Precalculus Examples

$u = 3 i - 2 j$ , $v = - 7 i - 7 j$

Step 1

Reorder $3 i$ and $- 2 j$ .

$u = - 2 j + 3 i$

$v = - 7 i - 7 j$

Step 2

Solve the equation for $i$ .

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Rewrite the equation as $- 7 i - 7 j = v$ .

$- 7 i - 7 j = v$

$u = - 2 j + 3 i$

Add $7 j$ to both sides of the equation.

$- 7 i = v + 7 j$

$u = - 2 j + 3 i$

Divide each term in $- 7 i = v + 7 j$ by $- 7$ and simplify.

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Divide each term in $- 7 i = v + 7 j$ by $- 7$ .

$\frac{- 7 i}{- 7} = \frac{v}{- 7} + \frac{7 j}{- 7}$

$u = - 2 j + 3 i$

Simplify the left side.

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Cancel the common factor of $- 7$ .

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Cancel the common factor.

$\frac{- 7 i}{- 7} = \frac{v}{- 7} + \frac{7 j}{- 7}$

$u = - 2 j + 3 i$

Divide $i$ by $1$ .

$i = \frac{v}{- 7} + \frac{7 j}{- 7}$

$u = - 2 j + 3 i$

$i = \frac{v}{- 7} + \frac{7 j}{- 7}$

$u = - 2 j + 3 i$

$i = \frac{v}{- 7} + \frac{7 j}{- 7}$

$u = - 2 j + 3 i$

Simplify the right side.

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Simplify each term.

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Move the negative in front of the fraction.

$i = - \frac{v}{7} + \frac{7 j}{- 7}$

$u = - 2 j + 3 i$

Cancel the common factor of $7$ and $- 7$ .

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Factor $7$ out of $7 j$ .

$i = - \frac{v}{7} + \frac{7 (j)}{- 7}$

$u = - 2 j + 3 i$

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

$i = - \frac{v}{7} - 1 \cdot j$

$u = - 2 j + 3 i$

$i = - \frac{v}{7} - 1 \cdot j$

$u = - 2 j + 3 i$

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

$i = - \frac{v}{7} - j$

$u = - 2 j + 3 i$

$i = - \frac{v}{7} - j$

$u = - 2 j + 3 i$

$i = - \frac{v}{7} - j$

$u = - 2 j + 3 i$

$i = - \frac{v}{7} - j$

$u = - 2 j + 3 i$

$i = - \frac{v}{7} - j$

$u = - 2 j + 3 i$

Step 3

Solve the equation for $j$ .

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Simplify $- 2 j + 3 (- \frac{v}{7} - j)$ .

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Simplify each term.

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Apply the distributive property.

$- 2 j + 3 (- \frac{v}{7}) + 3 (- j) = 3 (- \frac{v}{7} - j) - 2 j$

$i = - \frac{v}{7} - j$

Multiply $3 (- \frac{v}{7})$ .

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Multiply $- 1$ by $3$ .

$- 2 j - 3 \frac{v}{7} + 3 (- j) = 3 (- \frac{v}{7} - j) - 2 j$

$i = - \frac{v}{7} - j$

Combine $- 3$ and $\frac{v}{7}$ .

$- 2 j + \frac{- 3 v}{7} + 3 (- j) = 3 (- \frac{v}{7} - j) - 2 j$

$i = - \frac{v}{7} - j$

$- 2 j + \frac{- 3 v}{7} + 3 (- j) = 3 (- \frac{v}{7} - j) - 2 j$

$i = - \frac{v}{7} - j$

Multiply $- 1$ by $3$ .

$- 2 j + \frac{- 3 v}{7} - 3 j = 3 (- \frac{v}{7} - j) - 2 j$

$i = - \frac{v}{7} - j$

Move the negative in front of the fraction.

$- 2 j - \frac{3 v}{7} - 3 j = 3 (- \frac{v}{7} - j) - 2 j$

$i = - \frac{v}{7} - j$

$- 2 j - \frac{3 v}{7} - 3 j = 3 (- \frac{v}{7} - j) - 2 j$

$i = - \frac{v}{7} - j$

Subtract $3 j$ from $- 2 j$ .

$- 5 j - \frac{3 v}{7} = 3 (- \frac{v}{7} - j) - 2 j$

$i = - \frac{v}{7} - j$

$- 5 j - \frac{3 v}{7} = 3 (- \frac{v}{7} - j) - 2 j$

$i = - \frac{v}{7} - j$

Simplify $3 (- \frac{v}{7} - j) - 2 j$ .

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Simplify each term.

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Apply the distributive property.

$- 5 j - \frac{3 v}{7} = 3 (- \frac{v}{7}) + 3 (- j) - 2 j$

$i = - \frac{v}{7} - j$

Multiply $3 (- \frac{v}{7})$ .

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Multiply $- 1$ by $3$ .

$- 5 j - \frac{3 v}{7} = - 3 \frac{v}{7} + 3 (- j) - 2 j$

$i = - \frac{v}{7} - j$

Combine $- 3$ and $\frac{v}{7}$ .

$- 5 j - \frac{3 v}{7} = \frac{- 3 v}{7} + 3 (- j) - 2 j$

$i = - \frac{v}{7} - j$

$- 5 j - \frac{3 v}{7} = \frac{- 3 v}{7} + 3 (- j) - 2 j$

$i = - \frac{v}{7} - j$

Multiply $- 1$ by $3$ .

$- 5 j - \frac{3 v}{7} = \frac{- 3 v}{7} - 3 j - 2 j$

$i = - \frac{v}{7} - j$

Move the negative in front of the fraction.

$- 5 j - \frac{3 v}{7} = - \frac{3 v}{7} - 3 j - 2 j$

$i = - \frac{v}{7} - j$

$- 5 j - \frac{3 v}{7} = - \frac{3 v}{7} - 3 j - 2 j$

$i = - \frac{v}{7} - j$

Subtract $2 j$ from $- 3 j$ .

$- 5 j - \frac{3 v}{7} = - \frac{3 v}{7} - 5 j$

$i = - \frac{v}{7} - j$

$- 5 j - \frac{3 v}{7} = - \frac{3 v}{7} - 5 j$

$i = - \frac{v}{7} - j$

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

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Add $5 j$ to both sides of the equation.

$- 5 j - \frac{3 v}{7} + 5 j = - \frac{3 v}{7}$

$i = - \frac{v}{7} - j$

Combine the opposite terms in $- 5 j - \frac{3 v}{7} + 5 j$ .

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Add $- 5 j$ and $5 j$ .

$- \frac{3 v}{7} + 0 = - \frac{3 v}{7}$

$i = - \frac{v}{7} - j$

Add $- \frac{3 v}{7}$ and $0$ .

$- \frac{3 v}{7} = - \frac{3 v}{7}$

$i = - \frac{v}{7} - j$

$- \frac{3 v}{7} = - \frac{3 v}{7}$

$i = - \frac{v}{7} - j$

$- \frac{3 v}{7} = - \frac{3 v}{7}$

$i = - \frac{v}{7} - j$

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- 3 v = - 3 v$

$i = - \frac{v}{7} - j$

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

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

$- 3 v + 3 v = 0$

$i = - \frac{v}{7} - j$

Add $- 3 v$ and $3 v$ .

$0 = 0$

$i = - \frac{v}{7} - j$

$0 = 0$

$i = - \frac{v}{7} - j$

Since $0 = 0$ , the equation will always be true.

Always true

$i = - \frac{v}{7} - j$

Always true

$i = - \frac{v}{7} - j$

Step 4

Simplify the right side.

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Reorder $- \frac{v}{7}$ and $- j$ .

$i = - j - \frac{v}{7}$

Always true

$i = - j - \frac{v}{7}$

Always true