Precalculus Examples

$a = 5 i + 7 j$ , $b = - 4 i + 3 j$

Step 1

Solve the equation for $i$ .

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Step 1.1

Rewrite the equation as $5 i + 7 j = a$ .

$5 i + 7 j = a$

$b = - 4 i + 3 j$

Step 1.2

Subtract $7 j$ from both sides of the equation.

$5 i = a - 7 j$

$b = - 4 i + 3 j$

Step 1.3

Divide each term in $5 i = a - 7 j$ by $5$ and simplify.

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Step 1.3.1

Divide each term in $5 i = a - 7 j$ by $5$ .

$\frac{5 i}{5} = \frac{a}{5} + \frac{- 7 j}{5}$

$b = - 4 i + 3 j$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $5$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{5 i}{5} = \frac{a}{5} + \frac{- 7 j}{5}$

$b = - 4 i + 3 j$

Step 1.3.2.1.2

Divide $i$ by $1$ .

$i = \frac{a}{5} + \frac{- 7 j}{5}$

$b = - 4 i + 3 j$

$i = \frac{a}{5} + \frac{- 7 j}{5}$

$b = - 4 i + 3 j$

$i = \frac{a}{5} + \frac{- 7 j}{5}$

$b = - 4 i + 3 j$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Move the negative in front of the fraction.

$i = \frac{a}{5} - \frac{7 j}{5}$

$b = - 4 i + 3 j$

$i = \frac{a}{5} - \frac{7 j}{5}$

$b = - 4 i + 3 j$

$i = \frac{a}{5} - \frac{7 j}{5}$

$b = - 4 i + 3 j$

$i = \frac{a}{5} - \frac{7 j}{5}$

$b = - 4 i + 3 j$

Step 2

Simplify $- 4 (\frac{a}{5} - \frac{7 j}{5}) + 3 j$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$b = - 4 \frac{a}{5} - 4 (- \frac{7 j}{5}) + 3 j$

$i = \frac{a}{5} - \frac{7 j}{5}$

Step 2.1.2

Combine $- 4$ and $\frac{a}{5}$ .

$b = \frac{- 4 a}{5} - 4 (- \frac{7 j}{5}) + 3 j$

$i = \frac{a}{5} - \frac{7 j}{5}$

Step 2.1.3

Multiply $- 4 (- \frac{7 j}{5})$ .

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Step 2.1.3.1

Multiply $- 1$ by $- 4$ .

$b = \frac{- 4 a}{5} + 4 (\frac{7 j}{5}) + 3 j$

$i = \frac{a}{5} - \frac{7 j}{5}$

Step 2.1.3.2

Combine $4$ and $\frac{7 j}{5}$ .

$b = \frac{- 4 a}{5} + \frac{4 (7 j)}{5} + 3 j$

$i = \frac{a}{5} - \frac{7 j}{5}$

Step 2.1.3.3

Multiply $7$ by $4$ .

$b = \frac{- 4 a}{5} + \frac{28 j}{5} + 3 j$

$i = \frac{a}{5} - \frac{7 j}{5}$

$b = \frac{- 4 a}{5} + \frac{28 j}{5} + 3 j$

$i = \frac{a}{5} - \frac{7 j}{5}$

Step 2.1.4

Move the negative in front of the fraction.

$b = - \frac{4 a}{5} + \frac{28 j}{5} + 3 j$

$i = \frac{a}{5} - \frac{7 j}{5}$

$b = - \frac{4 a}{5} + \frac{28 j}{5} + 3 j$

$i = \frac{a}{5} - \frac{7 j}{5}$

Step 2.2

To write $3 j$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$b = - \frac{4 a}{5} + \frac{28 j}{5} + 3 j \cdot \frac{5}{5}$

$i = \frac{a}{5} - \frac{7 j}{5}$

Step 2.3

Combine $3 j$ and $\frac{5}{5}$ .

$b = - \frac{4 a}{5} + \frac{28 j}{5} + \frac{3 j \cdot 5}{5}$

$i = \frac{a}{5} - \frac{7 j}{5}$

Step 2.4

Combine the numerators over the common denominator.

$b = - \frac{4 a}{5} + \frac{28 j + 3 j \cdot 5}{5}$

$i = \frac{a}{5} - \frac{7 j}{5}$

Step 2.5

Combine the numerators over the common denominator.

$b = \frac{- 4 a + 28 j + 3 j \cdot 5}{5}$

$i = \frac{a}{5} - \frac{7 j}{5}$

Step 2.6

Multiply $5$ by $3$ .

$b = \frac{- 4 a + 28 j + 15 j}{5}$

$i = \frac{a}{5} - \frac{7 j}{5}$

Step 2.7

Add $28 j$ and $15 j$ .

$b = \frac{- 4 a + 43 j}{5}$

$i = \frac{a}{5} - \frac{7 j}{5}$

Step 2.8

Factor $- 1$ out of $- 4 a$ .

$b = \frac{- (4 a) + 43 j}{5}$

$i = \frac{a}{5} - \frac{7 j}{5}$

Step 2.9

Factor $- 1$ out of $43 j$ .

$b = \frac{- (4 a) - (- 43 j)}{5}$

$i = \frac{a}{5} - \frac{7 j}{5}$

Step 2.10

Factor $- 1$ out of $- (4 a) - (- 43 j)$ .

$b = \frac{- (4 a - 43 j)}{5}$

$i = \frac{a}{5} - \frac{7 j}{5}$

Step 2.11

Simplify the expression.

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Step 2.11.1

Rewrite $- (4 a - 43 j)$ as $- 1 (4 a - 43 j)$ .

$b = \frac{- 1 (4 a - 43 j)}{5}$

$i = \frac{a}{5} - \frac{7 j}{5}$

Step 2.11.2

Move the negative in front of the fraction.

$b = - \frac{4 a - 43 j}{5}$

$i = \frac{a}{5} - \frac{7 j}{5}$

$b = - \frac{4 a - 43 j}{5}$

$i = \frac{a}{5} - \frac{7 j}{5}$

$b = - \frac{4 a - 43 j}{5}$

$i = \frac{a}{5} - \frac{7 j}{5}$

Step 3

Solve the equation for $j$ .

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Step 3.1

Rewrite the equation as $5 (\frac{a}{5} - \frac{7 j}{5}) + 7 j = a$ .

$5 (\frac{a}{5} - \frac{7 j}{5}) + 7 j = a$

$i = \frac{a}{5} - \frac{7 j}{5}$

Step 3.2

Simplify $5 (\frac{a}{5} - \frac{7 j}{5}) + 7 j$ .

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Apply the distributive property.

$5 (\frac{a}{5}) + 5 (- \frac{7 j}{5}) + 7 j = a$

$i = \frac{a}{5} - \frac{7 j}{5}$

Step 3.2.1.2

Cancel the common factor of $5$ .

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Step 3.2.1.2.1

Cancel the common factor.

$5 (\frac{a}{5}) + 5 (- \frac{7 j}{5}) + 7 j = a$

$i = \frac{a}{5} - \frac{7 j}{5}$

Step 3.2.1.2.2

Rewrite the expression.

$a + 5 (- \frac{7 j}{5}) + 7 j = a$

$i = \frac{a}{5} - \frac{7 j}{5}$

$a + 5 (- \frac{7 j}{5}) + 7 j = a$

$i = \frac{a}{5} - \frac{7 j}{5}$

Step 3.2.1.3

Cancel the common factor of $5$ .

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Step 3.2.1.3.1

Move the leading negative in $- \frac{7 j}{5}$ into the numerator.

$a + 5 (\frac{- 7 j}{5}) + 7 j = a$

$i = \frac{a}{5} - \frac{7 j}{5}$

Step 3.2.1.3.2

Cancel the common factor.

$a + 5 (\frac{- 7 j}{5}) + 7 j = a$

$i = \frac{a}{5} - \frac{7 j}{5}$

Step 3.2.1.3.3

Rewrite the expression.

$a - 7 j + 7 j = a$

$i = \frac{a}{5} - \frac{7 j}{5}$

$a - 7 j + 7 j = a$

$i = \frac{a}{5} - \frac{7 j}{5}$

$a - 7 j + 7 j = a$

$i = \frac{a}{5} - \frac{7 j}{5}$

Step 3.2.2

Combine the opposite terms in $a - 7 j + 7 j$ .

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Step 3.2.2.1

Add $- 7 j$ and $7 j$ .

$a + 0 = a$

$i = \frac{a}{5} - \frac{7 j}{5}$

Step 3.2.2.2

Add $a$ and $0$ .

$a = a$

$i = \frac{a}{5} - \frac{7 j}{5}$

$a = a$

$i = \frac{a}{5} - \frac{7 j}{5}$

$a = a$

$i = \frac{a}{5} - \frac{7 j}{5}$

Step 3.3

The variable $j$ got canceled.

All real numbers

$i = \frac{a}{5} - \frac{7 j}{5}$

All real numbers

$i = \frac{a}{5} - \frac{7 j}{5}$