Algebra Examples

$\frac{1}{10} + \frac{1}{y} = \frac{3}{y + 5}$

Step 1

Subtract $\frac{3}{y + 5}$ from both sides of the equation.

$\frac{1}{10} + \frac{1}{y} - \frac{3}{y + 5} = 0$

Step 2

Simplify $\frac{1}{10} + \frac{1}{y} - \frac{3}{y + 5}$ .

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Step 2.1

To write $\frac{1}{10}$ as a fraction with a common denominator, multiply by $\frac{y + 5}{y + 5}$ .

$\frac{1}{y} + \frac{1}{10} \cdot \frac{y + 5}{y + 5} - \frac{3}{y + 5} = 0$

Step 2.2

To write $- \frac{3}{y + 5}$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$\frac{1}{y} + \frac{1}{10} \cdot \frac{y + 5}{y + 5} - \frac{3}{y + 5} \cdot \frac{10}{10} = 0$

Step 2.3

Write each expression with a common denominator of $10 (y + 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{1}{10}$ by $\frac{y + 5}{y + 5}$ .

$\frac{1}{y} + \frac{y + 5}{10 (y + 5)} - \frac{3}{y + 5} \cdot \frac{10}{10} = 0$

Step 2.3.2

Multiply $\frac{3}{y + 5}$ by $\frac{10}{10}$ .

$\frac{1}{y} + \frac{y + 5}{10 (y + 5)} - \frac{3 \cdot 10}{(y + 5) \cdot 10} = 0$

Step 2.3.3

Reorder the factors of $(y + 5) \cdot 10$ .

$\frac{1}{y} + \frac{y + 5}{10 (y + 5)} - \frac{3 \cdot 10}{10 (y + 5)} = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{1}{y} + \frac{y + 5 - 3 \cdot 10}{10 (y + 5)} = 0$

Step 2.5

Multiply $- 3$ by $10$ .

$\frac{1}{y} + \frac{y + 5 - 30}{10 (y + 5)} = 0$

Step 2.6

Subtract $30$ from $5$ .

$\frac{1}{y} + \frac{y - 25}{10 (y + 5)} = 0$

Step 2.7

To write $\frac{1}{y}$ as a fraction with a common denominator, multiply by $\frac{10 (y + 5)}{10 (y + 5)}$ .

$\frac{1}{y} \cdot \frac{10 (y + 5)}{10 (y + 5)} + \frac{y - 25}{10 (y + 5)} = 0$

Step 2.8

To write $\frac{y - 25}{10 (y + 5)}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{1}{y} \cdot \frac{10 (y + 5)}{10 (y + 5)} + \frac{y - 25}{10 (y + 5)} \cdot \frac{y}{y} = 0$

Step 2.9

Write each expression with a common denominator of $y \cdot 10 (y + 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.9.1

Multiply $\frac{1}{y}$ by $\frac{10 (y + 5)}{10 (y + 5)}$ .

$\frac{10 (y + 5)}{y (10 (y + 5))} + \frac{y - 25}{10 (y + 5)} \cdot \frac{y}{y} = 0$

Step 2.9.2

Multiply $\frac{y - 25}{10 (y + 5)}$ by $\frac{y}{y}$ .

$\frac{10 (y + 5)}{y (10 (y + 5))} + \frac{(y - 25) y}{10 (y + 5) y} = 0$

Step 2.9.3

Reorder the factors of $y (10 (y + 5))$ .

$\frac{10 (y + 5)}{10 y (y + 5)} + \frac{(y - 25) y}{10 (y + 5) y} = 0$

Step 2.9.4

Reorder the factors of $10 (y + 5) y$ .

$\frac{10 (y + 5)}{10 y (y + 5)} + \frac{(y - 25) y}{10 y (y + 5)} = 0$

Step 2.10

Combine the numerators over the common denominator.

$\frac{10 (y + 5) + (y - 25) y}{10 y (y + 5)} = 0$

Step 2.11

Simplify the numerator.

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Step 2.11.1

Apply the distributive property.

$\frac{10 y + 10 \cdot 5 + (y - 25) y}{10 y (y + 5)} = 0$

Step 2.11.2

Multiply $10$ by $5$ .

$\frac{10 y + 50 + (y - 25) y}{10 y (y + 5)} = 0$

Step 2.11.3

Apply the distributive property.

$\frac{10 y + 50 + y \cdot y - 25 y}{10 y (y + 5)} = 0$

Step 2.11.4

Multiply $y$ by $y$ .

$\frac{10 y + 50 + y^{2} - 25 y}{10 y (y + 5)} = 0$

Step 2.11.5

Subtract $25 y$ from $10 y$ .

$\frac{- 15 y + 50 + y^{2}}{10 y (y + 5)} = 0$

Step 2.11.6

Reorder terms.

$\frac{y^{2} - 15 y + 50}{10 y (y + 5)} = 0$

Step 2.11.7

Factor $y^{2} - 15 y + 50$ using the AC method.

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Step 2.11.7.1

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 $50$ and whose sum is $- 15$ .

$- 10, - 5$

Step 2.11.7.2

Write the factored form using these integers.

$\frac{(y - 10) (y - 5)}{10 y (y + 5)} = 0$

Step 3

Set the numerator equal to zero.

$(y - 10) (y - 5) = 0$

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 $0$ .

$y - 10 = 0$

$y - 5 = 0$

Step 4.2

Set $y - 10$ equal to $0$ and solve for $y$ .

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Step 4.2.1

Set $y - 10$ equal to $0$ .

$y - 10 = 0$

Step 4.2.2

Add $10$ to both sides of the equation.

$y = 10$

Step 4.3

Set $y - 5$ equal to $0$ and solve for $y$ .

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Step 4.3.1

Set $y - 5$ equal to $0$ .

$y - 5 = 0$

Step 4.3.2

Add $5$ to both sides of the equation.

$y = 5$

Step 4.4

The final solution is all the values that make $(y - 10) (y - 5) = 0$ true.

$y = 10, 5$