Solve by Factoring 1/24+1/y=2/(y+4)

$\frac{1}{24} + \frac{1}{y} = \frac{2}{y + 4}$

Step 1

Subtract $\frac{2}{y + 4}$ from both sides of the equation.

$\frac{1}{24} + \frac{1}{y} - \frac{2}{y + 4} = 0$

Step 2

Simplify $\frac{1}{24} + \frac{1}{y} - \frac{2}{y + 4}$ .

To write $\frac{1}{24}$ as a fraction with a common denominator, multiply by $\frac{y + 4}{y + 4}$ .

$\frac{1}{y} + \frac{1}{24} \cdot \frac{y + 4}{y + 4} - \frac{2}{y + 4} = 0$

To write $- \frac{2}{y + 4}$ as a fraction with a common denominator, multiply by $\frac{24}{24}$ .

$\frac{1}{y} + \frac{1}{24} \cdot \frac{y + 4}{y + 4} - \frac{2}{y + 4} \cdot \frac{24}{24} = 0$

Write each expression with a common denominator of $24 (y + 4)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{24}$ by $\frac{y + 4}{y + 4}$ .

$\frac{1}{y} + \frac{y + 4}{24 (y + 4)} - \frac{2}{y + 4} \cdot \frac{24}{24} = 0$

Multiply $\frac{2}{y + 4}$ by $\frac{24}{24}$ .

$\frac{1}{y} + \frac{y + 4}{24 (y + 4)} - \frac{2 \cdot 24}{(y + 4) \cdot 24} = 0$

Reorder the factors of $(y + 4) \cdot 24$ .

$\frac{1}{y} + \frac{y + 4}{24 (y + 4)} - \frac{2 \cdot 24}{24 (y + 4)} = 0$

Combine the numerators over the common denominator.

$\frac{1}{y} + \frac{y + 4 - 2 \cdot 24}{24 (y + 4)} = 0$

Multiply $- 2$ by $24$ .

$\frac{1}{y} + \frac{y + 4 - 48}{24 (y + 4)} = 0$

Subtract $48$ from $4$ .

$\frac{1}{y} + \frac{y - 44}{24 (y + 4)} = 0$

To write $\frac{1}{y}$ as a fraction with a common denominator, multiply by $\frac{24 (y + 4)}{24 (y + 4)}$ .

$\frac{1}{y} \cdot \frac{24 (y + 4)}{24 (y + 4)} + \frac{y - 44}{24 (y + 4)} = 0$

To write $\frac{y - 44}{24 (y + 4)}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{1}{y} \cdot \frac{24 (y + 4)}{24 (y + 4)} + \frac{y - 44}{24 (y + 4)} \cdot \frac{y}{y} = 0$

Write each expression with a common denominator of $y \cdot 24 (y + 4)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{y}$ by $\frac{24 (y + 4)}{24 (y + 4)}$ .

$\frac{24 (y + 4)}{y (24 (y + 4))} + \frac{y - 44}{24 (y + 4)} \cdot \frac{y}{y} = 0$

Multiply $\frac{y - 44}{24 (y + 4)}$ by $\frac{y}{y}$ .

$\frac{24 (y + 4)}{y (24 (y + 4))} + \frac{(y - 44) y}{24 (y + 4) y} = 0$

Reorder the factors of $y (24 (y + 4))$ .

$\frac{24 (y + 4)}{24 y (y + 4)} + \frac{(y - 44) y}{24 (y + 4) y} = 0$

Reorder the factors of $24 (y + 4) y$ .

$\frac{24 (y + 4)}{24 y (y + 4)} + \frac{(y - 44) y}{24 y (y + 4)} = 0$

Combine the numerators over the common denominator.

$\frac{24 (y + 4) + (y - 44) y}{24 y (y + 4)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{24 y + 24 \cdot 4 + (y - 44) y}{24 y (y + 4)} = 0$

Multiply $24$ by $4$ .

$\frac{24 y + 96 + (y - 44) y}{24 y (y + 4)} = 0$

Apply the distributive property.

$\frac{24 y + 96 + y \cdot y - 44 y}{24 y (y + 4)} = 0$

Multiply $y$ by $y$ .

$\frac{24 y + 96 + y^{2} - 44 y}{24 y (y + 4)} = 0$

Subtract $44 y$ from $24 y$ .

$\frac{- 20 y + 96 + y^{2}}{24 y (y + 4)} = 0$

Reorder terms.

$\frac{y^{2} - 20 y + 96}{24 y (y + 4)} = 0$

Factor $y^{2} - 20 y + 96$ 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 $96$ and whose sum is $- 20$ .

$- 12, - 8$

Write the factored form using these integers.

$\frac{(y - 12) (y - 8)}{24 y (y + 4)} = 0$

Step 3

Set the numerator equal to zero.

$(y - 12) (y - 8) = 0$

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 $0$ .

$y - 12 = 0$

$y - 8 = 0$

Set $y - 12$ equal to $0$ and solve for $y$ .

Set $y - 12$ equal to $0$ .

$y - 12 = 0$

Add $12$ to both sides of the equation.

$y = 12$

Set $y - 8$ equal to $0$ and solve for $y$ .

Set $y - 8$ equal to $0$ .

$y - 8 = 0$

Add $8$ to both sides of the equation.

$y = 8$

The final solution is all the values that make $(y - 12) (y - 8) = 0$ true.

$y = 12, 8$

Solve by Factoring 1/24+1/y=2/(y+4)

Related Questions

Support LCMGCF.com by disabling your adblocker.