Algebra Examples

$\frac{f}{f - g} - \frac{g}{g - f}$

Step 1

Factor $- 1$ out of $g$ .

$\frac{f}{f - g} - \frac{g}{- 1 (- g) - f}$

Step 2

Factor $- 1$ out of $- f$ .

$\frac{f}{f - g} - \frac{g}{- 1 (- g) - (f)}$

Step 3

Factor $- 1$ out of $- 1 (- g) - (f)$ .

$\frac{f}{f - g} - \frac{g}{- 1 (- g + f)}$

Step 4

Reorder terms.

$\frac{f}{f - g} - \frac{g}{- 1 (f - g)}$

Step 5

To write $\frac{f}{f - g}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

$\frac{f}{f - g} \cdot \frac{- 1}{- 1} - \frac{g}{- 1 (f - g)}$

Step 6

Write each expression with a common denominator of $(f - g) \cdot - 1$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.1

Multiply $\frac{f}{f - g}$ by $\frac{- 1}{- 1}$ .

$\frac{f \cdot - 1}{(f - g) \cdot - 1} - \frac{g}{- 1 (f - g)}$

Step 6.2

Reorder the factors of $(f - g) \cdot - 1$ .

$\frac{f \cdot - 1}{- (f - g)} - \frac{g}{- 1 (f - g)}$

Step 7

Combine the numerators over the common denominator.

$\frac{f \cdot - 1 - g}{- (f - g)}$

Step 8

Simplify the numerator.

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Step 8.1

Move $- 1$ to the left of $f$ .

$\frac{- 1 \cdot f - g}{- (f - g)}$

Step 8.2

Rewrite $- 1 f$ as $- f$ .

$\frac{- f - g}{- (f - g)}$

Step 9

Simplify with factoring out.

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Step 9.1

Move the negative in front of the fraction.

$- \frac{- f - g}{f - g}$

Step 9.2

Factor $- 1$ out of $- f$ .

$- \frac{- (f) - g}{f - g}$

Step 9.3

Factor $- 1$ out of $- g$ .

$- \frac{- (f) - (g)}{f - g}$

Step 9.4

Factor $- 1$ out of $- (f) - (g)$ .

$- \frac{- (f + g)}{f - g}$

Step 9.5

Simplify the expression.

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Step 9.5.1

Rewrite $- (f + g)$ as $- 1 (f + g)$ .

$- \frac{- 1 (f + g)}{f - g}$

Step 9.5.2

Move the negative in front of the fraction.

$- - \frac{f + g}{f - g}$

Step 9.5.3

Multiply $- 1$ by $- 1$ .

$1 \frac{f + g}{f - g}$

Step 9.5.4

Multiply $\frac{f + g}{f - g}$ by $1$ .

$\frac{f + g}{f - g}$