Pre-Algebra Examples

$\frac{y - 5}{y - 2} = \frac{1}{y^{2} + 9 y - 22}$

Step 1

Factor $y^{2} + 9 y - 22$ using the AC method.

Tap for more steps...

Step 1.1

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 $- 22$ and whose sum is $9$ .

$- 2, 11$

Step 1.2

Write the factored form using these integers.

$\frac{y - 5}{y - 2} = \frac{1}{(y - 2) (y + 11)}$

Step 2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.1

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

$y - 2, (y - 2) (y + 11)$

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 $1, 1$ 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 $y - 2$ is $y - 2$ itself.

$(y - 2) = y - 2$

$(y - 2)$ occurs $1$ time.

Step 2.6

The factor for $y + 11$ is $y + 11$ itself.

$(y + 11) = y + 11$

$(y + 11)$ occurs $1$ time.

Step 2.7

The LCM of $y - 2, y - 2, y + 11$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(y - 2) (y + 11)$

Step 3

Multiply each term in $\frac{y - 5}{y - 2} = \frac{1}{(y - 2) (y + 11)}$ by $(y - 2) (y + 11)$ to eliminate the fractions.

Tap for more steps...

Step 3.1

Multiply each term in $\frac{y - 5}{y - 2} = \frac{1}{(y - 2) (y + 11)}$ by $(y - 2) (y + 11)$ .

$\frac{y - 5}{y - 2} ((y - 2) (y + 11)) = \frac{1}{(y - 2) (y + 11)} ((y - 2) (y + 11))$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Cancel the common factor of $y - 2$ .

Tap for more steps...

Step 3.2.1.1

Cancel the common factor.

$\frac{y - 5}{y - 2} ((y - 2) (y + 11)) = \frac{1}{(y - 2) (y + 11)} ((y - 2) (y + 11))$

Step 3.2.1.2

Rewrite the expression.

$(y - 5) (y + 11) = \frac{1}{(y - 2) (y + 11)} ((y - 2) (y + 11))$

Step 3.2.2

Expand $(y - 5) (y + 11)$ using the FOIL Method.

Tap for more steps...

Step 3.2.2.1

Apply the distributive property.

$y (y + 11) - 5 (y + 11) = \frac{1}{(y - 2) (y + 11)} ((y - 2) (y + 11))$

Step 3.2.2.2

Apply the distributive property.

$y \cdot y + y \cdot 11 - 5 (y + 11) = \frac{1}{(y - 2) (y + 11)} ((y - 2) (y + 11))$

Step 3.2.2.3

Apply the distributive property.

$y \cdot y + y \cdot 11 - 5 y - 5 \cdot 11 = \frac{1}{(y - 2) (y + 11)} ((y - 2) (y + 11))$

Step 3.2.3

Simplify and combine like terms.

Tap for more steps...

Step 3.2.3.1

Simplify each term.

Tap for more steps...

Step 3.2.3.1.1

Multiply $y$ by $y$ .

$y^{2} + y \cdot 11 - 5 y - 5 \cdot 11 = \frac{1}{(y - 2) (y + 11)} ((y - 2) (y + 11))$

Step 3.2.3.1.2

Move $11$ to the left of $y$ .

$y^{2} + 11 \cdot y - 5 y - 5 \cdot 11 = \frac{1}{(y - 2) (y + 11)} ((y - 2) (y + 11))$

Step 3.2.3.1.3

Multiply $- 5$ by $11$ .

$y^{2} + 11 y - 5 y - 55 = \frac{1}{(y - 2) (y + 11)} ((y - 2) (y + 11))$

Step 3.2.3.2

Subtract $5 y$ from $11 y$ .

$y^{2} + 6 y - 55 = \frac{1}{(y - 2) (y + 11)} ((y - 2) (y + 11))$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Cancel the common factor of $(y - 2) (y + 11)$ .

Tap for more steps...

Step 3.3.1.1

Cancel the common factor.

$y^{2} + 6 y - 55 = \frac{1}{(y - 2) (y + 11)} ((y - 2) (y + 11))$

Step 3.3.1.2

Rewrite the expression.

$y^{2} + 6 y - 55 = 1$

Step 4

Solve the equation.

Tap for more steps...

Step 4.1

Move all terms to the left side of the equation and simplify.

Tap for more steps...

Step 4.1.1

Subtract $1$ from both sides of the equation.

$y^{2} + 6 y - 55 - 1 = 0$

Step 4.1.2

Subtract $1$ from $- 55$ .

$y^{2} + 6 y - 56 = 0$

Step 4.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.3

Substitute the values $a = 1$ , $b = 6$ , and $c = - 56$ into the quadratic formula and solve for $y$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (1 \cdot - 56)}}{2 \cdot 1}$

Step 4.4

Simplify.

Tap for more steps...

Step 4.4.1

Simplify the numerator.

Tap for more steps...

Step 4.4.1.1

Raise $6$ to the power of $2$ .

$y = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 56}}{2 \cdot 1}$

Step 4.4.1.2

Multiply $- 4 \cdot 1 \cdot - 56$ .

Tap for more steps...

Step 4.4.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 56}}{2 \cdot 1}$

Step 4.4.1.2.2

Multiply $- 4$ by $- 56$ .

$y = \frac{- 6 \pm \sqrt{36 + 224}}{2 \cdot 1}$

Step 4.4.1.3

Add $36$ and $224$ .

$y = \frac{- 6 \pm \sqrt{260}}{2 \cdot 1}$

Step 4.4.1.4

Rewrite $260$ as $2^{2} \cdot 65$ .

Tap for more steps...

Step 4.4.1.4.1

Factor $4$ out of $260$ .

$y = \frac{- 6 \pm \sqrt{4 (65)}}{2 \cdot 1}$

Step 4.4.1.4.2

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 6 \pm \sqrt{2^{2} \cdot 65}}{2 \cdot 1}$

Step 4.4.1.5

Pull terms out from under the radical.

$y = \frac{- 6 \pm 2 \sqrt{65}}{2 \cdot 1}$

Step 4.4.2

Multiply $2$ by $1$ .

$y = \frac{- 6 \pm 2 \sqrt{65}}{2}$

Step 4.4.3

Simplify $\frac{- 6 \pm 2 \sqrt{65}}{2}$ .

$y = - 3 \pm \sqrt{65}$

Step 4.5

The final answer is the combination of both solutions.

$y = - 3 + \sqrt{65}, - 3 - \sqrt{65}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$y = - 3 + \sqrt{65}, - 3 - \sqrt{65}$

Decimal Form:

$y = 5.06225774 \dots, - 11.06225774 \dots$