Solve by Factoring 1/18+1/y=2/(y+3)

$\frac{1}{18} + \frac{1}{y} = \frac{2}{y + 3}$

Step 1

Subtract $\frac{2}{y + 3}$ from both sides of the equation.

$\frac{1}{18} + \frac{1}{y} - \frac{2}{y + 3} = 0$

Step 2

Simplify $\frac{1}{18} + \frac{1}{y} - \frac{2}{y + 3}$ .

To write $\frac{1}{18}$ as a fraction with a common denominator, multiply by $\frac{y + 3}{y + 3}$ .

$\frac{1}{y} + \frac{1}{18} \cdot \frac{y + 3}{y + 3} - \frac{2}{y + 3} = 0$

To write $- \frac{2}{y + 3}$ as a fraction with a common denominator, multiply by $\frac{18}{18}$ .

$\frac{1}{y} + \frac{1}{18} \cdot \frac{y + 3}{y + 3} - \frac{2}{y + 3} \cdot \frac{18}{18} = 0$

Write each expression with a common denominator of $18 (y + 3)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{18}$ by $\frac{y + 3}{y + 3}$ .

$\frac{1}{y} + \frac{y + 3}{18 (y + 3)} - \frac{2}{y + 3} \cdot \frac{18}{18} = 0$

Multiply $\frac{2}{y + 3}$ by $\frac{18}{18}$ .

$\frac{1}{y} + \frac{y + 3}{18 (y + 3)} - \frac{2 \cdot 18}{(y + 3) \cdot 18} = 0$

Reorder the factors of $(y + 3) \cdot 18$ .

$\frac{1}{y} + \frac{y + 3}{18 (y + 3)} - \frac{2 \cdot 18}{18 (y + 3)} = 0$

Combine the numerators over the common denominator.

$\frac{1}{y} + \frac{y + 3 - 2 \cdot 18}{18 (y + 3)} = 0$

Multiply $- 2$ by $18$ .

$\frac{1}{y} + \frac{y + 3 - 36}{18 (y + 3)} = 0$

Subtract $36$ from $3$ .

$\frac{1}{y} + \frac{y - 33}{18 (y + 3)} = 0$

To write $\frac{1}{y}$ as a fraction with a common denominator, multiply by $\frac{18 (y + 3)}{18 (y + 3)}$ .

$\frac{1}{y} \cdot \frac{18 (y + 3)}{18 (y + 3)} + \frac{y - 33}{18 (y + 3)} = 0$

To write $\frac{y - 33}{18 (y + 3)}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{1}{y} \cdot \frac{18 (y + 3)}{18 (y + 3)} + \frac{y - 33}{18 (y + 3)} \cdot \frac{y}{y} = 0$

Write each expression with a common denominator of $y \cdot 18 (y + 3)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{y}$ by $\frac{18 (y + 3)}{18 (y + 3)}$ .

$\frac{18 (y + 3)}{y (18 (y + 3))} + \frac{y - 33}{18 (y + 3)} \cdot \frac{y}{y} = 0$

Multiply $\frac{y - 33}{18 (y + 3)}$ by $\frac{y}{y}$ .

$\frac{18 (y + 3)}{y (18 (y + 3))} + \frac{(y - 33) y}{18 (y + 3) y} = 0$

Reorder the factors of $y (18 (y + 3))$ .

$\frac{18 (y + 3)}{18 y (y + 3)} + \frac{(y - 33) y}{18 (y + 3) y} = 0$

Reorder the factors of $18 (y + 3) y$ .

$\frac{18 (y + 3)}{18 y (y + 3)} + \frac{(y - 33) y}{18 y (y + 3)} = 0$

Combine the numerators over the common denominator.

$\frac{18 (y + 3) + (y - 33) y}{18 y (y + 3)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{18 y + 18 \cdot 3 + (y - 33) y}{18 y (y + 3)} = 0$

Multiply $18$ by $3$ .

$\frac{18 y + 54 + (y - 33) y}{18 y (y + 3)} = 0$

Apply the distributive property.

$\frac{18 y + 54 + y \cdot y - 33 y}{18 y (y + 3)} = 0$

Multiply $y$ by $y$ .

$\frac{18 y + 54 + y^{2} - 33 y}{18 y (y + 3)} = 0$

Subtract $33 y$ from $18 y$ .

$\frac{- 15 y + 54 + y^{2}}{18 y (y + 3)} = 0$

Reorder terms.

$\frac{y^{2} - 15 y + 54}{18 y (y + 3)} = 0$

Factor $y^{2} - 15 y + 54$ 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 $54$ and whose sum is $- 15$ .

$- 9, - 6$

Write the factored form using these integers.

$\frac{(y - 9) (y - 6)}{18 y (y + 3)} = 0$

Step 3

Set the numerator equal to zero.

$(y - 9) (y - 6) = 0$

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 $0$ .

$y - 9 = 0$

$y - 6 = 0$

Set $y - 9$ equal to $0$ and solve for $y$ .

Set $y - 9$ equal to $0$ .

$y - 9 = 0$

Add $9$ to both sides of the equation.

$y = 9$

Set $y - 6$ equal to $0$ and solve for $y$ .

Set $y - 6$ equal to $0$ .

$y - 6 = 0$

Add $6$ to both sides of the equation.

$y = 6$

The final solution is all the values that make $(y - 9) (y - 6) = 0$ true.

$y = 9, 6$

Solve by Factoring 1/18+1/y=2/(y+3)

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