Algebra Examples

$\frac{1}{y} - \frac{1}{y^{2}} = - \frac{y - 1}{y}$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y, y^{2}, y$

Step 1.2

Since $y, y^{2}, y$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 1$ then find LCM for the variable part $y^{1}, y^{2}, y^{1}$ .

Step 1.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 1.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 1.5

The LCM of $1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 1.6

The factor for $y^{1}$ is $y$ itself.

$y^{1} = y$

$y$ occurs $1$ time.

Step 1.7

The factors for $y^{2}$ are $y \cdot y$ , which is $y$ multiplied by each other $2$ times.

$y^{2} = y \cdot y$

$y$ occurs $2$ times.

Step 1.8

The factor for $y^{1}$ is $y$ itself.

$y^{1} = y$

$y$ occurs $1$ time.

Step 1.9

The LCM of $y^{1}, y^{2}, y^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$y \cdot y$

Step 1.10

Multiply $y$ by $y$ .

$y^{2}$

Step 2

Multiply each term in $\frac{1}{y} - \frac{1}{y^{2}} = - \frac{y - 1}{y}$ by $y^{2}$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{1}{y} - \frac{1}{y^{2}} = - \frac{y - 1}{y}$ by $y^{2}$ .

$\frac{1}{y} y^{2} - \frac{1}{y^{2}} y^{2} = - \frac{y - 1}{y} y^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Cancel the common factor of $y$ .

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Step 2.2.1.1.1

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

$\frac{1}{y} (y \cdot y) - \frac{1}{y^{2}} y^{2} = - \frac{y - 1}{y} y^{2}$

Step 2.2.1.1.2

Cancel the common factor.

$\frac{1}{y} (y \cdot y) - \frac{1}{y^{2}} y^{2} = - \frac{y - 1}{y} y^{2}$

Step 2.2.1.1.3

Rewrite the expression.

$y - \frac{1}{y^{2}} y^{2} = - \frac{y - 1}{y} y^{2}$

Step 2.2.1.2

Cancel the common factor of $y^{2}$ .

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Step 2.2.1.2.1

Move the leading negative in $- \frac{1}{y^{2}}$ into the numerator.

$y + \frac{- 1}{y^{2}} y^{2} = - \frac{y - 1}{y} y^{2}$

Step 2.2.1.2.2

Cancel the common factor.

$y + \frac{- 1}{y^{2}} y^{2} = - \frac{y - 1}{y} y^{2}$

Step 2.2.1.2.3

Rewrite the expression.

$y - 1 = - \frac{y - 1}{y} y^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $y$ .

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Step 2.3.1.1

Move the leading negative in $- \frac{y - 1}{y}$ into the numerator.

$y - 1 = \frac{- (y - 1)}{y} y^{2}$

Step 2.3.1.2

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

$y - 1 = \frac{- (y - 1)}{y} (y \cdot y)$

Step 2.3.1.3

Cancel the common factor.

$y - 1 = \frac{- (y - 1)}{y} (y \cdot y)$

Step 2.3.1.4

Rewrite the expression.

$y - 1 = - (y - 1) y$

Step 2.3.2

Apply the distributive property.

$y - 1 = (- y - - 1) y$

Step 2.3.3

Multiply $- 1$ by $- 1$ .

$y - 1 = (- y + 1) y$

Step 2.3.4

Apply the distributive property.

$y - 1 = - y \cdot y + 1 y$

Step 2.3.5

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

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Step 2.3.5.1

Move $y$ .

$y - 1 = - (y \cdot y) + 1 y$

Step 2.3.5.2

Multiply $y$ by $y$ .

$y - 1 = - y^{2} + 1 y$

Step 2.3.6

Multiply $y$ by $1$ .

$y - 1 = - y^{2} + y$

Step 3

Solve the equation.

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Step 3.1

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- y^{2} + y = y - 1$

Step 3.2

Move all terms containing $y$ to the left side of the equation.

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Step 3.2.1

Subtract $y$ from both sides of the equation.

$- y^{2} + y - y = - 1$

Step 3.2.2

Combine the opposite terms in $- y^{2} + y - y$ .

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Step 3.2.2.1

Subtract $y$ from $y$ .

$- y^{2} + 0 = - 1$

Step 3.2.2.2

Add $- y^{2}$ and $0$ .

$- y^{2} = - 1$

Step 3.3

Divide each term in $- y^{2} = - 1$ by $- 1$ and simplify.

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Step 3.3.1

Divide each term in $- y^{2} = - 1$ by $- 1$ .

$\frac{- y^{2}}{- 1} = \frac{- 1}{- 1}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Dividing two negative values results in a positive value.

$\frac{y^{2}}{1} = \frac{- 1}{- 1}$

Step 3.3.2.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{- 1}{- 1}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $- 1$ by $- 1$ .

$y^{2} = 1$

Step 3.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{1}$

Step 3.5

Any root of $1$ is $1$ .

$y = \pm 1$

Step 3.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.6.1

First, use the positive value of the $\pm$ to find the first solution.

$y = 1$

Step 3.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - 1$

Step 3.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = 1, - 1$