Solve by Factoring 14/(y+7)+2/y=(9y-7)/(y^2-49)

$\frac{14}{y + 7} + \frac{2}{y} = \frac{9 y - 7}{y^{2} - 49}$

Step 1

Subtract $\frac{9 y - 7}{y^{2} - 49}$ from both sides of the equation.

$\frac{14}{y + 7} + \frac{2}{y} - \frac{9 y - 7}{y^{2} - 49} = 0$

Step 2

Simplify $\frac{14}{y + 7} + \frac{2}{y} - \frac{9 y - 7}{y^{2} - 49}$ .

Simplify the denominator.

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

$\frac{14}{y + 7} + \frac{2}{y} - \frac{9 y - 7}{y^{2} - 7^{2}} = 0$

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 $b = 7$ .

$\frac{14}{y + 7} + \frac{2}{y} - \frac{9 y - 7}{(y + 7) (y - 7)} = 0$

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

$\frac{14}{y + 7} \cdot \frac{y}{y} + \frac{2}{y} - \frac{9 y - 7}{(y + 7) (y - 7)} = 0$

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

$\frac{14}{y + 7} \cdot \frac{y}{y} + \frac{2}{y} \cdot \frac{y + 7}{y + 7} - \frac{9 y - 7}{(y + 7) (y - 7)} = 0$

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

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

$\frac{14 y}{(y + 7) y} + \frac{2}{y} \cdot \frac{y + 7}{y + 7} - \frac{9 y - 7}{(y + 7) (y - 7)} = 0$

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

$\frac{14 y}{(y + 7) y} + \frac{2 (y + 7)}{y (y + 7)} - \frac{9 y - 7}{(y + 7) (y - 7)} = 0$

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

$\frac{14 y}{y (y + 7)} + \frac{2 (y + 7)}{y (y + 7)} - \frac{9 y - 7}{(y + 7) (y - 7)} = 0$

Combine the numerators over the common denominator.

$\frac{14 y + 2 (y + 7)}{y (y + 7)} - \frac{9 y - 7}{(y + 7) (y - 7)} = 0$

Simplify the numerator.

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

Factor $2$ out of $14 y$ .

$\frac{2 (7 y) + 2 (y + 7)}{y (y + 7)} - \frac{9 y - 7}{(y + 7) (y - 7)} = 0$

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

$\frac{2 (7 y + y + 7)}{y (y + 7)} - \frac{9 y - 7}{(y + 7) (y - 7)} = 0$

Add $7 y$ and $y$ .

$\frac{2 (8 y + 7)}{y (y + 7)} - \frac{9 y - 7}{(y + 7) (y - 7)} = 0$

To write $\frac{2 (8 y + 7)}{y (y + 7)}$ as a fraction with a common denominator, multiply by $\frac{y - 7}{y - 7}$ .

$\frac{2 (8 y + 7)}{y (y + 7)} \cdot \frac{y - 7}{y - 7} - \frac{9 y - 7}{(y + 7) (y - 7)} = 0$

To write $- \frac{9 y - 7}{(y + 7) (y - 7)}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{2 (8 y + 7)}{y (y + 7)} \cdot \frac{y - 7}{y - 7} - \frac{9 y - 7}{(y + 7) (y - 7)} \cdot \frac{y}{y} = 0$

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

Multiply $\frac{2 (8 y + 7)}{y (y + 7)}$ by $\frac{y - 7}{y - 7}$ .

$\frac{2 (8 y + 7) (y - 7)}{y (y + 7) (y - 7)} - \frac{9 y - 7}{(y + 7) (y - 7)} \cdot \frac{y}{y} = 0$

Multiply $\frac{9 y - 7}{(y + 7) (y - 7)}$ by $\frac{y}{y}$ .

$\frac{2 (8 y + 7) (y - 7)}{y (y + 7) (y - 7)} - \frac{(9 y - 7) y}{(y + 7) (y - 7) y} = 0$

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

$\frac{2 (8 y + 7) (y - 7)}{y (y + 7) (y - 7)} - \frac{(9 y - 7) y}{y (y + 7) (y - 7)} = 0$

Combine the numerators over the common denominator.

$\frac{2 (8 y + 7) (y - 7) - (9 y - 7) y}{y (y + 7) (y - 7)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{(2 (8 y) + 2 \cdot 7) (y - 7) - (9 y - 7) y}{y (y + 7) (y - 7)} = 0$

Multiply $8$ by $2$ .

$\frac{(16 y + 2 \cdot 7) (y - 7) - (9 y - 7) y}{y (y + 7) (y - 7)} = 0$

Multiply $2$ by $7$ .

$\frac{(16 y + 14) (y - 7) - (9 y - 7) y}{y (y + 7) (y - 7)} = 0$

Expand $(16 y + 14) (y - 7)$ using the FOIL Method.

Apply the distributive property.

$\frac{16 y (y - 7) + 14 (y - 7) - (9 y - 7) y}{y (y + 7) (y - 7)} = 0$

Apply the distributive property.

$\frac{16 y \cdot y + 16 y \cdot - 7 + 14 (y - 7) - (9 y - 7) y}{y (y + 7) (y - 7)} = 0$

Apply the distributive property.

$\frac{16 y \cdot y + 16 y \cdot - 7 + 14 y + 14 \cdot - 7 - (9 y - 7) y}{y (y + 7) (y - 7)} = 0$

Simplify and combine like terms.

Simplify each term.

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

Move $y$ .

$\frac{16 (y \cdot y) + 16 y \cdot - 7 + 14 y + 14 \cdot - 7 - (9 y - 7) y}{y (y + 7) (y - 7)} = 0$

Multiply $y$ by $y$ .

$\frac{16 y^{2} + 16 y \cdot - 7 + 14 y + 14 \cdot - 7 - (9 y - 7) y}{y (y + 7) (y - 7)} = 0$

Multiply $- 7$ by $16$ .

$\frac{16 y^{2} - 112 y + 14 y + 14 \cdot - 7 - (9 y - 7) y}{y (y + 7) (y - 7)} = 0$

Multiply $14$ by $- 7$ .

$\frac{16 y^{2} - 112 y + 14 y - 98 - (9 y - 7) y}{y (y + 7) (y - 7)} = 0$

Add $- 112 y$ and $14 y$ .

$\frac{16 y^{2} - 98 y - 98 - (9 y - 7) y}{y (y + 7) (y - 7)} = 0$

Apply the distributive property.

$\frac{16 y^{2} - 98 y - 98 + (- (9 y) - - 7) y}{y (y + 7) (y - 7)} = 0$

Multiply $9$ by $- 1$ .

$\frac{16 y^{2} - 98 y - 98 + (- 9 y - - 7) y}{y (y + 7) (y - 7)} = 0$

Multiply $- 1$ by $- 7$ .

$\frac{16 y^{2} - 98 y - 98 + (- 9 y + 7) y}{y (y + 7) (y - 7)} = 0$

Apply the distributive property.

$\frac{16 y^{2} - 98 y - 98 - 9 y \cdot y + 7 y}{y (y + 7) (y - 7)} = 0$

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

Move $y$ .

$\frac{16 y^{2} - 98 y - 98 - 9 (y \cdot y) + 7 y}{y (y + 7) (y - 7)} = 0$

Multiply $y$ by $y$ .

$\frac{16 y^{2} - 98 y - 98 - 9 y^{2} + 7 y}{y (y + 7) (y - 7)} = 0$

Subtract $9 y^{2}$ from $16 y^{2}$ .

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

Add $- 98 y$ and $7 y$ .

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

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

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

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

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

Factor $7$ out of $- 91 y$ .

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

Factor $7$ out of $- 98$ .

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

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

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

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

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

Factor $y^{2} - 13 y - 14$ using the AC method.

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 $- 14$ and whose sum is $- 13$ .

$- 14, 1$

Write the factored form using these integers.

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

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

Step 3

Set the numerator equal to zero.

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

Step 4

Solve the equation for $y$ .

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

$y - 14 = 0$

$y + 1 = 0$

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

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

$y - 14 = 0$

Add $14$ to both sides of the equation.

$y = 14$

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

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

$y + 1 = 0$

Subtract $1$ from both sides of the equation.

$y = - 1$

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

$y = 14, - 1$

Solve by Factoring 14/(y+7)+2/y=(9y-7)/(y^2-49)

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