Solve for b 3a+(4b+2)i=9-6i

$3 a + (4 b + 2) i = 9 - 6 i$

Step 1

Apply the distributive property.

$3 a + 4 b i + 2 i = 9 - 6 i$

Step 2

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

Subtract $3 a$ from both sides of the equation.

$4 b i + 2 i = 9 - 6 i - 3 a$

Subtract $2 i$ from both sides of the equation.

$4 b i = 9 - 6 i - 3 a - 2 i$

Subtract $2 i$ from $- 6 i$ .

$4 b i = 9 - 3 a - 8 i$

Step 3

Divide each term in $4 b i = 9 - 3 a - 8 i$ by $4 i$ and simplify.

Divide each term in $4 b i = 9 - 3 a - 8 i$ by $4 i$ .

$\frac{4 b i}{4 i} = \frac{9}{4 i} + \frac{- 3 a}{4 i} + \frac{- 8 i}{4 i}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 b i}{4 i} = \frac{9}{4 i} + \frac{- 3 a}{4 i} + \frac{- 8 i}{4 i}$

Rewrite the expression.

$\frac{b i}{i} = \frac{9}{4 i} + \frac{- 3 a}{4 i} + \frac{- 8 i}{4 i}$

Cancel the common factor of $i$ .

Cancel the common factor.

$\frac{b i}{i} = \frac{9}{4 i} + \frac{- 3 a}{4 i} + \frac{- 8 i}{4 i}$

Divide $b$ by $1$ .

$b = \frac{9}{4 i} + \frac{- 3 a}{4 i} + \frac{- 8 i}{4 i}$

Simplify the right side.

Simplify each term.

Multiply the numerator and denominator of $\frac{9}{4 i}$ by the conjugate of $4 i$ to make the denominator real.

$b = \frac{9}{4 i} \cdot \frac{i}{i} + \frac{- 3 a}{4 i} + \frac{- 8 i}{4 i}$

Multiply.

Combine.

$b = \frac{9 i}{4 i i} + \frac{- 3 a}{4 i} + \frac{- 8 i}{4 i}$

Simplify the denominator.

Add parentheses.

$b = \frac{9 i}{4 (i i)} + \frac{- 3 a}{4 i} + \frac{- 8 i}{4 i}$

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

$b = \frac{9 i}{4 (i^{1} i)} + \frac{- 3 a}{4 i} + \frac{- 8 i}{4 i}$

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

$b = \frac{9 i}{4 (i^{1} i^{1})} + \frac{- 3 a}{4 i} + \frac{- 8 i}{4 i}$

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

$b = \frac{9 i}{4 i^{1 + 1}} + \frac{- 3 a}{4 i} + \frac{- 8 i}{4 i}$

Add $1$ and $1$ .

$b = \frac{9 i}{4 i^{2}} + \frac{- 3 a}{4 i} + \frac{- 8 i}{4 i}$

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

$b = \frac{9 i}{4 \cdot - 1} + \frac{- 3 a}{4 i} + \frac{- 8 i}{4 i}$

Multiply $4$ by $- 1$ .

$b = \frac{9 i}{- 4} + \frac{- 3 a}{4 i} + \frac{- 8 i}{4 i}$

Move the negative in front of the fraction.

$b = - \frac{9 i}{4} + \frac{- 3 a}{4 i} + \frac{- 8 i}{4 i}$

Multiply the numerator and denominator of $\frac{- 3 a}{4 i}$ by the conjugate of $4 i$ to make the denominator real.

$b = - \frac{9 i}{4} + \frac{- 3 a}{4 i} \cdot \frac{i}{i} + \frac{- 8 i}{4 i}$

Multiply.

Combine.

$b = - \frac{9 i}{4} + \frac{- 3 a i}{4 i i} + \frac{- 8 i}{4 i}$

Simplify the denominator.

Add parentheses.

$b = - \frac{9 i}{4} + \frac{- 3 a i}{4 (i i)} + \frac{- 8 i}{4 i}$

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

$b = - \frac{9 i}{4} + \frac{- 3 a i}{4 (i^{1} i)} + \frac{- 8 i}{4 i}$

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

$b = - \frac{9 i}{4} + \frac{- 3 a i}{4 (i^{1} i^{1})} + \frac{- 8 i}{4 i}$

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

$b = - \frac{9 i}{4} + \frac{- 3 a i}{4 i^{1 + 1}} + \frac{- 8 i}{4 i}$

Add $1$ and $1$ .

$b = - \frac{9 i}{4} + \frac{- 3 a i}{4 i^{2}} + \frac{- 8 i}{4 i}$

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

$b = - \frac{9 i}{4} + \frac{- 3 a i}{4 \cdot - 1} + \frac{- 8 i}{4 i}$

Multiply $4$ by $- 1$ .

$b = - \frac{9 i}{4} + \frac{- 3 a i}{- 4} + \frac{- 8 i}{4 i}$

Dividing two negative values results in a positive value.

$b = - \frac{9 i}{4} + \frac{(3) a i}{4} + \frac{- 8 i}{4 i}$

Cancel the common factor of $- 8$ and $4$ .

Factor $4$ out of $- 8 i$ .

$b = - \frac{9 i}{4} + \frac{3 a i}{4} + \frac{4 (- 2 i)}{4 i}$

Cancel the common factors.

Factor $4$ out of $4 i$ .

$b = - \frac{9 i}{4} + \frac{3 a i}{4} + \frac{4 (- 2 i)}{4 (i)}$

Cancel the common factor.

$b = - \frac{9 i}{4} + \frac{3 a i}{4} + \frac{4 (- 2 i)}{4 i}$

Rewrite the expression.

$b = - \frac{9 i}{4} + \frac{3 a i}{4} + \frac{- 2 i}{i}$

Cancel the common factor of $i$ .

Cancel the common factor.

$b = - \frac{9 i}{4} + \frac{3 a i}{4} + \frac{- 2 i}{i}$

Divide $- 2$ by $1$ .

$b = - \frac{9 i}{4} + \frac{3 a i}{4} - 2$

Solve for b 3a+(4b+2)i=9-6i

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