Algebra Examples

$\frac{y + 3}{y + 1} \div \frac{y^{2} + 4 y + 3}{- 5 - 5 y}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{y + 3}{y + 1} \cdot \frac{- 5 - 5 y}{y^{2} + 4 y + 3}$

Step 2

Factor $5$ out of $- 5 - 5 y$ .

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Step 2.1

Factor $5$ out of $- 5$ .

$\frac{y + 3}{y + 1} \cdot \frac{5 (- 1) - 5 y}{y^{2} + 4 y + 3}$

Step 2.2

Factor $5$ out of $- 5 y$ .

$\frac{y + 3}{y + 1} \cdot \frac{5 (- 1) + 5 (- y)}{y^{2} + 4 y + 3}$

Step 2.3

Factor $5$ out of $5 (- 1) + 5 (- y)$ .

$\frac{y + 3}{y + 1} \cdot \frac{5 (- 1 - y)}{y^{2} + 4 y + 3}$

Step 3

Factor $y^{2} + 4 y + 3$ using the AC method.

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Step 3.1

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 $3$ and whose sum is $4$ .

$1, 3$

Step 3.2

Write the factored form using these integers.

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $y + 3$ .

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Step 4.1.1

Factor $y + 3$ out of $(y + 1) (y + 3)$ .

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

$\frac{1}{y + 1} \cdot \frac{5 (- 1 - y)}{y + 1}$

Step 4.2

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

$\frac{5 (- 1 - y)}{(y + 1) (y + 1)}$

Step 5

Raise $y + 1$ to the power of $1$ .

$\frac{5 (- 1 - y)}{{(y + 1)}^{1} (y + 1)}$

Step 6

Raise $y + 1$ to the power of $1$ .

$\frac{5 (- 1 - y)}{{(y + 1)}^{1} {(y + 1)}^{1}}$

Step 7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{5 (- 1 - y)}{{(y + 1)}^{1 + 1}}$

Step 8

Reduce the expression by cancelling the common factors.

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Step 8.1

Add $1$ and $1$ .

$\frac{5 (- 1 - y)}{{(y + 1)}^{2}}$

Step 8.2

Cancel the common factor of $- 1 - y$ and ${(y + 1)}^{2}$ .

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Step 8.2.1

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

$\frac{5 (- 1 (1) - y)}{{(y + 1)}^{2}}$

Step 8.2.2

Factor $- 1$ out of $- y$ .

$\frac{5 (- 1 (1) - (y))}{{(y + 1)}^{2}}$

Step 8.2.3

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

$\frac{5 (- 1 (1 + y))}{{(y + 1)}^{2}}$

Step 8.2.4

Reorder terms.

$\frac{5 (- 1 (y + 1))}{{(y + 1)}^{2}}$

Step 8.2.5

Factor $y + 1$ out of $5 (- 1 (y + 1))$ .

$\frac{(y + 1) (5 \cdot (- 1))}{{(y + 1)}^{2}}$

Step 8.2.6

Cancel the common factors.

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Step 8.2.6.1

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

$\frac{(y + 1) (5 \cdot (- 1))}{(y + 1) (y + 1)}$

Step 8.2.6.2

Cancel the common factor.

$\frac{(y + 1) (5 \cdot (- 1))}{(y + 1) (y + 1)}$

Step 8.2.6.3

Rewrite the expression.

$\frac{5 \cdot (- 1)}{y + 1}$

Step 9

Multiply $5$ by $- 1$ .

$\frac{- 5}{y + 1}$

Step 10

Move the negative in front of the fraction.

$- \frac{5}{y + 1}$