Basic Math Examples

$\frac{5}{a} + \frac{3}{a + 1} - \frac{7}{a - 3}$

Step 1

Find the common denominator.

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Step 1.1

Multiply $\frac{5}{a}$ by $\frac{(a + 1) (a - 3)}{(a + 1) (a - 3)}$ .

$\frac{5}{a} \cdot \frac{(a + 1) (a - 3)}{(a + 1) (a - 3)} + \frac{3}{a + 1} - \frac{7}{a - 3}$

Step 1.2

Multiply $\frac{5}{a}$ by $\frac{(a + 1) (a - 3)}{(a + 1) (a - 3)}$ .

$\frac{5 ((a + 1) (a - 3))}{a ((a + 1) (a - 3))} + \frac{3}{a + 1} - \frac{7}{a - 3}$

Step 1.3

Multiply $\frac{3}{a + 1}$ by $\frac{a (a - 3)}{a (a - 3)}$ .

$\frac{5 ((a + 1) (a - 3))}{a ((a + 1) (a - 3))} + \frac{3}{a + 1} \cdot \frac{a (a - 3)}{a (a - 3)} - \frac{7}{a - 3}$

Step 1.4

Multiply $\frac{3}{a + 1}$ by $\frac{a (a - 3)}{a (a - 3)}$ .

$\frac{5 ((a + 1) (a - 3))}{a ((a + 1) (a - 3))} + \frac{3 (a (a - 3))}{(a + 1) (a (a - 3))} - \frac{7}{a - 3}$

Step 1.5

Multiply $\frac{7}{a - 3}$ by $\frac{a (a + 1)}{a (a + 1)}$ .

$\frac{5 ((a + 1) (a - 3))}{a ((a + 1) (a - 3))} + \frac{3 (a (a - 3))}{(a + 1) (a (a - 3))} - (\frac{7}{a - 3} \cdot \frac{a (a + 1)}{a (a + 1)})$

Step 1.6

Multiply $\frac{7}{a - 3}$ by $\frac{a (a + 1)}{a (a + 1)}$ .

$\frac{5 ((a + 1) (a - 3))}{a ((a + 1) (a - 3))} + \frac{3 (a (a - 3))}{(a + 1) (a (a - 3))} - \frac{7 (a (a + 1))}{(a - 3) (a (a + 1))}$

Step 1.7

Reorder the factors of $a ((a + 1) (a - 3))$ .

$\frac{5 ((a + 1) (a - 3))}{a (a + 1) (a - 3)} + \frac{3 (a (a - 3))}{(a + 1) (a (a - 3))} - \frac{7 (a (a + 1))}{(a - 3) (a (a + 1))}$

Step 1.8

Reorder the factors of $(a + 1) (a (a - 3))$ .

$\frac{5 ((a + 1) (a - 3))}{a (a + 1) (a - 3)} + \frac{3 (a (a - 3))}{a (a + 1) (a - 3)} - \frac{7 (a (a + 1))}{(a - 3) (a (a + 1))}$

Step 1.9

Reorder the factors of $(a - 3) (a (a + 1))$ .

$\frac{5 ((a + 1) (a - 3))}{a (a + 1) (a - 3)} + \frac{3 (a (a - 3))}{a (a + 1) (a - 3)} - \frac{7 (a (a + 1))}{a (a + 1) (a - 3)}$

Step 2

Combine the numerators over the common denominator.

$\frac{5 ((a + 1) (a - 3)) + 3 (a (a - 3)) - 7 (a (a + 1))}{a (a + 1) (a - 3)}$

Step 3

Simplify each term.

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Step 3.1

Expand $(a + 1) (a - 3)$ using the FOIL Method.

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Step 3.1.1

Apply the distributive property.

$\frac{5 (a (a - 3) + 1 (a - 3)) + 3 (a (a - 3)) - 7 (a (a + 1))}{a (a + 1) (a - 3)}$

Step 3.1.2

Apply the distributive property.

$\frac{5 (a \cdot a + a \cdot - 3 + 1 (a - 3)) + 3 (a (a - 3)) - 7 (a (a + 1))}{a (a + 1) (a - 3)}$

Step 3.1.3

Apply the distributive property.

$\frac{5 (a \cdot a + a \cdot - 3 + 1 a + 1 \cdot - 3) + 3 (a (a - 3)) - 7 (a (a + 1))}{a (a + 1) (a - 3)}$

Step 3.2

Simplify and combine like terms.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $a$ by $a$ .

$\frac{5 (a^{2} + a \cdot - 3 + 1 a + 1 \cdot - 3) + 3 (a (a - 3)) - 7 (a (a + 1))}{a (a + 1) (a - 3)}$

Step 3.2.1.2

Move $- 3$ to the left of $a$ .

$\frac{5 (a^{2} - 3 \cdot a + 1 a + 1 \cdot - 3) + 3 (a (a - 3)) - 7 (a (a + 1))}{a (a + 1) (a - 3)}$

Step 3.2.1.3

Multiply $a$ by $1$ .

$\frac{5 (a^{2} - 3 a + a + 1 \cdot - 3) + 3 (a (a - 3)) - 7 (a (a + 1))}{a (a + 1) (a - 3)}$

Step 3.2.1.4

Multiply $- 3$ by $1$ .

$\frac{5 (a^{2} - 3 a + a - 3) + 3 (a (a - 3)) - 7 (a (a + 1))}{a (a + 1) (a - 3)}$

Step 3.2.2

Add $- 3 a$ and $a$ .

$\frac{5 (a^{2} - 2 a - 3) + 3 (a (a - 3)) - 7 (a (a + 1))}{a (a + 1) (a - 3)}$

Step 3.3

Apply the distributive property.

$\frac{5 a^{2} + 5 (- 2 a) + 5 \cdot - 3 + 3 (a (a - 3)) - 7 (a (a + 1))}{a (a + 1) (a - 3)}$

Step 3.4

Simplify.

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Step 3.4.1

Multiply $- 2$ by $5$ .

$\frac{5 a^{2} - 10 a + 5 \cdot - 3 + 3 (a (a - 3)) - 7 (a (a + 1))}{a (a + 1) (a - 3)}$

Step 3.4.2

Multiply $5$ by $- 3$ .

$\frac{5 a^{2} - 10 a - 15 + 3 (a (a - 3)) - 7 (a (a + 1))}{a (a + 1) (a - 3)}$

Step 3.5

Apply the distributive property.

$\frac{5 a^{2} - 10 a - 15 + 3 (a \cdot a + a \cdot - 3) - 7 (a (a + 1))}{a (a + 1) (a - 3)}$

Step 3.6

Multiply $a$ by $a$ .

$\frac{5 a^{2} - 10 a - 15 + 3 (a^{2} + a \cdot - 3) - 7 (a (a + 1))}{a (a + 1) (a - 3)}$

Step 3.7

Move $- 3$ to the left of $a$ .

$\frac{5 a^{2} - 10 a - 15 + 3 (a^{2} - 3 \cdot a) - 7 (a (a + 1))}{a (a + 1) (a - 3)}$

Step 3.8

Apply the distributive property.

$\frac{5 a^{2} - 10 a - 15 + 3 a^{2} + 3 (- 3 a) - 7 (a (a + 1))}{a (a + 1) (a - 3)}$

Step 3.9

Multiply $- 3$ by $3$ .

$\frac{5 a^{2} - 10 a - 15 + 3 a^{2} - 9 a - 7 (a (a + 1))}{a (a + 1) (a - 3)}$

Step 3.10

Apply the distributive property.

$\frac{5 a^{2} - 10 a - 15 + 3 a^{2} - 9 a - 7 (a \cdot a + a \cdot 1)}{a (a + 1) (a - 3)}$

Step 3.11

Multiply $a$ by $a$ .

$\frac{5 a^{2} - 10 a - 15 + 3 a^{2} - 9 a - 7 (a^{2} + a \cdot 1)}{a (a + 1) (a - 3)}$

Step 3.12

Multiply $a$ by $1$ .

$\frac{5 a^{2} - 10 a - 15 + 3 a^{2} - 9 a - 7 (a^{2} + a)}{a (a + 1) (a - 3)}$

Step 3.13

Apply the distributive property.

$\frac{5 a^{2} - 10 a - 15 + 3 a^{2} - 9 a - 7 a^{2} - 7 a}{a (a + 1) (a - 3)}$

Step 4

Simplify by adding terms.

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Step 4.1

Add $5 a^{2}$ and $3 a^{2}$ .

$\frac{8 a^{2} - 10 a - 15 - 9 a - 7 a^{2} - 7 a}{a (a + 1) (a - 3)}$

Step 4.2

Subtract $7 a^{2}$ from $8 a^{2}$ .

$\frac{a^{2} - 10 a - 15 - 9 a - 7 a}{a (a + 1) (a - 3)}$

Step 4.3

Subtract $9 a$ from $- 10 a$ .

$\frac{a^{2} - 19 a - 15 - 7 a}{a (a + 1) (a - 3)}$

Step 4.4

Subtract $7 a$ from $- 19 a$ .

$\frac{a^{2} - 26 a - 15}{a (a + 1) (a - 3)}$