Basic Math Examples

$\frac{- 4}{y + 2} + \frac{y}{7 - y} + \frac{y^{2} + 14}{y^{2} - 5 y - 14}$

Step 1

Simplify each term.

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Step 1.1

Move the negative in front of the fraction.

$- \frac{4}{y + 2} + \frac{y}{7 - y} + \frac{y^{2} + 14}{y^{2} - 5 y - 14}$

Step 1.2

Factor $y^{2} - 5 y - 14$ using the AC method.

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Step 1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 14$ and whose sum is $- 5$ .

$- 7, 2$

Step 1.2.2

Write the factored form using these integers.

$- \frac{4}{y + 2} + \frac{y}{7 - y} + \frac{y^{2} + 14}{(y - 7) (y + 2)}$

Step 2

To write $- \frac{4}{y + 2}$ as a fraction with a common denominator, multiply by $\frac{7 - y}{7 - y}$ .

$- \frac{4}{y + 2} \cdot \frac{7 - y}{7 - y} + \frac{y}{7 - y} + \frac{y^{2} + 14}{(y - 7) (y + 2)}$

Step 3

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

$- \frac{4}{y + 2} \cdot \frac{7 - y}{7 - y} + \frac{y}{7 - y} \cdot \frac{y + 2}{y + 2} + \frac{y^{2} + 14}{(y - 7) (y + 2)}$

Step 4

Write each expression with a common denominator of $(y + 2) (7 - y)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{4}{y + 2}$ by $\frac{7 - y}{7 - y}$ .

$- \frac{4 (7 - y)}{(y + 2) (7 - y)} + \frac{y}{7 - y} \cdot \frac{y + 2}{y + 2} + \frac{y^{2} + 14}{(y - 7) (y + 2)}$

Step 4.2

Multiply $\frac{y}{7 - y}$ by $\frac{y + 2}{y + 2}$ .

$- \frac{4 (7 - y)}{(y + 2) (7 - y)} + \frac{y (y + 2)}{(7 - y) (y + 2)} + \frac{y^{2} + 14}{(y - 7) (y + 2)}$

Step 4.3

Reorder the factors of $(y + 2) (7 - y)$ .

$- \frac{4 (7 - y)}{(7 - y) (y + 2)} + \frac{y (y + 2)}{(7 - y) (y + 2)} + \frac{y^{2} + 14}{(y - 7) (y + 2)}$

Step 5

Combine the numerators over the common denominator.

$\frac{- 4 (7 - y) + y (y + 2)}{(7 - y) (y + 2)} + \frac{y^{2} + 14}{(y - 7) (y + 2)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{- 4 \cdot 7 - 4 (- y) + y (y + 2)}{(7 - y) (y + 2)} + \frac{y^{2} + 14}{(y - 7) (y + 2)}$

Step 6.2

Multiply $- 4$ by $7$ .

$\frac{- 28 - 4 (- y) + y (y + 2)}{(7 - y) (y + 2)} + \frac{y^{2} + 14}{(y - 7) (y + 2)}$

Step 6.3

Multiply $- 1$ by $- 4$ .

$\frac{- 28 + 4 y + y (y + 2)}{(7 - y) (y + 2)} + \frac{y^{2} + 14}{(y - 7) (y + 2)}$

Step 6.4

Apply the distributive property.

$\frac{- 28 + 4 y + y \cdot y + y \cdot 2}{(7 - y) (y + 2)} + \frac{y^{2} + 14}{(y - 7) (y + 2)}$

Step 6.5

Multiply $y$ by $y$ .

$\frac{- 28 + 4 y + y^{2} + y \cdot 2}{(7 - y) (y + 2)} + \frac{y^{2} + 14}{(y - 7) (y + 2)}$

Step 6.6

Move $2$ to the left of $y$ .

$\frac{- 28 + 4 y + y^{2} + 2 \cdot y}{(7 - y) (y + 2)} + \frac{y^{2} + 14}{(y - 7) (y + 2)}$

Step 6.7

Add $4 y$ and $2 y$ .

$\frac{- 28 + y^{2} + 6 y}{(7 - y) (y + 2)} + \frac{y^{2} + 14}{(y - 7) (y + 2)}$

Step 6.8

Reorder terms.

$\frac{y^{2} + 6 y - 28}{(7 - y) (y + 2)} + \frac{y^{2} + 14}{(y - 7) (y + 2)}$

Step 7

Simplify terms.

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Step 7.1

Factor $- 1$ out of $y$ .

$\frac{y^{2} + 6 y - 28}{(7 - y) (y + 2)} + \frac{y^{2} + 14}{(- 1 (- y) - 7) (y + 2)}$

Step 7.2

Rewrite $- 7$ as $- 1 (7)$ .

$\frac{y^{2} + 6 y - 28}{(7 - y) (y + 2)} + \frac{y^{2} + 14}{(- 1 (- y) - 1 (7)) (y + 2)}$

Step 7.3

Factor $- 1$ out of $- 1 (- y) - 1 (7)$ .

$\frac{y^{2} + 6 y - 28}{(7 - y) (y + 2)} + \frac{y^{2} + 14}{- 1 (- y + 7) (y + 2)}$

Step 7.4

Simplify the expression.

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Step 7.4.1

Move a negative from the denominator of $\frac{y^{2} + 14}{- 1 (- y + 7) (y + 2)}$ to the numerator.

$\frac{y^{2} + 6 y - 28}{(7 - y) (y + 2)} + \frac{- (y^{2} + 14)}{(- y + 7) (y + 2)}$

Step 7.4.2

Reorder terms.

$\frac{y^{2} + 6 y - 28}{(- y + 7) (y + 2)} + \frac{- (y^{2} + 14)}{(- y + 7) (y + 2)}$

Step 7.5

Combine the numerators over the common denominator.

$\frac{y^{2} + 6 y - 28 - (y^{2} + 14)}{(- y + 7) (y + 2)}$

Step 8

Simplify the numerator.

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Step 8.1

Apply the distributive property.

$\frac{y^{2} + 6 y - 28 - y^{2} - 1 \cdot 14}{(- y + 7) (y + 2)}$

Step 8.2

Multiply $- 1$ by $14$ .

$\frac{y^{2} + 6 y - 28 - y^{2} - 14}{(- y + 7) (y + 2)}$

Step 8.3

Subtract $y^{2}$ from $y^{2}$ .

$\frac{6 y - 28 + 0 - 14}{(- y + 7) (y + 2)}$

Step 8.4

Add $6 y$ and $0$ .

$\frac{6 y - 28 - 14}{(- y + 7) (y + 2)}$

Step 8.5

Subtract $14$ from $- 28$ .

$\frac{6 y - 42}{(- y + 7) (y + 2)}$

Step 8.6

Factor $6$ out of $6 y - 42$ .

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Step 8.6.1

Factor $6$ out of $6 y$ .

$\frac{6 (y) - 42}{(- y + 7) (y + 2)}$

Step 8.6.2

Factor $6$ out of $- 42$ .

$\frac{6 y + 6 \cdot - 7}{(- y + 7) (y + 2)}$

Step 8.6.3

Factor $6$ out of $6 y + 6 \cdot - 7$ .

$\frac{6 (y - 7)}{(- y + 7) (y + 2)}$

Step 9

Cancel the common factor of $y - 7$ and $- y + 7$ .

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Step 9.1

Factor $- 1$ out of $y$ .

$\frac{6 (- 1 (- y) - 7)}{(- y + 7) (y + 2)}$

Step 9.2

Rewrite $- 7$ as $- 1 (7)$ .

$\frac{6 (- 1 (- y) - 1 (7))}{(- y + 7) (y + 2)}$

Step 9.3

Factor $- 1$ out of $- 1 (- y) - 1 (7)$ .

$\frac{6 (- 1 (- y + 7))}{(- y + 7) (y + 2)}$

Step 9.4

Cancel the common factor.

$\frac{6 (- 1 (- y + 7))}{(- y + 7) (y + 2)}$

Step 9.5

Rewrite the expression.

$\frac{6 \cdot (- 1)}{y + 2}$

Step 10

Multiply $6$ by $- 1$ .

$\frac{- 6}{y + 2}$

Step 11

Move the negative in front of the fraction.

$- \frac{6}{y + 2}$