Algebra Examples

$(4 y - 5) + (- x - 3) i = 7 - 8 i$

Step 1

Apply the distributive property.

$4 y - 5 - x i - 3 i = 7 - 8 i$

Step 2

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

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Step 2.1

Subtract $4 y$ from both sides of the equation.

$- 5 - x i - 3 i = 7 - 8 i - 4 y$

Step 2.2

Add $5$ to both sides of the equation.

$- x i - 3 i = 7 - 8 i - 4 y + 5$

Step 2.3

Add $3 i$ to both sides of the equation.

$- x i = 7 - 8 i - 4 y + 5 + 3 i$

Step 2.4

Add $7$ and $5$ .

$- x i = 12 - 8 i - 4 y + 3 i$

Step 2.5

Add $- 8 i$ and $3 i$ .

$- x i = 12 - 4 y - 5 i$

Step 3

Divide each term in $- x i = 12 - 4 y - 5 i$ by $- i$ and simplify.

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Step 3.1

Divide each term in $- x i = 12 - 4 y - 5 i$ by $- i$ .

$\frac{- x i}{- i} = \frac{12}{- i} + \frac{- 4 y}{- i} + \frac{- 5 i}{- i}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{x i}{i} = \frac{12}{- i} + \frac{- 4 y}{- i} + \frac{- 5 i}{- i}$

Step 3.2.2

Cancel the common factor of $i$ .

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Step 3.2.2.1

Cancel the common factor.

$\frac{x i}{i} = \frac{12}{- i} + \frac{- 4 y}{- i} + \frac{- 5 i}{- i}$

Step 3.2.2.2

Divide $x$ by $1$ .

$x = \frac{12}{- i} + \frac{- 4 y}{- i} + \frac{- 5 i}{- i}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply the numerator and denominator of $\frac{12}{- i}$ by the conjugate of $- i$ to make the denominator real.

$x = \frac{12}{- i} \cdot \frac{i}{i} + \frac{- 4 y}{- i} + \frac{- 5 i}{- i}$

Step 3.3.1.2

Multiply.

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Step 3.3.1.2.1

Combine.

$x = \frac{12 i}{- i i} + \frac{- 4 y}{- i} + \frac{- 5 i}{- i}$

Step 3.3.1.2.2

Simplify the denominator.

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Step 3.3.1.2.2.1

Add parentheses.

$x = \frac{12 i}{- (i i)} + \frac{- 4 y}{- i} + \frac{- 5 i}{- i}$

Step 3.3.1.2.2.2

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

$x = \frac{12 i}{- (i^{1} i)} + \frac{- 4 y}{- i} + \frac{- 5 i}{- i}$

Step 3.3.1.2.2.3

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

$x = \frac{12 i}{- (i^{1} i^{1})} + \frac{- 4 y}{- i} + \frac{- 5 i}{- i}$

Step 3.3.1.2.2.4

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

$x = \frac{12 i}{- i^{1 + 1}} + \frac{- 4 y}{- i} + \frac{- 5 i}{- i}$

Step 3.3.1.2.2.5

Add $1$ and $1$ .

$x = \frac{12 i}{- i^{2}} + \frac{- 4 y}{- i} + \frac{- 5 i}{- i}$

Step 3.3.1.2.2.6

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

$x = \frac{12 i}{- - 1} + \frac{- 4 y}{- i} + \frac{- 5 i}{- i}$

Step 3.3.1.3

Multiply $- 1$ by $- 1$ .

$x = \frac{12 i}{1} + \frac{- 4 y}{- i} + \frac{- 5 i}{- i}$

Step 3.3.1.4

Divide $12 i$ by $1$ .

$x = 12 i + \frac{- 4 y}{- i} + \frac{- 5 i}{- i}$

Step 3.3.1.5

Multiply the numerator and denominator of $\frac{- 4 y}{- i}$ by the conjugate of $- i$ to make the denominator real.

$x = 12 i + \frac{- 4 y}{- i} \cdot \frac{i}{i} + \frac{- 5 i}{- i}$

Step 3.3.1.6

Multiply.

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Step 3.3.1.6.1

Combine.

$x = 12 i + \frac{- 4 y i}{- i i} + \frac{- 5 i}{- i}$

Step 3.3.1.6.2

Simplify the denominator.

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Step 3.3.1.6.2.1

Add parentheses.

$x = 12 i + \frac{- 4 y i}{- (i i)} + \frac{- 5 i}{- i}$

Step 3.3.1.6.2.2

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

$x = 12 i + \frac{- 4 y i}{- (i^{1} i)} + \frac{- 5 i}{- i}$

Step 3.3.1.6.2.3

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

$x = 12 i + \frac{- 4 y i}{- (i^{1} i^{1})} + \frac{- 5 i}{- i}$

Step 3.3.1.6.2.4

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

$x = 12 i + \frac{- 4 y i}{- i^{1 + 1}} + \frac{- 5 i}{- i}$

Step 3.3.1.6.2.5

Add $1$ and $1$ .

$x = 12 i + \frac{- 4 y i}{- i^{2}} + \frac{- 5 i}{- i}$

Step 3.3.1.6.2.6

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

$x = 12 i + \frac{- 4 y i}{- - 1} + \frac{- 5 i}{- i}$

Step 3.3.1.7

Multiply $- 1$ by $- 1$ .

$x = 12 i + \frac{- 4 y i}{1} + \frac{- 5 i}{- i}$

Step 3.3.1.8

Divide $- 4 y i$ by $1$ .

$x = 12 i - 4 y i + \frac{- 5 i}{- i}$

Step 3.3.1.9

Cancel the common factor of $i$ .

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Step 3.3.1.9.1

Cancel the common factor.

$x = 12 i - 4 y i + \frac{- 5 i}{- i}$

Step 3.3.1.9.2

Rewrite the expression.

$x = 12 i - 4 y i + \frac{- 5}{- 1}$

Step 3.3.1.9.3

Move the negative one from the denominator of $\frac{- 5}{- 1}$ .

$x = 12 i - 4 y i - 1 \cdot - 5$

Step 3.3.1.10

Rewrite $- 1 \cdot - 5$ as $- - 5$ .

$x = 12 i - 4 y i - - 5$

Step 3.3.1.11

Multiply $- 1$ by $- 5$ .

$x = 12 i - 4 y i + 5$