.3.3.3

Simplify the denominator.

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Step 4.3.3.3.3.1

Add parentheses.

$z = \frac{0}{- (i i)}$

Step 4.3.3.3.3.2

Raise $i$ to the power of $1$ .

$z = \frac{0}{- (i^{1} i)}$

Step 4.3.3.3.3.3

Raise $i$ to the power of $1$ .

$z = \frac{0}{- (i^{1} i^{1})}$

Step 4.3.3.3.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$z = \frac{0}{- i^{1 + 1}}$

Step 4.3.3.3.3.5

Add $1$ and $1$ .

$z = \frac{0}{- i^{2}}$

Step 4.3.3.3.3.6

Rewrite $i^{2}$ as $- 1$ .

$z = \frac{0}{- - 1}$

Step 4.3.3.4

Multiply $- 1$ by $- 1$ .

$z = \frac{0}{1}$

Step 4.3.3.5

Divide $0$ by $1$ .

$z = 0$

Step 5

Solve $1 - i z = - (1 + i z)$ for $z$ .

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Step 5.1

Simplify $- (1 + i z)$ .

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Step 5.1.1

Rewrite.

$1 - i z = 0 + 0 - (1 + i z)$

Step 5.1.2

Simplify by adding zeros.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.1.4

Multiply $- 1$ by $1$ .

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

Step 5.2

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

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Step 5.2.1

Add $i z$ to both sides of the equation.

$1 - i z + i z = - 1$

Step 5.2.2

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

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Step 5.2.2.1

Add $- i z$ and $i z$ .

$1 + 0 = - 1$

Step 5.2.2.2

Add $1$ and $0$ .

$1 = - 1$

Step 5.3

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

No solution

Step 6

List all of the solutions.

$z = 0$

Step 7

Exclude the solutions that do not make $| 1 + i z | = | 1 - i z |$ true.

$No solution$