Precalculus Examples

$a = 4 i - 4 j$ , $b = 4 i + 5 j$

Step 1

Reorder $4 i$ and $- 4 j$ .

$a = - 4 j + 4 i$

$b = 4 i + 5 j$

Step 2

Solve the equation for $i$ .

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Step 2.1

Rewrite the equation as $4 i + 5 j = b$ .

$4 i + 5 j = b$

$a = - 4 j + 4 i$

Step 2.2

Subtract $5 j$ from both sides of the equation.

$4 i = b - 5 j$

$a = - 4 j + 4 i$

Step 2.3

Divide each term in $4 i = b - 5 j$ by $4$ and simplify.

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Step 2.3.1

Divide each term in $4 i = b - 5 j$ by $4$ .

$\frac{4 i}{4} = \frac{b}{4} + \frac{- 5 j}{4}$

$a = - 4 j + 4 i$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $4$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{4 i}{4} = \frac{b}{4} + \frac{- 5 j}{4}$

$a = - 4 j + 4 i$

Step 2.3.2.1.2

Divide $i$ by $1$ .

$i = \frac{b}{4} + \frac{- 5 j}{4}$

$a = - 4 j + 4 i$

$i = \frac{b}{4} + \frac{- 5 j}{4}$

$a = - 4 j + 4 i$

$i = \frac{b}{4} + \frac{- 5 j}{4}$

$a = - 4 j + 4 i$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Move the negative in front of the fraction.

$i = \frac{b}{4} - \frac{5 j}{4}$

$a = - 4 j + 4 i$

$i = \frac{b}{4} - \frac{5 j}{4}$

$a = - 4 j + 4 i$

$i = \frac{b}{4} - \frac{5 j}{4}$

$a = - 4 j + 4 i$

$i = \frac{b}{4} - \frac{5 j}{4}$

$a = - 4 j + 4 i$

Step 3

Solve the equation for $j$ .

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Step 3.1

Simplify $- 4 j + 4 (\frac{b}{4} - \frac{5 j}{4})$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Apply the distributive property.

$- 4 j + 4 (\frac{b}{4}) + 4 (- \frac{5 j}{4}) = 4 (\frac{b}{4} - \frac{5 j}{4}) - 4 j$

$a = - 4 j + 4 i$

Step 3.1.1.2

Cancel the common factor of $4$ .

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Step 3.1.1.2.1

Cancel the common factor.

$- 4 j + 4 (\frac{b}{4}) + 4 (- \frac{5 j}{4}) = 4 (\frac{b}{4} - \frac{5 j}{4}) - 4 j$

$a = - 4 j + 4 i$

Step 3.1.1.2.2

Rewrite the expression.

$- 4 j + b + 4 (- \frac{5 j}{4}) = 4 (\frac{b}{4} - \frac{5 j}{4}) - 4 j$

$a = - 4 j + 4 i$

$- 4 j + b + 4 (- \frac{5 j}{4}) = 4 (\frac{b}{4} - \frac{5 j}{4}) - 4 j$

$a = - 4 j + 4 i$

Step 3.1.1.3

Cancel the common factor of $4$ .

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Step 3.1.1.3.1

Move the leading negative in $- \frac{5 j}{4}$ into the numerator.

$- 4 j + b + 4 (\frac{- 5 j}{4}) = 4 (\frac{b}{4} - \frac{5 j}{4}) - 4 j$

$a = - 4 j + 4 i$

Step 3.1.1.3.2

Cancel the common factor.

$- 4 j + b + 4 (\frac{- 5 j}{4}) = 4 (\frac{b}{4} - \frac{5 j}{4}) - 4 j$

$a = - 4 j + 4 i$

Step 3.1.1.3.3

Rewrite the expression.

$- 4 j + b - 5 j = 4 (\frac{b}{4} - \frac{5 j}{4}) - 4 j$

$a = - 4 j + 4 i$

$- 4 j + b - 5 j = 4 (\frac{b}{4} - \frac{5 j}{4}) - 4 j$

$a = - 4 j + 4 i$

$- 4 j + b - 5 j = 4 (\frac{b}{4} - \frac{5 j}{4}) - 4 j$

$a = - 4 j + 4 i$

Step 3.1.2

Subtract $5 j$ from $- 4 j$ .

$b - 9 j = 4 (\frac{b}{4} - \frac{5 j}{4}) - 4 j$

$a = - 4 j + 4 i$

$b - 9 j = 4 (\frac{b}{4} - \frac{5 j}{4}) - 4 j$

$a = - 4 j + 4 i$

Step 3.2

Simplify $4 (\frac{b}{4} - \frac{5 j}{4}) - 4 j$ .

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Apply the distributive property.

$b - 9 j = 4 (\frac{b}{4}) + 4 (- \frac{5 j}{4}) - 4 j$

$a = - 4 j + 4 i$

Step 3.2.1.2

Cancel the common factor of $4$ .

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Step 3.2.1.2.1

Cancel the common factor.

$b - 9 j = 4 (\frac{b}{4}) + 4 (- \frac{5 j}{4}) - 4 j$

$a = - 4 j + 4 i$

Step 3.2.1.2.2

Rewrite the expression.

$b - 9 j = b + 4 (- \frac{5 j}{4}) - 4 j$

$a = - 4 j + 4 i$

$b - 9 j = b + 4 (- \frac{5 j}{4}) - 4 j$

$a = - 4 j + 4 i$

Step 3.2.1.3

Cancel the common factor of $4$ .

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Step 3.2.1.3.1

Move the leading negative in $- \frac{5 j}{4}$ into the numerator.

$b - 9 j = b + 4 (\frac{- 5 j}{4}) - 4 j$

$a = - 4 j + 4 i$

Step 3.2.1.3.2

Cancel the common factor.

$b - 9 j = b + 4 (\frac{- 5 j}{4}) - 4 j$

$a = - 4 j + 4 i$

Step 3.2.1.3.3

Rewrite the expression.

$b - 9 j = b - 5 j - 4 j$

$a = - 4 j + 4 i$

$b - 9 j = b - 5 j - 4 j$

$a = - 4 j + 4 i$

$b - 9 j = b - 5 j - 4 j$

$a = - 4 j + 4 i$

Step 3.2.2

Subtract $4 j$ from $- 5 j$ .

$b - 9 j = b - 9 j$

$a = - 4 j + 4 i$

$b - 9 j = b - 9 j$

$a = - 4 j + 4 i$

Step 3.3

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

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Step 3.3.1

Add $9 j$ to both sides of the equation.

$b - 9 j + 9 j = b$

$a = - 4 j + 4 i$

Step 3.3.2

Combine the opposite terms in $b - 9 j + 9 j$ .

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Step 3.3.2.1

Add $- 9 j$ and $9 j$ .

$b + 0 = b$

$a = - 4 j + 4 i$

Step 3.3.2.2

Add $b$ and $0$ .

$b = b$

$a = - 4 j + 4 i$

$b = b$

$a = - 4 j + 4 i$

$b = b$

$a = - 4 j + 4 i$

Step 3.4

The variable $j$ got canceled.

All real numbers

$a = - 4 j + 4 i$

All real numbers

$a = - 4 j + 4 i$