Linear Algebra Examples

$\frac{f + 3}{7 f} - \frac{3 f}{f + 4}$

Step 1

To write $\frac{f + 3}{7 f}$ as a fraction with a common denominator, multiply by $\frac{f + 4}{f + 4}$ .

$\frac{f + 3}{7 f} \cdot \frac{f + 4}{f + 4} - \frac{3 f}{f + 4}$

Step 2

To write $- \frac{3 f}{f + 4}$ as a fraction with a common denominator, multiply by $\frac{7 f}{7 f}$ .

$\frac{f + 3}{7 f} \cdot \frac{f + 4}{f + 4} - \frac{3 f}{f + 4} \cdot \frac{7 f}{7 f}$

Step 3

Write each expression with a common denominator of $7 f (f + 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{f + 3}{7 f}$ by $\frac{f + 4}{f + 4}$ .

$\frac{(f + 3) (f + 4)}{7 f (f + 4)} - \frac{3 f}{f + 4} \cdot \frac{7 f}{7 f}$

Step 3.2

Multiply $\frac{3 f}{f + 4}$ by $\frac{7 f}{7 f}$ .

$\frac{(f + 3) (f + 4)}{7 f (f + 4)} - \frac{3 f (7 f)}{(f + 4) (7 f)}$

Step 3.3

Reorder the factors of $(f + 4) (7 f)$ .

$\frac{(f + 3) (f + 4)}{7 f (f + 4)} - \frac{3 f (7 f)}{7 f (f + 4)}$

Step 4

Combine the numerators over the common denominator.

$\frac{(f + 3) (f + 4) - 3 f (7 f)}{7 f (f + 4)}$

Step 5

Simplify the numerator.

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Step 5.1

Expand $(f + 3) (f + 4)$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

$\frac{f (f + 4) + 3 (f + 4) - 3 f (7 f)}{7 f (f + 4)}$

Step 5.1.2

Apply the distributive property.

$\frac{f \cdot f + f \cdot 4 + 3 (f + 4) - 3 f (7 f)}{7 f (f + 4)}$

Step 5.1.3

Apply the distributive property.

$\frac{f \cdot f + f \cdot 4 + 3 f + 3 \cdot 4 - 3 f (7 f)}{7 f (f + 4)}$

Step 5.2

Simplify and combine like terms.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $f$ by $f$ .

$\frac{f^{2} + f \cdot 4 + 3 f + 3 \cdot 4 - 3 f (7 f)}{7 f (f + 4)}$

Step 5.2.1.2

Move $4$ to the left of $f$ .

$\frac{f^{2} + 4 \cdot f + 3 f + 3 \cdot 4 - 3 f (7 f)}{7 f (f + 4)}$

Step 5.2.1.3

Multiply $3$ by $4$ .

$\frac{f^{2} + 4 f + 3 f + 12 - 3 f (7 f)}{7 f (f + 4)}$

Step 5.2.2

Add $4 f$ and $3 f$ .

$\frac{f^{2} + 7 f + 12 - 3 f (7 f)}{7 f (f + 4)}$

Step 5.3

Rewrite using the commutative property of multiplication.

$\frac{f^{2} + 7 f + 12 - 3 \cdot 7 f \cdot f}{7 f (f + 4)}$

Step 5.4

Multiply $f$ by $f$ by adding the exponents.

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Step 5.4.1

Move $f$ .

$\frac{f^{2} + 7 f + 12 - 3 \cdot 7 (f \cdot f)}{7 f (f + 4)}$

Step 5.4.2

Multiply $f$ by $f$ .

$\frac{f^{2} + 7 f + 12 - 3 \cdot 7 f^{2}}{7 f (f + 4)}$

Step 5.5

Multiply $- 3$ by $7$ .

$\frac{f^{2} + 7 f + 12 - 21 f^{2}}{7 f (f + 4)}$

Step 5.6

Subtract $21 f^{2}$ from $f^{2}$ .

$\frac{- 20 f^{2} + 7 f + 12}{7 f (f + 4)}$

Step 6

Simplify with factoring out.

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Step 6.1

Factor $- 1$ out of $- 20 f^{2}$ .

$\frac{- (20 f^{2}) + 7 f + 12}{7 f (f + 4)}$

Step 6.2

Factor $- 1$ out of $7 f$ .

$\frac{- (20 f^{2}) - (- 7 f) + 12}{7 f (f + 4)}$

Step 6.3

Factor $- 1$ out of $- (20 f^{2}) - (- 7 f)$ .

$\frac{- (20 f^{2} - 7 f) + 12}{7 f (f + 4)}$

Step 6.4

Rewrite $12$ as $- 1 (- 12)$ .

$\frac{- (20 f^{2} - 7 f) - 1 (- 12)}{7 f (f + 4)}$

Step 6.5

Factor $- 1$ out of $- (20 f^{2} - 7 f) - 1 (- 12)$ .

$\frac{- (20 f^{2} - 7 f - 12)}{7 f (f + 4)}$

Step 6.6

Simplify the expression.

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Step 6.6.1

Rewrite $- (20 f^{2} - 7 f - 12)$ as $- 1 (20 f^{2} - 7 f - 12)$ .

$\frac{- 1 (20 f^{2} - 7 f - 12)}{7 f (f + 4)}$

Step 6.6.2

Move the negative in front of the fraction.

$- \frac{20 f^{2} - 7 f - 12}{7 f (f + 4)}$