Basic Math Examples

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

Step 1

Factor each term.

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Step 1.1

Apply the distributive property.

$- \frac{1}{y + 4} = - 1 y - 1 \cdot 4$

Step 1.2

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

$- \frac{1}{y + 4} = - y - 1 \cdot 4$

Step 1.3

Multiply $- 1$ by $4$ .

$- \frac{1}{y + 4} = - y - 4$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y + 4, 1, 1$

Step 2.2

Remove parentheses.

$y + 4, 1, 1$

Step 2.3

The LCM of one and any expression is the expression.

$y + 4$

Step 3

Multiply each term in $- \frac{1}{y + 4} = - y - 4$ by $y + 4$ to eliminate the fractions.

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Step 3.1

Multiply each term in $- \frac{1}{y + 4} = - y - 4$ by $y + 4$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $y + 4$ .

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Step 3.2.1.1

Move the leading negative in $- \frac{1}{y + 4}$ into the numerator.

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

Step 3.2.1.2

Cancel the common factor.

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

Step 3.2.1.3

Rewrite the expression.

$- 1 = - y (y + 4) - 4 (y + 4)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Apply the distributive property.

$- 1 = - y \cdot y - y \cdot 4 - 4 (y + 4)$

Step 3.3.1.2

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

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Step 3.3.1.2.1

Move $y$ .

$- 1 = - (y \cdot y) - y \cdot 4 - 4 (y + 4)$

Step 3.3.1.2.2

Multiply $y$ by $y$ .

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

Step 3.3.1.3

Multiply $4$ by $- 1$ .

$- 1 = - y^{2} - 4 y - 4 (y + 4)$

Step 3.3.1.4

Apply the distributive property.

$- 1 = - y^{2} - 4 y - 4 y - 4 \cdot 4$

Step 3.3.1.5

Multiply $- 4$ by $4$ .

$- 1 = - y^{2} - 4 y - 4 y - 16$

Step 3.3.2

Subtract $4 y$ from $- 4 y$ .

$- 1 = - y^{2} - 8 y - 16$

Step 4

Solve the equation.

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Step 4.1

Rewrite the equation as $- y^{2} - 8 y - 16 = - 1$ .

$- y^{2} - 8 y - 16 = - 1$

Step 4.2

Add $1$ to both sides of the equation.

$- y^{2} - 8 y - 16 + 1 = 0$

Step 4.3

Add $- 16$ and $1$ .

$- y^{2} - 8 y - 15 = 0$

Step 4.4

Factor the left side of the equation.

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Step 4.4.1

Factor $- 1$ out of $- y^{2} - 8 y - 15$ .

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Step 4.4.1.1

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

$- (y^{2}) - 8 y - 15 = 0$

Step 4.4.1.2

Factor $- 1$ out of $- 8 y$ .

$- (y^{2}) - (8 y) - 15 = 0$

Step 4.4.1.3

Rewrite $- 15$ as $- 1 (15)$ .

$- (y^{2}) - (8 y) - 1 \cdot 15 = 0$

Step 4.4.1.4

Factor $- 1$ out of $- (y^{2}) - (8 y)$ .

$- (y^{2} + 8 y) - 1 \cdot 15 = 0$

Step 4.4.1.5

Factor $- 1$ out of $- (y^{2} + 8 y) - 1 (15)$ .

$- (y^{2} + 8 y + 15) = 0$

Step 4.4.2

Factor.

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Step 4.4.2.1

Factor $y^{2} + 8 y + 15$ using the AC method.

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Step 4.4.2.1.1

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 $15$ and whose sum is $8$ .

$3, 5$

Step 4.4.2.1.2

Write the factored form using these integers.

$- ((y + 3) (y + 5)) = 0$

Step 4.4.2.2

Remove unnecessary parentheses.

$- (y + 3) (y + 5) = 0$

Step 4.5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y + 3 = 0$

$y + 5 = 0$

Step 4.6

Set $y + 3$ equal to $0$ and solve for $y$ .

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Step 4.6.1

Set $y + 3$ equal to $0$ .

$y + 3 = 0$

Step 4.6.2

Subtract $3$ from both sides of the equation.

$y = - 3$

Step 4.7

Set $y + 5$ equal to $0$ and solve for $y$ .

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Step 4.7.1

Set $y + 5$ equal to $0$ .

$y + 5 = 0$

Step 4.7.2

Subtract $5$ from both sides of the equation.

$y = - 5$

Step 4.8

The final solution is all the values that make $- (y + 3) (y + 5) = 0$ true.

$y = - 3, - 5$