Basic Math Examples

$\frac{7}{y + 2} - \frac{2 y + 1}{y}$

Step 1

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

$\frac{7}{y + 2} \cdot \frac{y}{y} - \frac{2 y + 1}{y}$

Step 2

To write $- \frac{2 y + 1}{y}$ as a fraction with a common denominator, multiply by $\frac{y + 2}{y + 2}$ .

$\frac{7}{y + 2} \cdot \frac{y}{y} - \frac{2 y + 1}{y} \cdot \frac{y + 2}{y + 2}$

Step 3

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

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Step 3.1

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

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

Step 3.2

Multiply $\frac{2 y + 1}{y}$ by $\frac{y + 2}{y + 2}$ .

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $2$ by $- 1$ .

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

Step 5.3

Multiply $- 1$ by $1$ .

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

Step 5.4

Expand $(- 2 y - 1) (y + 2)$ using the FOIL Method.

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Step 5.4.1

Apply the distributive property.

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

Step 5.4.2

Apply the distributive property.

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

Step 5.4.3

Apply the distributive property.

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

Step 5.5

Simplify and combine like terms.

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Step 5.5.1

Simplify each term.

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Step 5.5.1.1

Multiply $y$ by $y$ by adding the exponents.

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Step 5.5.1.1.1

Move $y$ .

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

Step 5.5.1.1.2

Multiply $y$ by $y$ .

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

Step 5.5.1.2

Multiply $2$ by $- 2$ .

$\frac{7 y - 2 y^{2} - 4 y - 1 y - 1 \cdot 2}{y (y + 2)}$

Step 5.5.1.3

Rewrite $- 1 y$ as $- y$ .

$\frac{7 y - 2 y^{2} - 4 y - y - 1 \cdot 2}{y (y + 2)}$

Step 5.5.1.4

Multiply $- 1$ by $2$ .

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

Step 5.5.2

Subtract $y$ from $- 4 y$ .

$\frac{7 y - 2 y^{2} - 5 y - 2}{y (y + 2)}$

Step 5.6

Subtract $5 y$ from $7 y$ .

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

Step 5.7

Factor $2$ out of $- 2 y^{2} + 2 y - 2$ .

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Step 5.7.1

Factor $2$ out of $- 2 y^{2}$ .

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

Step 5.7.2

Factor $2$ out of $2 y$ .

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

Step 5.7.3

Factor $2$ out of $- 2$ .

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

Step 5.7.4

Factor $2$ out of $2 (- y^{2}) + 2 (y)$ .

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

Step 5.7.5

Factor $2$ out of $2 (- y^{2} + y) + 2 (- 1)$ .

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

Step 6

Simplify with factoring out.

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Step 6.1

Factor $- 1$ out of $- y^{2}$ .

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

Step 6.2

Factor $- 1$ out of $y$ .

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

Step 6.3

Factor $- 1$ out of $- (y^{2}) - 1 (- y)$ .

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

Step 6.4

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

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

Step 6.5

Factor $- 1$ out of $- (y^{2} - y) - 1 (1)$ .

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

Step 6.6

Simplify the expression.

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Step 6.6.1

Rewrite $- (y^{2} - y + 1)$ as $- 1 (y^{2} - y + 1)$ .

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

Step 6.6.2

Move the negative in front of the fraction.

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