Linear Algebra Examples

$- \frac{1}{2} + \frac{1}{5} w = - \frac{3}{7}$

Step 1

Combine $\frac{1}{5}$ and $w$ .

$- \frac{1}{2} + \frac{w}{5} = - \frac{3}{7}$

Step 2

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

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Step 2.1

Add $\frac{1}{2}$ to both sides of the equation.

$\frac{w}{5} = - \frac{3}{7} + \frac{1}{2}$

Step 2.2

To write $- \frac{3}{7}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{w}{5} = - \frac{3}{7} \cdot \frac{2}{2} + \frac{1}{2}$

Step 2.3

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{w}{5} = - \frac{3}{7} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{7}{7}$

Step 2.4

Write each expression with a common denominator of $14$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{3}{7}$ by $\frac{2}{2}$ .

$\frac{w}{5} = - \frac{3 \cdot 2}{7 \cdot 2} + \frac{1}{2} \cdot \frac{7}{7}$

Step 2.4.2

Multiply $7$ by $2$ .

$\frac{w}{5} = - \frac{3 \cdot 2}{14} + \frac{1}{2} \cdot \frac{7}{7}$

Step 2.4.3

Multiply $\frac{1}{2}$ by $\frac{7}{7}$ .

$\frac{w}{5} = - \frac{3 \cdot 2}{14} + \frac{7}{2 \cdot 7}$

Step 2.4.4

Multiply $2$ by $7$ .

$\frac{w}{5} = - \frac{3 \cdot 2}{14} + \frac{7}{14}$

Step 2.5

Combine the numerators over the common denominator.

$\frac{w}{5} = \frac{- 3 \cdot 2 + 7}{14}$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 3$ by $2$ .

$\frac{w}{5} = \frac{- 6 + 7}{14}$

Step 2.6.2

Add $- 6$ and $7$ .

$\frac{w}{5} = \frac{1}{14}$

Step 3

Multiply both sides of the equation by $5$ .

$5 \frac{w}{5} = 5 (\frac{1}{14})$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $5$ .

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Step 4.1.1.1

Cancel the common factor.

$5 \frac{w}{5} = 5 (\frac{1}{14})$

Step 4.1.1.2

Rewrite the expression.

$w = 5 (\frac{1}{14})$

Step 4.2

Simplify the right side.

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Step 4.2.1

Combine $5$ and $\frac{1}{14}$ .

$w = \frac{5}{14}$

Step 5

The domain is the set of all valid $w$ values.

${w | w = \frac{5}{14}}$

Step 6