Basic Math Examples

$\frac{\frac{y + 2}{y - 1} - \frac{y - 3}{y - 2}}{y + 2}$

Step 1

Multiply the numerator and denominator of the fraction by $(y - 1) (y - 2)$ .

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Step 1.1

Multiply $\frac{\frac{y + 2}{y - 1} - \frac{y - 3}{y - 2}}{y + 2}$ by $\frac{(y - 1) (y - 2)}{(y - 1) (y - 2)}$ .

$\frac{(y - 1) (y - 2)}{(y - 1) (y - 2)} \cdot \frac{\frac{y + 2}{y - 1} - \frac{y - 3}{y - 2}}{y + 2}$

Step 1.2

Combine.

$\frac{(y - 1) (y - 2) (\frac{y + 2}{y - 1} - \frac{y - 3}{y - 2})}{(y - 1) (y - 2) (y + 2)}$

Step 2

Apply the distributive property.

$\frac{(y - 1) (y - 2) \frac{y + 2}{y - 1} + (y - 1) (y - 2) (- \frac{y - 3}{y - 2})}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $y - 1$ .

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Step 3.1.1

Cancel the common factor.

$\frac{(y - 1) (y - 2) \frac{y + 2}{y - 1} + (y - 1) (y - 2) (- \frac{y - 3}{y - 2})}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 3.1.2

Rewrite the expression.

$\frac{(y - 2) (y + 2) + (y - 1) (y - 2) (- \frac{y - 3}{y - 2})}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 3.2

Cancel the common factor of $y - 2$ .

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Step 3.2.1

Move the leading negative in $- \frac{y - 3}{y - 2}$ into the numerator.

$\frac{(y - 2) (y + 2) + (y - 1) (y - 2) \frac{- (y - 3)}{y - 2}}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 3.2.2

Factor $y - 2$ out of $(y - 1) (y - 2)$ .

$\frac{(y - 2) (y + 2) + (y - 2) (y - 1) \frac{- (y - 3)}{y - 2}}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 3.2.3

Cancel the common factor.

$\frac{(y - 2) (y + 2) + (y - 2) (y - 1) \frac{- (y - 3)}{y - 2}}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 3.2.4

Rewrite the expression.

$\frac{(y - 2) (y + 2) + (y - 1) (- (y - 3))}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 4

Simplify the numerator.

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Step 4.1

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

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Step 4.1.1

Apply the distributive property.

$\frac{y (y + 2) - 2 (y + 2) + (y - 1) (- (y - 3))}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 4.1.2

Apply the distributive property.

$\frac{y \cdot y + y \cdot 2 - 2 (y + 2) + (y - 1) (- (y - 3))}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 4.1.3

Apply the distributive property.

$\frac{y \cdot y + y \cdot 2 - 2 y - 2 \cdot 2 + (y - 1) (- (y - 3))}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 4.2

Combine the opposite terms in $y \cdot y + y \cdot 2 - 2 y - 2 \cdot 2$ .

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Step 4.2.1

Reorder the factors in the terms $y \cdot 2$ and $- 2 y$ .

$\frac{y \cdot y + 2 y - 2 y - 2 \cdot 2 + (y - 1) (- (y - 3))}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 4.2.2

Subtract $2 y$ from $2 y$ .

$\frac{y \cdot y + 0 - 2 \cdot 2 + (y - 1) (- (y - 3))}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 4.2.3

Add $y \cdot y$ and $0$ .

$\frac{y \cdot y - 2 \cdot 2 + (y - 1) (- (y - 3))}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 4.3

Simplify each term.

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Step 4.3.1

Multiply $y$ by $y$ .

$\frac{y^{2} - 2 \cdot 2 + (y - 1) (- (y - 3))}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 4.3.2

Multiply $- 2$ by $2$ .

$\frac{y^{2} - 4 + (y - 1) (- (y - 3))}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 4.4

Apply the distributive property.

$\frac{y^{2} - 4 + (y - 1) (- y - - 3)}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 4.5

Multiply $- 1$ by $- 3$ .

$\frac{y^{2} - 4 + (y - 1) (- y + 3)}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 4.6

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

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Step 4.6.1

Apply the distributive property.

$\frac{y^{2} - 4 + y (- y + 3) - 1 (- y + 3)}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 4.6.2

Apply the distributive property.

$\frac{y^{2} - 4 + y (- y) + y \cdot 3 - 1 (- y + 3)}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 4.6.3

Apply the distributive property.

$\frac{y^{2} - 4 + y (- y) + y \cdot 3 - 1 (- y) - 1 \cdot 3}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 4.7

Simplify and combine like terms.

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Step 4.7.1

Simplify each term.

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Step 4.7.1.1

Rewrite using the commutative property of multiplication.

$\frac{y^{2} - 4 - y \cdot y + y \cdot 3 - 1 (- y) - 1 \cdot 3}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 4.7.1.2

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

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Step 4.7.1.2.1

Move $y$ .

$\frac{y^{2} - 4 - (y \cdot y) + y \cdot 3 - 1 (- y) - 1 \cdot 3}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 4.7.1.2.2

Multiply $y$ by $y$ .

$\frac{y^{2} - 4 - y^{2} + y \cdot 3 - 1 (- y) - 1 \cdot 3}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 4.7.1.3

Move $3$ to the left of $y$ .

$\frac{y^{2} - 4 - y^{2} + 3 \cdot y - 1 (- y) - 1 \cdot 3}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 4.7.1.4

Multiply $- 1 (- y)$ .

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Step 4.7.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{y^{2} - 4 - y^{2} + 3 y + 1 y - 1 \cdot 3}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 4.7.1.4.2

Multiply $y$ by $1$ .

$\frac{y^{2} - 4 - y^{2} + 3 y + y - 1 \cdot 3}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 4.7.1.5

Multiply $- 1$ by $3$ .

$\frac{y^{2} - 4 - y^{2} + 3 y + y - 3}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 4.7.2

Add $3 y$ and $y$ .

$\frac{y^{2} - 4 - y^{2} + 4 y - 3}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 4.8

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

$\frac{0 - 4 + 4 y - 3}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 4.9

Subtract $4$ from $0$ .

$\frac{- 4 + 4 y - 3}{(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2}$

Step 4.10

Subtract $3$ from $- 4$ .

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

Step 5

Factor $(y - 1) (y - 2)$ out of $(y - 1) (y - 2) y + (y - 1) (y - 2) \cdot 2$ .

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Step 5.1

Factor $(y - 1) (y - 2)$ out of $(y - 1) (y - 2) y$ .

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

Step 5.2

Factor $(y - 1) (y - 2)$ out of $(y - 1) (y - 2) \cdot 2$ .

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

Step 5.3

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

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