Algebra Examples

$4 x + 2 i = 8 + y i$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$4 x + 2 i = 8 + (0) \cdot i$

Step 1.2

Solve the equation.

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Step 1.2.1

Simplify $8 + (0) \cdot i$ .

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Step 1.2.1.1

Multiply $0$ by $i$ .

$4 x + 2 i = 8 + 0$

Step 1.2.1.2

Add $8$ and $0$ .

$4 x + 2 i = 8$

Step 1.2.2

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

$4 x = 8 - 2 i$

Step 1.2.3

Divide each term in $4 x = 8 - 2 i$ by $4$ and simplify.

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Step 1.2.3.1

Divide each term in $4 x = 8 - 2 i$ by $4$ .

$\frac{4 x}{4} = \frac{8}{4} + \frac{- 2 i}{4}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $4$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{8}{4} + \frac{- 2 i}{4}$

Step 1.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{8}{4} + \frac{- 2 i}{4}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Simplify each term.

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Step 1.2.3.3.1.1

Divide $8$ by $4$ .

$x = 2 + \frac{- 2 i}{4}$

Step 1.2.3.3.1.2

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

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Step 1.2.3.3.1.2.1

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

$x = 2 + \frac{2 (- i)}{4}$

Step 1.2.3.3.1.2.2

Cancel the common factors.

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Step 1.2.3.3.1.2.2.1

Factor $2$ out of $4$ .

$x = 2 + \frac{2 (- i)}{2 (2)}$

Step 1.2.3.3.1.2.2.2

Cancel the common factor.

$x = 2 + \frac{2 (- i)}{2 \cdot 2}$

Step 1.2.3.3.1.2.2.3

Rewrite the expression.

$x = 2 + \frac{- i}{2}$

Step 1.2.3.3.1.3

Move the negative in front of the fraction.

$x = 2 - \frac{i}{2}$

Step 1.3

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

x-intercept(s): $None$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$4 (0) + 2 i = 8 + y i$

Step 2.2

Solve the equation.

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Step 2.2.1

Rewrite the equation as $8 + y i = 4 (0) + 2 i$ .

$8 + y i = 4 (0) + 2 i$

Step 2.2.2

Simplify $4 (0) + 2 i$ .

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Step 2.2.2.1

Multiply $4$ by $0$ .

$8 + y i = 0 + 2 i$

Step 2.2.2.2

Add $0$ and $2 i$ .

$8 + y i = 2 i$

Step 2.2.3

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

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Step 2.2.3.1

Subtract $8$ from both sides of the equation.

$y i = 2 i - 8$

Step 2.2.3.2

Reorder $2 i$ and $- 8$ .

$y i = - 8 + 2 i$

Step 2.2.4

Divide each term in $y i = - 8 + 2 i$ by $i$ and simplify.

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Step 2.2.4.1

Divide each term in $y i = - 8 + 2 i$ by $i$ .

$\frac{y i}{i} = \frac{- 8}{i} + \frac{2 i}{i}$

Step 2.2.4.2

Simplify the left side.

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Step 2.2.4.2.1

Cancel the common factor of $i$ .

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Step 2.2.4.2.1.1

Cancel the common factor.

$\frac{y i}{i} = \frac{- 8}{i} + \frac{2 i}{i}$

Step 2.2.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 8}{i} + \frac{2 i}{i}$

Step 2.2.4.3

Simplify the right side.

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Step 2.2.4.3.1

Simplify each term.

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Step 2.2.4.3.1.1

Multiply the numerator and denominator of $\frac{- 8}{i}$ by the conjugate of $i$ to make the denominator real.

$y = \frac{- 8}{i} \cdot \frac{i}{i} + \frac{2 i}{i}$

Step 2.2.4.3.1.2

Multiply.

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Step 2.2.4.3.1.2.1

Combine.

$y = \frac{- 8 i}{i i} + \frac{2 i}{i}$

Step 2.2.4.3.1.2.2

Simplify the denominator.

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Step 2.2.4.3.1.2.2.1

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

$y = \frac{- 8 i}{i^{1} i} + \frac{2 i}{i}$

Step 2.2.4.3.1.2.2.2

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

$y = \frac{- 8 i}{i^{1} i^{1}} + \frac{2 i}{i}$

Step 2.2.4.3.1.2.2.3

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

$y = \frac{- 8 i}{i^{1 + 1}} + \frac{2 i}{i}$

Step 2.2.4.3.1.2.2.4

Add $1$ and $1$ .

$y = \frac{- 8 i}{i^{2}} + \frac{2 i}{i}$

Step 2.2.4.3.1.2.2.5

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

$y = \frac{- 8 i}{- 1} + \frac{2 i}{i}$

Step 2.2.4.3.1.3

Move the negative one from the denominator of $\frac{- 8 i}{- 1}$ .

$y = - 1 \cdot (- 8 i) + \frac{2 i}{i}$

Step 2.2.4.3.1.4

Rewrite $- 1 \cdot (- 8 i)$ as $- (- 8 i)$ .

$y = - (- 8 i) + \frac{2 i}{i}$

Step 2.2.4.3.1.5

Multiply $- 8$ by $- 1$ .

$y = 8 i + \frac{2 i}{i}$

Step 2.2.4.3.1.6

Cancel the common factor of $i$ .

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Step 2.2.4.3.1.6.1

Cancel the common factor.

$y = 8 i + \frac{2 i}{i}$

Step 2.2.4.3.1.6.2

Divide $2$ by $1$ .

$y = 8 i + 2$

Step 2.2.4.3.2

Reorder $8 i$ and $2$ .

$y = 2 + 8 i$

Step 2.3

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

y-intercept(s): $None$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $None$

Step 4