Algebra Examples

$y + \frac{y^{2} - 5}{y^{2} - 1} = \frac{y^{2} + y + 2}{y + 1}$

Step 1

Factor each term.

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Step 1.1

Rewrite $1$ as $1^{2}$ .

$y + \frac{y^{2} - 5}{y^{2} - 1^{2}} = \frac{y^{2} + y + 2}{y + 1}$

Step 1.2

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 $b = 1$ .

$y + \frac{y^{2} - 5}{(y + 1) (y - 1)} = \frac{y^{2} + y + 2}{y + 1}$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

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

$1, (y + 1) (y - 1), y + 1$

Step 2.2

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 2.3

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

Not prime

Step 2.4

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 2.5

The factor for $y + 1$ is $y + 1$ itself.

$(y + 1) = y + 1$

$(y + 1)$ occurs $1$ time.

Step 2.6

The factor for $y - 1$ is $y - 1$ itself.

$(y - 1) = y - 1$

$(y - 1)$ occurs $1$ time.

Step 2.7

The factor for $y + 1$ is $y + 1$ itself.

$(y + 1) = y + 1$

$(y + 1)$ occurs $1$ time.

Step 2.8

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

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

Step 3

Multiply each term in $y + \frac{y^{2} - 5}{(y + 1) (y - 1)} = \frac{y^{2} + y + 2}{y + 1}$ by $(y + 1) (y - 1)$ to eliminate the fractions.

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Step 3.1

Multiply each term in $y + \frac{y^{2} - 5}{(y + 1) (y - 1)} = \frac{y^{2} + y + 2}{y + 1}$ by $(y + 1) (y - 1)$ .

$y ((y + 1) (y - 1)) + \frac{y^{2} - 5}{(y + 1) (y - 1)} ((y + 1) (y - 1)) = \frac{y^{2} + y + 2}{y + 1} ((y + 1) (y - 1))$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Expand $(y + 1) (y - 1)$ using the FOIL Method.

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Step 3.2.1.1.1

Apply the distributive property.

$y (y (y - 1) + 1 (y - 1)) + \frac{y^{2} - 5}{(y + 1) (y - 1)} ((y + 1) (y - 1)) = \frac{y^{2} + y + 2}{y + 1} ((y + 1) (y - 1))$

Step 3.2.1.1.2

Apply the distributive property.

$y (y \cdot y + y \cdot - 1 + 1 (y - 1)) + \frac{y^{2} - 5}{(y + 1) (y - 1)} ((y + 1) (y - 1)) = \frac{y^{2} + y + 2}{y + 1} ((y + 1) (y - 1))$

Step 3.2.1.1.3

Apply the distributive property.

$y (y \cdot y + y \cdot - 1 + 1 y + 1 \cdot - 1) + \frac{y^{2} - 5}{(y + 1) (y - 1)} ((y + 1) (y - 1)) = \frac{y^{2} + y + 2}{y + 1} ((y + 1) (y - 1))$

Step 3.2.1.2

Simplify and combine like terms.

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Step 3.2.1.2.1

Simplify each term.

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Step 3.2.1.2.1.1

Multiply $y$ by $y$ .

$y (y^{2} + y \cdot - 1 + 1 y + 1 \cdot - 1) + \frac{y^{2} - 5}{(y + 1) (y - 1)} ((y + 1) (y - 1)) = \frac{y^{2} + y + 2}{y + 1} ((y + 1) (y - 1))$

Step 3.2.1.2.1.2

Move $- 1$ to the left of $y$ .

$y (y^{2} - 1 \cdot y + 1 y + 1 \cdot - 1) + \frac{y^{2} - 5}{(y + 1) (y - 1)} ((y + 1) (y - 1)) = \frac{y^{2} + y + 2}{y + 1} ((y + 1) (y - 1))$

Step 3.2.1.2.1.3

Rewrite $- 1 y$ as $- y$ .

$y (y^{2} - y + 1 y + 1 \cdot - 1) + \frac{y^{2} - 5}{(y + 1) (y - 1)} ((y + 1) (y - 1)) = \frac{y^{2} + y + 2}{y + 1} ((y + 1) (y - 1))$

Step 3.2.1.2.1.4

Multiply $y$ by $1$ .

$y (y^{2} - y + y + 1 \cdot - 1) + \frac{y^{2} - 5}{(y + 1) (y - 1)} ((y + 1) (y - 1)) = \frac{y^{2} + y + 2}{y + 1} ((y + 1) (y - 1))$

Step 3.2.1.2.1.5

Multiply $- 1$ by $1$ .

$y (y^{2} - y + y - 1) + \frac{y^{2} - 5}{(y + 1) (y - 1)} ((y + 1) (y - 1)) = \frac{y^{2} + y + 2}{y + 1} ((y + 1) (y - 1))$

Step 3.2.1.2.2

Add $- y$ and $y$ .

$y (y^{2} + 0 - 1) + \frac{y^{2} - 5}{(y + 1) (y - 1)} ((y + 1) (y - 1)) = \frac{y^{2} + y + 2}{y + 1} ((y + 1) (y - 1))$

Step 3.2.1.2.3

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

$y (y^{2} - 1) + \frac{y^{2} - 5}{(y + 1) (y - 1)} ((y + 1) (y - 1)) = \frac{y^{2} + y + 2}{y + 1} ((y + 1) (y - 1))$

Step 3.2.1.3

Apply the distributive property.

$y \cdot y^{2} + y \cdot - 1 + \frac{y^{2} - 5}{(y + 1) (y - 1)} ((y + 1) (y - 1)) = \frac{y^{2} + y + 2}{y + 1} ((y + 1) (y - 1))$

Step 3.2.1.4

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 3.2.1.4.1

Multiply $y$ by $y^{2}$ .

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Step 3.2.1.4.1.1

Raise $y$ to the power of $1$ .

$y^{1} y^{2} + y \cdot - 1 + \frac{y^{2} - 5}{(y + 1) (y - 1)} ((y + 1) (y - 1)) = \frac{y^{2} + y + 2}{y + 1} ((y + 1) (y - 1))$

Step 3.2.1.4.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y^{1 + 2} + y \cdot - 1 + \frac{y^{2} - 5}{(y + 1) (y - 1)} ((y + 1) (y - 1)) = \frac{y^{2} + y + 2}{y + 1} ((y + 1) (y - 1))$

Step 3.2.1.4.2

Add $1$ and $2$ .

$y^{3} + y \cdot - 1 + \frac{y^{2} - 5}{(y + 1) (y - 1)} ((y + 1) (y - 1)) = \frac{y^{2} + y + 2}{y + 1} ((y + 1) (y - 1))$

Step 3.2.1.5

Move $- 1$ to the left of $y$ .

$y^{3} - 1 \cdot y + \frac{y^{2} - 5}{(y + 1) (y - 1)} ((y + 1) (y - 1)) = \frac{y^{2} + y + 2}{y + 1} ((y + 1) (y - 1))$

Step 3.2.1.6

Rewrite $- 1 y$ as $- y$ .

$y^{3} - y + \frac{y^{2} - 5}{(y + 1) (y - 1)} ((y + 1) (y - 1)) = \frac{y^{2} + y + 2}{y + 1} ((y + 1) (y - 1))$

Step 3.2.1.7

Cancel the common factor of $(y + 1) (y - 1)$ .

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Step 3.2.1.7.1

Cancel the common factor.

$y^{3} - y + \frac{y^{2} - 5}{(y + 1) (y - 1)} ((y + 1) (y - 1)) = \frac{y^{2} + y + 2}{y + 1} ((y + 1) (y - 1))$

Step 3.2.1.7.2

Rewrite the expression.

$y^{3} - y + y^{2} - 5 = \frac{y^{2} + y + 2}{y + 1} ((y + 1) (y - 1))$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $y + 1$ .

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Step 3.3.1.1

Cancel the common factor.

$y^{3} - y + y^{2} - 5 = \frac{y^{2} + y + 2}{y + 1} ((y + 1) (y - 1))$

Step 3.3.1.2

Rewrite the expression.

$y^{3} - y + y^{2} - 5 = (y^{2} + y + 2) (y - 1)$

Step 3.3.2

Expand $(y^{2} + y + 2) (y - 1)$ by multiplying each term in the first expression by each term in the second expression.

$y^{3} - y + y^{2} - 5 = y^{2} y + y^{2} \cdot - 1 + y \cdot y + y \cdot - 1 + 2 y + 2 \cdot - 1$

Step 3.3.3

Simplify terms.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 3.3.3.1.1.1

Multiply $y^{2}$ by $y$ .

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Step 3.3.3.1.1.1.1

Raise $y$ to the power of $1$ .

$y^{3} - y + y^{2} - 5 = y^{2} y^{1} + y^{2} \cdot - 1 + y \cdot y + y \cdot - 1 + 2 y + 2 \cdot - 1$

Step 3.3.3.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y^{3} - y + y^{2} - 5 = y^{2 + 1} + y^{2} \cdot - 1 + y \cdot y + y \cdot - 1 + 2 y + 2 \cdot - 1$

Step 3.3.3.1.1.2

Add $2$ and $1$ .

$y^{3} - y + y^{2} - 5 = y^{3} + y^{2} \cdot - 1 + y \cdot y + y \cdot - 1 + 2 y + 2 \cdot - 1$

Step 3.3.3.1.2

Move $- 1$ to the left of $y^{2}$ .

$y^{3} - y + y^{2} - 5 = y^{3} - 1 \cdot y^{2} + y \cdot y + y \cdot - 1 + 2 y + 2 \cdot - 1$

Step 3.3.3.1.3

Rewrite $- 1 y^{2}$ as $- y^{2}$ .

$y^{3} - y + y^{2} - 5 = y^{3} - y^{2} + y \cdot y + y \cdot - 1 + 2 y + 2 \cdot - 1$

Step 3.3.3.1.4

Multiply $y$ by $y$ .

$y^{3} - y + y^{2} - 5 = y^{3} - y^{2} + y^{2} + y \cdot - 1 + 2 y + 2 \cdot - 1$

Step 3.3.3.1.5

Move $- 1$ to the left of $y$ .

$y^{3} - y + y^{2} - 5 = y^{3} - y^{2} + y^{2} - 1 \cdot y + 2 y + 2 \cdot - 1$

Step 3.3.3.1.6

Rewrite $- 1 y$ as $- y$ .

$y^{3} - y + y^{2} - 5 = y^{3} - y^{2} + y^{2} - y + 2 y + 2 \cdot - 1$

Step 3.3.3.1.7

Multiply $2$ by $- 1$ .

$y^{3} - y + y^{2} - 5 = y^{3} - y^{2} + y^{2} - y + 2 y - 2$

Step 3.3.3.2

Simplify by adding terms.

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Step 3.3.3.2.1

Combine the opposite terms in $y^{3} - y^{2} + y^{2} - y + 2 y - 2$ .

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Step 3.3.3.2.1.1

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

$y^{3} - y + y^{2} - 5 = y^{3} + 0 - y + 2 y - 2$

Step 3.3.3.2.1.2

Add $y^{3}$ and $0$ .

$y^{3} - y + y^{2} - 5 = y^{3} - y + 2 y - 2$

Step 3.3.3.2.2

Add $- y$ and $2 y$ .

$y^{3} - y + y^{2} - 5 = y^{3} + y - 2$

Step 4

Solve the equation.

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Step 4.1

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

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Step 4.1.1

Subtract $y^{3}$ from both sides of the equation.

$y^{3} - y + y^{2} - 5 - y^{3} = y - 2$

Step 4.1.2

Subtract $y$ from both sides of the equation.

$y^{3} - y + y^{2} - 5 - y^{3} - y = - 2$

Step 4.1.3

Combine the opposite terms in $y^{3} - y + y^{2} - 5 - y^{3} - y$ .

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Step 4.1.3.1

Subtract $y^{3}$ from $y^{3}$ .

$- y + y^{2} - 5 + 0 - y = - 2$

Step 4.1.3.2

Add $- y + y^{2} - 5$ and $0$ .

$- y + y^{2} - 5 - y = - 2$

Step 4.1.4

Subtract $y$ from $- y$ .

$- 2 y + y^{2} - 5 = - 2$

Step 4.2

Add $2$ to both sides of the equation.

$- 2 y + y^{2} - 5 + 2 = 0$

Step 4.3

Add $- 5$ and $2$ .

$- 2 y + y^{2} - 3 = 0$

Step 4.4

Factor the left side of the equation.

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Step 4.4.1

Let $u = y$ . Substitute $u$ for all occurrences of $y$ .

$- 2 u + u^{2} - 3 = 0$

Step 4.4.2

Factor $- 2 u + u^{2} - 3$ using the AC method.

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Step 4.4.2.1

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 4.4.2.2

Write the factored form using these integers.

$(u - 3) (u + 1) = 0$

Step 4.4.3

Replace all occurrences of $u$ with $y$ .

$(y - 3) (y + 1) = 0$

Step 4.5

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

$y - 3 = 0$

$y + 1 = 0$

Step 4.6

Set $y - 3$ equal to $0$ and solve for $y$ .

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Step 4.6.1

Set $y - 3$ equal to $0$ .

$y - 3 = 0$

Step 4.6.2

Add $3$ to both sides of the equation.

$y = 3$

Step 4.7

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

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Step 4.7.1

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

$y + 1 = 0$

Step 4.7.2

Subtract $1$ from both sides of the equation.

$y = - 1$

Step 4.8

The final solution is all the values that make $(y - 3) (y + 1) = 0$ true.

$y = 3, - 1$

Step 5

Exclude the solutions that do not make $y + \frac{y^{2} - 5}{y^{2} - 1} = \frac{y^{2} + y + 2}{y + 1}$ true.

$y = 3$