Basic Math Examples

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

Step 1

Factor each term.

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Step 1.1

Move the negative in front of the fraction.

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

Step 1.2

Apply the distributive property.

$- \frac{7}{y - 2} - y - 1 \cdot 2 = 4$

Step 1.3

Multiply $- 1$ by $2$ .

$- \frac{7}{y - 2} - y - 2 = 4$

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.

$y - 2, 1, 1, 1$

Step 2.2

Remove parentheses.

$y - 2, 1, 1, 1$

Step 2.3

The LCM of one and any expression is the expression.

$y - 2$

Step 3

Multiply each term in $- \frac{7}{y - 2} - y - 2 = 4$ by $y - 2$ to eliminate the fractions.

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Step 3.1

Multiply each term in $- \frac{7}{y - 2} - y - 2 = 4$ by $y - 2$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Cancel the common factor of $y - 2$ .

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Step 3.2.1.1.1

Move the leading negative in $- \frac{7}{y - 2}$ into the numerator.

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

Step 3.2.1.1.2

Cancel the common factor.

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

Step 3.2.1.1.3

Rewrite the expression.

$- 7 - y (y - 2) - 2 (y - 2) = 4 (y - 2)$

Step 3.2.1.2

Apply the distributive property.

$- 7 - y \cdot y - y \cdot - 2 - 2 (y - 2) = 4 (y - 2)$

Step 3.2.1.3

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

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Step 3.2.1.3.1

Move $y$ .

$- 7 - (y \cdot y) - y \cdot - 2 - 2 (y - 2) = 4 (y - 2)$

Step 3.2.1.3.2

Multiply $y$ by $y$ .

$- 7 - y^{2} - y \cdot - 2 - 2 (y - 2) = 4 (y - 2)$

Step 3.2.1.4

Multiply $- 2$ by $- 1$ .

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

Step 3.2.1.5

Apply the distributive property.

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

Step 3.2.1.6

Multiply $- 2$ by $- 2$ .

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

Step 3.2.2

Simplify by adding terms.

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Step 3.2.2.1

Combine the opposite terms in $- 7 - y^{2} + 2 y - 2 y + 4$ .

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Step 3.2.2.1.1

Subtract $2 y$ from $2 y$ .

$- 7 - y^{2} + 0 + 4 = 4 (y - 2)$

Step 3.2.2.1.2

Add $- 7 - y^{2}$ and $0$ .

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

Step 3.2.2.2

Add $- 7$ and $4$ .

$- y^{2} - 3 = 4 (y - 2)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Apply the distributive property.

$- y^{2} - 3 = 4 y + 4 \cdot - 2$

Step 3.3.2

Multiply $4$ by $- 2$ .

$- y^{2} - 3 = 4 y - 8$

Step 4

Solve the equation.

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Step 4.1

Subtract $4 y$ from both sides of the equation.

$- y^{2} - 3 - 4 y = - 8$

Step 4.2

Add $8$ to both sides of the equation.

$- y^{2} - 3 - 4 y + 8 = 0$

Step 4.3

Add $- 3$ and $8$ .

$- y^{2} - 4 y + 5 = 0$

Step 4.4

Factor the left side of the equation.

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Step 4.4.1

Factor $- 1$ out of $- y^{2} - 4 y + 5$ .

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Step 4.4.1.1

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

$- (y^{2}) - 4 y + 5 = 0$

Step 4.4.1.2

Factor $- 1$ out of $- 4 y$ .

$- (y^{2}) - (4 y) + 5 = 0$

Step 4.4.1.3

Rewrite $5$ as $- 1 (- 5)$ .

$- (y^{2}) - (4 y) - 1 \cdot - 5 = 0$

Step 4.4.1.4

Factor $- 1$ out of $- (y^{2}) - (4 y)$ .

$- (y^{2} + 4 y) - 1 \cdot - 5 = 0$

Step 4.4.1.5

Factor $- 1$ out of $- (y^{2} + 4 y) - 1 (- 5)$ .

$- (y^{2} + 4 y - 5) = 0$

Step 4.4.2

Factor.

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Step 4.4.2.1

Factor $y^{2} + 4 y - 5$ 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 $- 5$ and whose sum is $4$ .

$- 1, 5$

Step 4.4.2.1.2

Write the factored form using these integers.

$- ((y - 1) (y + 5)) = 0$

Step 4.4.2.2

Remove unnecessary parentheses.

$- (y - 1) (y + 5) = 0$

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 $0$ .

$y - 1 = 0$

$y + 5 = 0$

Step 4.6

Set $y - 1$ equal to $0$ and solve for $y$ .

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Step 4.6.1

Set $y - 1$ equal to $0$ .

$y - 1 = 0$

Step 4.6.2

Add $1$ to both sides of the equation.

$y = 1$

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 $- (y - 1) (y + 5) = 0$ true.

$y = 1, - 5$