Algebra Examples

$\frac{14}{y - 7} - \frac{2}{y} = \frac{19 y + 7}{y^{2} - 49}$

Step 1

Subtract $\frac{19 y + 7}{y^{2} - 49}$ from both sides of the equation.

$\frac{14}{y - 7} - \frac{2}{y} - \frac{19 y + 7}{y^{2} - 49} = 0$

Step 2

Simplify $\frac{14}{y - 7} - \frac{2}{y} - \frac{19 y + 7}{y^{2} - 49}$ .

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Step 2.1

Simplify the denominator.

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Step 2.1.1

Rewrite $49$ as $7^{2}$ .

$\frac{14}{y - 7} - \frac{2}{y} - \frac{19 y + 7}{y^{2} - 7^{2}} = 0$

Step 2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 7$ .

$\frac{14}{y - 7} - \frac{2}{y} - \frac{19 y + 7}{(y + 7) (y - 7)} = 0$

Step 2.2

To write $\frac{14}{y - 7}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{14}{y - 7} \cdot \frac{y}{y} - \frac{2}{y} - \frac{19 y + 7}{(y + 7) (y - 7)} = 0$

Step 2.3

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

$\frac{14}{y - 7} \cdot \frac{y}{y} - \frac{2}{y} \cdot \frac{y - 7}{y - 7} - \frac{19 y + 7}{(y + 7) (y - 7)} = 0$

Step 2.4

Write each expression with a common denominator of $(y - 7) y$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{14}{y - 7}$ by $\frac{y}{y}$ .

$\frac{14 y}{(y - 7) y} - \frac{2}{y} \cdot \frac{y - 7}{y - 7} - \frac{19 y + 7}{(y + 7) (y - 7)} = 0$

Step 2.4.2

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

$\frac{14 y}{(y - 7) y} - \frac{2 (y - 7)}{y (y - 7)} - \frac{19 y + 7}{(y + 7) (y - 7)} = 0$

Step 2.4.3

Reorder the factors of $(y - 7) y$ .

$\frac{14 y}{y (y - 7)} - \frac{2 (y - 7)}{y (y - 7)} - \frac{19 y + 7}{(y + 7) (y - 7)} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{14 y - 2 (y - 7)}{y (y - 7)} - \frac{19 y + 7}{(y + 7) (y - 7)} = 0$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Factor $2$ out of $14 y - 2 (y - 7)$ .

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Step 2.6.1.1

Factor $2$ out of $14 y$ .

$\frac{2 (7 y) - 2 (y - 7)}{y (y - 7)} - \frac{19 y + 7}{(y + 7) (y - 7)} = 0$

Step 2.6.1.2

Factor $2$ out of $- 2 (y - 7)$ .

$\frac{2 (7 y) + 2 (- (y - 7))}{y (y - 7)} - \frac{19 y + 7}{(y + 7) (y - 7)} = 0$

Step 2.6.1.3

Factor $2$ out of $2 (7 y) + 2 (- (y - 7))$ .

$\frac{2 (7 y - (y - 7))}{y (y - 7)} - \frac{19 y + 7}{(y + 7) (y - 7)} = 0$

Step 2.6.2

Apply the distributive property.

$\frac{2 (7 y - y - - 7)}{y (y - 7)} - \frac{19 y + 7}{(y + 7) (y - 7)} = 0$

Step 2.6.3

Multiply $- 1$ by $- 7$ .

$\frac{2 (7 y - y + 7)}{y (y - 7)} - \frac{19 y + 7}{(y + 7) (y - 7)} = 0$

Step 2.6.4

Subtract $y$ from $7 y$ .

$\frac{2 (6 y + 7)}{y (y - 7)} - \frac{19 y + 7}{(y + 7) (y - 7)} = 0$

Step 2.7

To write $\frac{2 (6 y + 7)}{y (y - 7)}$ as a fraction with a common denominator, multiply by $\frac{y + 7}{y + 7}$ .

$\frac{2 (6 y + 7)}{y (y - 7)} \cdot \frac{y + 7}{y + 7} - \frac{19 y + 7}{(y + 7) (y - 7)} = 0$

Step 2.8

To write $- \frac{19 y + 7}{(y + 7) (y - 7)}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{2 (6 y + 7)}{y (y - 7)} \cdot \frac{y + 7}{y + 7} - \frac{19 y + 7}{(y + 7) (y - 7)} \cdot \frac{y}{y} = 0$

Step 2.9

Write each expression with a common denominator of $y (y - 7) (y + 7)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.9.1

Multiply $\frac{2 (6 y + 7)}{y (y - 7)}$ by $\frac{y + 7}{y + 7}$ .

$\frac{2 (6 y + 7) (y + 7)}{y (y - 7) (y + 7)} - \frac{19 y + 7}{(y + 7) (y - 7)} \cdot \frac{y}{y} = 0$

Step 2.9.2

Multiply $\frac{19 y + 7}{(y + 7) (y - 7)}$ by $\frac{y}{y}$ .

$\frac{2 (6 y + 7) (y + 7)}{y (y - 7) (y + 7)} - \frac{(19 y + 7) y}{(y + 7) (y - 7) y} = 0$

Step 2.9.3

Reorder the factors of $y (y - 7) (y + 7)$ .

$\frac{2 (6 y + 7) (y + 7)}{y (y + 7) (y - 7)} - \frac{(19 y + 7) y}{(y + 7) (y - 7) y} = 0$

Step 2.9.4

Reorder the factors of $(y + 7) (y - 7) y$ .

$\frac{2 (6 y + 7) (y + 7)}{y (y + 7) (y - 7)} - \frac{(19 y + 7) y}{y (y + 7) (y - 7)} = 0$

Step 2.10

Combine the numerators over the common denominator.

$\frac{2 (6 y + 7) (y + 7) - (19 y + 7) y}{y (y + 7) (y - 7)} = 0$

Step 2.11

Simplify the numerator.

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Step 2.11.1

Apply the distributive property.

$\frac{(2 (6 y) + 2 \cdot 7) (y + 7) - (19 y + 7) y}{y (y + 7) (y - 7)} = 0$

Step 2.11.2

Multiply $6$ by $2$ .

$\frac{(12 y + 2 \cdot 7) (y + 7) - (19 y + 7) y}{y (y + 7) (y - 7)} = 0$

Step 2.11.3

Multiply $2$ by $7$ .

$\frac{(12 y + 14) (y + 7) - (19 y + 7) y}{y (y + 7) (y - 7)} = 0$

Step 2.11.4

Expand $(12 y + 14) (y + 7)$ using the FOIL Method.

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Step 2.11.4.1

Apply the distributive property.

$\frac{12 y (y + 7) + 14 (y + 7) - (19 y + 7) y}{y (y + 7) (y - 7)} = 0$

Step 2.11.4.2

Apply the distributive property.

$\frac{12 y \cdot y + 12 y \cdot 7 + 14 (y + 7) - (19 y + 7) y}{y (y + 7) (y - 7)} = 0$

Step 2.11.4.3

Apply the distributive property.

$\frac{12 y \cdot y + 12 y \cdot 7 + 14 y + 14 \cdot 7 - (19 y + 7) y}{y (y + 7) (y - 7)} = 0$

Step 2.11.5

Simplify and combine like terms.

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Step 2.11.5.1

Simplify each term.

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Step 2.11.5.1.1

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

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Step 2.11.5.1.1.1

Move $y$ .

$\frac{12 (y \cdot y) + 12 y \cdot 7 + 14 y + 14 \cdot 7 - (19 y + 7) y}{y (y + 7) (y - 7)} = 0$

Step 2.11.5.1.1.2

Multiply $y$ by $y$ .

$\frac{12 y^{2} + 12 y \cdot 7 + 14 y + 14 \cdot 7 - (19 y + 7) y}{y (y + 7) (y - 7)} = 0$

Step 2.11.5.1.2

Multiply $7$ by $12$ .

$\frac{12 y^{2} + 84 y + 14 y + 14 \cdot 7 - (19 y + 7) y}{y (y + 7) (y - 7)} = 0$

Step 2.11.5.1.3

Multiply $14$ by $7$ .

$\frac{12 y^{2} + 84 y + 14 y + 98 - (19 y + 7) y}{y (y + 7) (y - 7)} = 0$

Step 2.11.5.2

Add $84 y$ and $14 y$ .

$\frac{12 y^{2} + 98 y + 98 - (19 y + 7) y}{y (y + 7) (y - 7)} = 0$

Step 2.11.6

Apply the distributive property.

$\frac{12 y^{2} + 98 y + 98 + (- (19 y) - 1 \cdot 7) y}{y (y + 7) (y - 7)} = 0$

Step 2.11.7

Multiply $19$ by $- 1$ .

$\frac{12 y^{2} + 98 y + 98 + (- 19 y - 1 \cdot 7) y}{y (y + 7) (y - 7)} = 0$

Step 2.11.8

Multiply $- 1$ by $7$ .

$\frac{12 y^{2} + 98 y + 98 + (- 19 y - 7) y}{y (y + 7) (y - 7)} = 0$

Step 2.11.9

Apply the distributive property.

$\frac{12 y^{2} + 98 y + 98 - 19 y \cdot y - 7 y}{y (y + 7) (y - 7)} = 0$

Step 2.11.10

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

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Step 2.11.10.1

Move $y$ .

$\frac{12 y^{2} + 98 y + 98 - 19 (y \cdot y) - 7 y}{y (y + 7) (y - 7)} = 0$

Step 2.11.10.2

Multiply $y$ by $y$ .

$\frac{12 y^{2} + 98 y + 98 - 19 y^{2} - 7 y}{y (y + 7) (y - 7)} = 0$

Step 2.11.11

Subtract $19 y^{2}$ from $12 y^{2}$ .

$\frac{- 7 y^{2} + 98 y + 98 - 7 y}{y (y + 7) (y - 7)} = 0$

Step 2.11.12

Subtract $7 y$ from $98 y$ .

$\frac{- 7 y^{2} + 91 y + 98}{y (y + 7) (y - 7)} = 0$

Step 2.11.13

Rewrite $- 7 y^{2} + 91 y + 98$ in a factored form.

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Step 2.11.13.1

Factor $7$ out of $- 7 y^{2} + 91 y + 98$ .

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Step 2.11.13.1.1

Factor $7$ out of $- 7 y^{2}$ .

$\frac{7 (- y^{2}) + 91 y + 98}{y (y + 7) (y - 7)} = 0$

Step 2.11.13.1.2

Factor $7$ out of $91 y$ .

$\frac{7 (- y^{2}) + 7 (13 y) + 98}{y (y + 7) (y - 7)} = 0$

Step 2.11.13.1.3

Factor $7$ out of $98$ .

$\frac{7 (- y^{2}) + 7 (13 y) + 7 (14)}{y (y + 7) (y - 7)} = 0$

Step 2.11.13.1.4

Factor $7$ out of $7 (- y^{2}) + 7 (13 y)$ .

$\frac{7 (- y^{2} + 13 y) + 7 (14)}{y (y + 7) (y - 7)} = 0$

Step 2.11.13.1.5

Factor $7$ out of $7 (- y^{2} + 13 y) + 7 (14)$ .

$\frac{7 (- y^{2} + 13 y + 14)}{y (y + 7) (y - 7)} = 0$

Step 2.11.13.2

Factor by grouping.

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Step 2.11.13.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 14 = - 14$ and whose sum is $b = 13$ .

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Step 2.11.13.2.1.1

Factor $13$ out of $13 y$ .

$\frac{7 (- y^{2} + 13 (y) + 14)}{y (y + 7) (y - 7)} = 0$

Step 2.11.13.2.1.2

Rewrite $13$ as $- 1$ plus $14$

$\frac{7 (- y^{2} + (- 1 + 14) y + 14)}{y (y + 7) (y - 7)} = 0$

Step 2.11.13.2.1.3

Apply the distributive property.

$\frac{7 (- y^{2} - 1 y + 14 y + 14)}{y (y + 7) (y - 7)} = 0$

Step 2.11.13.2.2

Factor out the greatest common factor from each group.

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Step 2.11.13.2.2.1

Group the first two terms and the last two terms.

$\frac{7 ((- y^{2} - 1 y) + 14 y + 14)}{y (y + 7) (y - 7)} = 0$

Step 2.11.13.2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{7 (y (- y - 1) - 14 (- y - 1))}{y (y + 7) (y - 7)} = 0$

Step 2.11.13.2.3

Factor the polynomial by factoring out the greatest common factor, $- y - 1$ .

$\frac{7 ((- y - 1) (y - 14))}{y (y + 7) (y - 7)} = 0$

$\frac{7 (- y - 1) (y - 14)}{y (y + 7) (y - 7)} = 0$

Step 2.12

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

$\frac{7 (- (y) - 1) (y - 14)}{y (y + 7) (y - 7)} = 0$

Step 2.13

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

$\frac{7 (- (y) - 1 (1)) (y - 14)}{y (y + 7) (y - 7)} = 0$

Step 2.14

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

$\frac{7 (- (y + 1)) (y - 14)}{y (y + 7) (y - 7)} = 0$

Step 2.15

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

$\frac{7 (- 1 (y + 1)) (y - 14)}{y (y + 7) (y - 7)} = 0$

Step 2.16

Move the negative in front of the fraction.

$- \frac{(7 (y + 1)) (y - 14)}{y (y + 7) (y - 7)} = 0$

Step 2.17

Reorder factors in $- \frac{(7 (y + 1)) (y - 14)}{y (y + 7) (y - 7)}$ .

$- \frac{7 (y + 1) (y - 14)}{y (y + 7) (y - 7)} = 0$

Step 3

Set the numerator equal to zero.

$7 (y + 1) (y - 14) = 0$

Step 4

Solve the equation for $y$ .

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Step 4.1

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

$y + 1 = 0$

$y - 14 = 0$

Step 4.2

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

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Step 4.2.1

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

$y + 1 = 0$

Step 4.2.2

Subtract $1$ from both sides of the equation.

$y = - 1$

Step 4.3

Set $y - 14$ equal to $0$ and solve for $y$ .

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Step 4.3.1

Set $y - 14$ equal to $0$ .

$y - 14 = 0$

Step 4.3.2

Add $14$ to both sides of the equation.

$y = 14$

Step 4.4

The final solution is all the values that make $7 (y + 1) (y - 14) = 0$ true.

$y = - 1, 14$