Algebra Examples

$\frac{7}{w + 4} = \frac{5}{w + 4} - \frac{6}{w}$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $\frac{5}{w + 4}$ from both sides of the equation.

$\frac{7}{w + 4} - \frac{5}{w + 4} = - \frac{6}{w}$

Step 1.2

Add $\frac{6}{w}$ to both sides of the equation.

$\frac{7}{w + 4} - \frac{5}{w + 4} + \frac{6}{w} = 0$

Step 2

Simplify $\frac{7}{w + 4} - \frac{5}{w + 4} + \frac{6}{w}$ .

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Step 2.1

Combine the numerators over the common denominator.

$\frac{7 - 5}{w + 4} + \frac{6}{w} = 0$

Step 2.2

Subtract $5$ from $7$ .

$\frac{2}{w + 4} + \frac{6}{w} = 0$

Step 2.3

To write $\frac{2}{w + 4}$ as a fraction with a common denominator, multiply by $\frac{w}{w}$ .

$\frac{2}{w + 4} \cdot \frac{w}{w} + \frac{6}{w} = 0$

Step 2.4

To write $\frac{6}{w}$ as a fraction with a common denominator, multiply by $\frac{w + 4}{w + 4}$ .

$\frac{2}{w + 4} \cdot \frac{w}{w} + \frac{6}{w} \cdot \frac{w + 4}{w + 4} = 0$

Step 2.5

Write each expression with a common denominator of $(w + 4) w$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.5.1

Multiply $\frac{2}{w + 4}$ by $\frac{w}{w}$ .

$\frac{2 w}{(w + 4) w} + \frac{6}{w} \cdot \frac{w + 4}{w + 4} = 0$

Step 2.5.2

Multiply $\frac{6}{w}$ by $\frac{w + 4}{w + 4}$ .

$\frac{2 w}{(w + 4) w} + \frac{6 (w + 4)}{w (w + 4)} = 0$

Step 2.5.3

Reorder the factors of $(w + 4) w$ .

$\frac{2 w}{w (w + 4)} + \frac{6 (w + 4)}{w (w + 4)} = 0$

Step 2.6

Combine the numerators over the common denominator.

$\frac{2 w + 6 (w + 4)}{w (w + 4)} = 0$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Factor $2$ out of $2 w + 6 (w + 4)$ .

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Step 2.7.1.1

Factor $2$ out of $2 w$ .

$\frac{2 (w) + 6 (w + 4)}{w (w + 4)} = 0$

Step 2.7.1.2

Factor $2$ out of $6 (w + 4)$ .

$\frac{2 (w) + 2 (3 (w + 4))}{w (w + 4)} = 0$

Step 2.7.1.3

Factor $2$ out of $2 (w) + 2 (3 (w + 4))$ .

$\frac{2 (w + 3 (w + 4))}{w (w + 4)} = 0$

Step 2.7.2

Apply the distributive property.

$\frac{2 (w + 3 w + 3 \cdot 4)}{w (w + 4)} = 0$

Step 2.7.3

Multiply $3$ by $4$ .

$\frac{2 (w + 3 w + 12)}{w (w + 4)} = 0$

Step 2.7.4

Add $w$ and $3 w$ .

$\frac{2 (4 w + 12)}{w (w + 4)} = 0$

Step 2.7.5

Factor $4$ out of $4 w + 12$ .

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Step 2.7.5.1

Factor $4$ out of $4 w$ .

$\frac{2 (4 (w) + 12)}{w (w + 4)} = 0$

Step 2.7.5.2

Factor $4$ out of $12$ .

$\frac{2 (4 w + 4 \cdot 3)}{w (w + 4)} = 0$

Step 2.7.5.3

Factor $4$ out of $4 w + 4 \cdot 3$ .

$\frac{2 (4 (w + 3))}{w (w + 4)} = 0$

$\frac{2 \cdot 4 (w + 3)}{w (w + 4)} = 0$

Step 2.7.6

Multiply $2$ by $4$ .

$\frac{8 (w + 3)}{w (w + 4)} = 0$

Step 3

Set the numerator equal to zero.

$8 (w + 3) = 0$

Step 4

Solve the equation for $w$ .

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Step 4.1

Divide each term in $8 (w + 3) = 0$ by $8$ and simplify.

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Step 4.1.1

Divide each term in $8 (w + 3) = 0$ by $8$ .

$\frac{8 (w + 3)}{8} = \frac{0}{8}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $8$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{8 (w + 3)}{8} = \frac{0}{8}$

Step 4.1.2.1.2

Divide $w + 3$ by $1$ .

$w + 3 = \frac{0}{8}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $8$ .

$w + 3 = 0$

Step 4.2

Subtract $3$ from both sides of the equation.

$w = - 3$