Precalculus Examples

$\frac{- 7}{y + 8} + \frac{6}{7 - y} = \frac{(- 8) y - 9}{y^{2} + y - 56}$

Step 1

Factor each term.

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Step 1.1

Move the negative in front of the fraction.

$- \frac{7}{y + 8} + \frac{6}{7 - y} = \frac{- 8 y - 9}{y^{2} + y - 56}$

Step 1.2

Factor $y^{2} + y - 56$ using the AC method.

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Step 1.2.1

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 $- 56$ and whose sum is $1$ .

$- 7, 8$

Step 1.2.2

Write the factored form using these integers.

$- \frac{7}{y + 8} + \frac{6}{7 - y} = \frac{- 8 y - 9}{(y - 7) (y + 8)}$

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.

$y + 8, 7 - y, (y - 7) (y + 8)$

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 $y + 8$ is $y + 8$ itself.

$(y + 8) = y + 8$

$(y + 8)$ occurs $1$ time.

Step 2.6

The factor for $7 - y$ is $7 - y$ itself.

$(7 - y) = 7 - y$

$(7 - y)$ occurs $1$ time.

Step 2.7

The factor for $y - 7$ is $y - 7$ itself.

$(y - 7) = y - 7$

$(y - 7)$ occurs $1$ time.

Step 2.8

The factor for $y + 8$ is $y + 8$ itself.

$(y + 8) = y + 8$

$(y + 8)$ occurs $1$ time.

Step 2.9

The LCM of $y + 8, 7 - y, y - 7, y + 8$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(y + 8) (7 - y) (y - 7)$

Step 3

Multiply each term in $- \frac{7}{y + 8} + \frac{6}{7 - y} = \frac{- 8 y - 9}{(y - 7) (y + 8)}$ by $(y + 8) (7 - y) (y - 7)$ to eliminate the fractions.

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Step 3.1

Multiply each term in $- \frac{7}{y + 8} + \frac{6}{7 - y} = \frac{- 8 y - 9}{(y - 7) (y + 8)}$ by $(y + 8) (7 - y) (y - 7)$ .

$- \frac{7}{y + 8} ((y + 8) (7 - y) (y - 7)) + \frac{6}{7 - y} ((y + 8) (7 - y) (y - 7)) = \frac{- 8 y - 9}{(y - 7) (y + 8)} ((y + 8) (7 - y) (y - 7))$

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

Cancel the common factor of $y + 8$ .

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Step 3.2.1.1.1

Move the leading negative in $- \frac{7}{y + 8}$ into the numerator.

$\frac{- 7}{y + 8} ((y + 8) (7 - y) (y - 7)) + \frac{6}{7 - y} ((y + 8) (7 - y) (y - 7)) = \frac{- 8 y - 9}{(y - 7) (y + 8)} ((y + 8) (7 - y) (y - 7))$

Step 3.2.1.1.2

Factor $y + 8$ out of $(y + 8) (7 - y) (y - 7)$ .

$\frac{- 7}{y + 8} ((y + 8) ((7 - y) (y - 7))) + \frac{6}{7 - y} ((y + 8) (7 - y) (y - 7)) = \frac{- 8 y - 9}{(y - 7) (y + 8)} ((y + 8) (7 - y) (y - 7))$

Step 3.2.1.1.3

Cancel the common factor.

$\frac{- 7}{y + 8} ((y + 8) ((7 - y) (y - 7))) + \frac{6}{7 - y} ((y + 8) (7 - y) (y - 7)) = \frac{- 8 y - 9}{(y - 7) (y + 8)} ((y + 8) (7 - y) (y - 7))$

Step 3.2.1.1.4

Rewrite the expression.

$- 7 ((7 - y) (y - 7)) + \frac{6}{7 - y} ((y + 8) (7 - y) (y - 7)) = \frac{- 8 y - 9}{(y - 7) (y + 8)} ((y + 8) (7 - y) (y - 7))$

Step 3.2.1.2

Rewrite $7$ as $- 1 (- 7)$ .

$- 7 ((- 1 (- 7) - y) (y - 7)) + \frac{6}{7 - y} ((y + 8) (7 - y) (y - 7)) = \frac{- 8 y - 9}{(y - 7) (y + 8)} ((y + 8) (7 - y) (y - 7))$

Step 3.2.1.3

Factor $- 1$ out of $- y$ .

$- 7 ((- 1 (- 7) - (y)) (y - 7)) + \frac{6}{7 - y} ((y + 8) (7 - y) (y - 7)) = \frac{- 8 y - 9}{(y - 7) (y + 8)} ((y + 8) (7 - y) (y - 7))$

Step 3.2.1.4

Factor $- 1$ out of $- 1 (- 7) - (y)$ .

$- 7 (- 1 (- 7 + y) (y - 7)) + \frac{6}{7 - y} ((y + 8) (7 - y) (y - 7)) = \frac{- 8 y - 9}{(y - 7) (y + 8)} ((y + 8) (7 - y) (y - 7))$

Step 3.2.1.5

Reorder terms.

$- 7 (- 1 (y - 7) (y - 7)) + \frac{6}{7 - y} ((y + 8) (7 - y) (y - 7)) = \frac{- 8 y - 9}{(y - 7) (y + 8)} ((y + 8) (7 - y) (y - 7))$

Step 3.2.1.6

Raise $y - 7$ to the power of $1$ .

$- 7 (- 1 ({(y - 7)}^{1} (y - 7))) + \frac{6}{7 - y} ((y + 8) (7 - y) (y - 7)) = \frac{- 8 y - 9}{(y - 7) (y + 8)} ((y + 8) (7 - y) (y - 7))$

Step 3.2.1.7

Raise $y - 7$ to the power of $1$ .

$- 7 (- 1 ({(y - 7)}^{1} {(y - 7)}^{1})) + \frac{6}{7 - y} ((y + 8) (7 - y) (y - 7)) = \frac{- 8 y - 9}{(y - 7) (y + 8)} ((y + 8) (7 - y) (y - 7))$

Step 3.2.1.8

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

$- 7 (- 1 {(y - 7)}^{1 + 1}) + \frac{6}{7 - y} ((y + 8) (7 - y) (y - 7)) = \frac{- 8 y - 9}{(y - 7) (y + 8)} ((y + 8) (7 - y) (y - 7))$

Step 3.2.1.9

Add $1$ and $1$ .

$- 7 (- 1 {(y - 7)}^{2}) + \frac{6}{7 - y} ((y + 8) (7 - y) (y - 7)) = \frac{- 8 y - 9}{(y - 7) (y + 8)} ((y + 8) (7 - y) (y - 7))$

Step 3.2.1.10

Multiply $- 1$ by $- 7$ .

$7 {(y - 7)}^{2} + \frac{6}{7 - y} ((y + 8) (7 - y) (y - 7)) = \frac{- 8 y - 9}{(y - 7) (y + 8)} ((y + 8) (7 - y) (y - 7))$

Step 3.2.1.11

Cancel the common factor of $7 - y$ .

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Step 3.2.1.11.1

Factor $7 - y$ out of $(y + 8) (7 - y) (y - 7)$ .

$7 {(y - 7)}^{2} + \frac{6}{7 - y} ((7 - y) ((y + 8) (y - 7))) = \frac{- 8 y - 9}{(y - 7) (y + 8)} ((y + 8) (7 - y) (y - 7))$

Step 3.2.1.11.2

Cancel the common factor.

$7 {(y - 7)}^{2} + \frac{6}{7 - y} ((7 - y) ((y + 8) (y - 7))) = \frac{- 8 y - 9}{(y - 7) (y + 8)} ((y + 8) (7 - y) (y - 7))$

Step 3.2.1.11.3

Rewrite the expression.

$7 {(y - 7)}^{2} + 6 ((y + 8) (y - 7)) = \frac{- 8 y - 9}{(y - 7) (y + 8)} ((y + 8) (7 - y) (y - 7))$

Step 3.2.1.12

Expand $(y + 8) (y - 7)$ using the FOIL Method.

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Step 3.2.1.12.1

Apply the distributive property.

$7 {(y - 7)}^{2} + 6 (y (y - 7) + 8 (y - 7)) = \frac{- 8 y - 9}{(y - 7) (y + 8)} ((y + 8) (7 - y) (y - 7))$

Step 3.2.1.12.2

Apply the distributive property.

$7 {(y - 7)}^{2} + 6 (y \cdot y + y \cdot - 7 + 8 (y - 7)) = \frac{- 8 y - 9}{(y - 7) (y + 8)} ((y + 8) (7 - y) (y - 7))$

Step 3.2.1.12.3

Apply the distributive property.

$7 {(y - 7)}^{2} + 6 (y \cdot y + y \cdot - 7 + 8 y + 8 \cdot - 7) = \frac{- 8 y - 9}{(y - 7) (y + 8)} ((y + 8) (7 - y) (y - 7))$

Step 3.2.1.13

Simplify and combine like terms.

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Step 3.2.1.13.1

Simplify each term.

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Step 3.2.1.13.1.1

Multiply $y$ by $y$ .

$7 {(y - 7)}^{2} + 6 (y^{2} + y \cdot - 7 + 8 y + 8 \cdot - 7) = \frac{- 8 y - 9}{(y - 7) (y + 8)} ((y + 8) (7 - y) (y - 7))$

Step 3.2.1.13.1.2

Move $- 7$ to the left of $y$ .

$7 {(y - 7)}^{2} + 6 (y^{2} - 7 \cdot y + 8 y + 8 \cdot - 7) = \frac{- 8 y - 9}{(y - 7) (y + 8)} ((y + 8) (7 - y) (y - 7))$

Step 3.2.1.13.1.3

Multiply $8$ by $- 7$ .

$7 {(y - 7)}^{2} + 6 (y^{2} - 7 y + 8 y - 56) = \frac{- 8 y - 9}{(y - 7) (y + 8)} ((y + 8) (7 - y) (y - 7))$

Step 3.2.1.13.2

Add $- 7 y$ and $8 y$ .

$7 {(y - 7)}^{2} + 6 (y^{2} + y - 56) = \frac{- 8 y - 9}{(y - 7) (y + 8)} ((y + 8) (7 - y) (y - 7))$

Step 3.2.1.14

Apply the distributive property.

$7 {(y - 7)}^{2} + 6 y^{2} + 6 y + 6 \cdot - 56 = \frac{- 8 y - 9}{(y - 7) (y + 8)} ((y + 8) (7 - y) (y - 7))$

Step 3.2.1.15

Multiply $6$ by $- 56$ .

$7 {(y - 7)}^{2} + 6 y^{2} + 6 y - 336 = \frac{- 8 y - 9}{(y - 7) (y + 8)} ((y + 8) (7 - y) (y - 7))$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $(y + 8) (y - 7)$ .

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Step 3.3.1.1

Factor $(y + 8) (y - 7)$ out of $(y - 7) (y + 8)$ .

$7 {(y - 7)}^{2} + 6 y^{2} + 6 y - 336 = \frac{- 8 y - 9}{(y + 8) (y - 7) (1)} ((y + 8) (7 - y) (y - 7))$

Step 3.3.1.2

Factor $(y + 8) (y - 7)$ out of $(y + 8) (7 - y) (y - 7)$ .

$7 {(y - 7)}^{2} + 6 y^{2} + 6 y - 336 = \frac{- 8 y - 9}{(y + 8) (y - 7) (1)} ((y + 8) (y - 7) (7 - y))$

Step 3.3.1.3

Cancel the common factor.

$7 {(y - 7)}^{2} + 6 y^{2} + 6 y - 336 = \frac{- 8 y - 9}{(y + 8) (y - 7) \cdot 1} ((y + 8) (y - 7) (7 - y))$

Step 3.3.1.4

Rewrite the expression.

$7 {(y - 7)}^{2} + 6 y^{2} + 6 y - 336 = (- 8 y - 9) (7 - y)$

Step 3.3.2

Expand $(- 8 y - 9) (7 - y)$ using the FOIL Method.

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Step 3.3.2.1

Apply the distributive property.

$7 {(y - 7)}^{2} + 6 y^{2} + 6 y - 336 = - 8 y (7 - y) - 9 (7 - y)$

Step 3.3.2.2

Apply the distributive property.

$7 {(y - 7)}^{2} + 6 y^{2} + 6 y - 336 = - 8 y \cdot 7 - 8 y (- y) - 9 (7 - y)$

Step 3.3.2.3

Apply the distributive property.

$7 {(y - 7)}^{2} + 6 y^{2} + 6 y - 336 = - 8 y \cdot 7 - 8 y (- y) - 9 \cdot 7 - 9 (- y)$

Step 3.3.3

Simplify and combine like terms.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Multiply $7$ by $- 8$ .

$7 {(y - 7)}^{2} + 6 y^{2} + 6 y - 336 = - 56 y - 8 y (- y) - 9 \cdot 7 - 9 (- y)$

Step 3.3.3.1.2

Rewrite using the commutative property of multiplication.

$7 {(y - 7)}^{2} + 6 y^{2} + 6 y - 336 = - 56 y - 8 \cdot - 1 y \cdot y - 9 \cdot 7 - 9 (- y)$

Step 3.3.3.1.3

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

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Step 3.3.3.1.3.1

Move $y$ .

$7 {(y - 7)}^{2} + 6 y^{2} + 6 y - 336 = - 56 y - 8 \cdot - 1 (y \cdot y) - 9 \cdot 7 - 9 (- y)$

Step 3.3.3.1.3.2

Multiply $y$ by $y$ .

$7 {(y - 7)}^{2} + 6 y^{2} + 6 y - 336 = - 56 y - 8 \cdot - 1 y^{2} - 9 \cdot 7 - 9 (- y)$

Step 3.3.3.1.4

Multiply $- 8$ by $- 1$ .

$7 {(y - 7)}^{2} + 6 y^{2} + 6 y - 336 = - 56 y + 8 y^{2} - 9 \cdot 7 - 9 (- y)$

Step 3.3.3.1.5

Multiply $- 9$ by $7$ .

$7 {(y - 7)}^{2} + 6 y^{2} + 6 y - 336 = - 56 y + 8 y^{2} - 63 - 9 (- y)$

Step 3.3.3.1.6

Multiply $- 1$ by $- 9$ .

$7 {(y - 7)}^{2} + 6 y^{2} + 6 y - 336 = - 56 y + 8 y^{2} - 63 + 9 y$

Step 3.3.3.2

Add $- 56 y$ and $9 y$ .

$7 {(y - 7)}^{2} + 6 y^{2} + 6 y - 336 = - 47 y + 8 y^{2} - 63$

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

Add $47 y$ to both sides of the equation.

$7 {(y - 7)}^{2} + 6 y^{2} + 6 y - 336 + 47 y = 8 y^{2} - 63$

Step 4.1.2

Subtract $8 y^{2}$ from both sides of the equation.

$7 {(y - 7)}^{2} + 6 y^{2} + 6 y - 336 + 47 y - 8 y^{2} = - 63$

Step 4.1.3

Simplify each term.

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Step 4.1.3.1

Rewrite ${(y - 7)}^{2}$ as $(y - 7) (y - 7)$ .

$7 ((y - 7) (y - 7)) + 6 y^{2} + 6 y - 336 + 47 y - 8 y^{2} = - 63$

Step 4.1.3.2

Expand $(y - 7) (y - 7)$ using the FOIL Method.

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Step 4.1.3.2.1

Apply the distributive property.

$7 (y (y - 7) - 7 (y - 7)) + 6 y^{2} + 6 y - 336 + 47 y - 8 y^{2} = - 63$

Step 4.1.3.2.2

Apply the distributive property.

$7 (y \cdot y + y \cdot - 7 - 7 (y - 7)) + 6 y^{2} + 6 y - 336 + 47 y - 8 y^{2} = - 63$

Step 4.1.3.2.3

Apply the distributive property.

$7 (y \cdot y + y \cdot - 7 - 7 y - 7 \cdot - 7) + 6 y^{2} + 6 y - 336 + 47 y - 8 y^{2} = - 63$

Step 4.1.3.3

Simplify and combine like terms.

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Step 4.1.3.3.1

Simplify each term.

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Step 4.1.3.3.1.1

Multiply $y$ by $y$ .

$7 (y^{2} + y \cdot - 7 - 7 y - 7 \cdot - 7) + 6 y^{2} + 6 y - 336 + 47 y - 8 y^{2} = - 63$

Step 4.1.3.3.1.2

Move $- 7$ to the left of $y$ .

$7 (y^{2} - 7 \cdot y - 7 y - 7 \cdot - 7) + 6 y^{2} + 6 y - 336 + 47 y - 8 y^{2} = - 63$

Step 4.1.3.3.1.3

Multiply $- 7$ by $- 7$ .

$7 (y^{2} - 7 y - 7 y + 49) + 6 y^{2} + 6 y - 336 + 47 y - 8 y^{2} = - 63$

Step 4.1.3.3.2

Subtract $7 y$ from $- 7 y$ .

$7 (y^{2} - 14 y + 49) + 6 y^{2} + 6 y - 336 + 47 y - 8 y^{2} = - 63$

Step 4.1.3.4

Apply the distributive property.

$7 y^{2} + 7 (- 14 y) + 7 \cdot 49 + 6 y^{2} + 6 y - 336 + 47 y - 8 y^{2} = - 63$

Step 4.1.3.5

Simplify.

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Step 4.1.3.5.1

Multiply $- 14$ by $7$ .

$7 y^{2} - 98 y + 7 \cdot 49 + 6 y^{2} + 6 y - 336 + 47 y - 8 y^{2} = - 63$

Step 4.1.3.5.2

Multiply $7$ by $49$ .

$7 y^{2} - 98 y + 343 + 6 y^{2} + 6 y - 336 + 47 y - 8 y^{2} = - 63$

Step 4.1.4

Add $7 y^{2}$ and $6 y^{2}$ .

$13 y^{2} - 98 y + 343 + 6 y - 336 + 47 y - 8 y^{2} = - 63$

Step 4.1.5

Subtract $8 y^{2}$ from $13 y^{2}$ .

$5 y^{2} - 98 y + 343 + 6 y - 336 + 47 y = - 63$

Step 4.1.6

Add $- 98 y$ and $6 y$ .

$5 y^{2} - 92 y + 343 - 336 + 47 y = - 63$

Step 4.1.7

Add $- 92 y$ and $47 y$ .

$5 y^{2} - 45 y + 343 - 336 = - 63$

Step 4.1.8

Subtract $336$ from $343$ .

$5 y^{2} - 45 y + 7 = - 63$

Step 4.2

Add $63$ to both sides of the equation.

$5 y^{2} - 45 y + 7 + 63 = 0$

Step 4.3

Add $7$ and $63$ .

$5 y^{2} - 45 y + 70 = 0$

Step 4.4

Factor the left side of the equation.

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Step 4.4.1

Factor $5$ out of $5 y^{2} - 45 y + 70$ .

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Step 4.4.1.1

Factor $5$ out of $5 y^{2}$ .

$5 (y^{2}) - 45 y + 70 = 0$

Step 4.4.1.2

Factor $5$ out of $- 45 y$ .

$5 (y^{2}) + 5 (- 9 y) + 70 = 0$

Step 4.4.1.3

Factor $5$ out of $70$ .

$5 y^{2} + 5 (- 9 y) + 5 \cdot 14 = 0$

Step 4.4.1.4

Factor $5$ out of $5 y^{2} + 5 (- 9 y)$ .

$5 (y^{2} - 9 y) + 5 \cdot 14 = 0$

Step 4.4.1.5

Factor $5$ out of $5 (y^{2} - 9 y) + 5 \cdot 14$ .

$5 (y^{2} - 9 y + 14) = 0$

Step 4.4.2

Factor.

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Step 4.4.2.1

Factor $y^{2} - 9 y + 14$ using the AC method.

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Step 4.4.2.1.1

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 $14$ and whose sum is $- 9$ .

$- 7, - 2$

Step 4.4.2.1.2

Write the factored form using these integers.

$5 ((y - 7) (y - 2)) = 0$

Step 4.4.2.2

Remove unnecessary parentheses.

$5 (y - 7) (y - 2) = 0$

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 $0$ .

$y - 7 = 0$

$y - 2 = 0$

Step 4.6

Set $y - 7$ equal to $0$ and solve for $y$ .

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Step 4.6.1

Set $y - 7$ equal to $0$ .

$y - 7 = 0$

Step 4.6.2

Add $7$ to both sides of the equation.

$y = 7$

Step 4.7

Set $y - 2$ equal to $0$ and solve for $y$ .

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Step 4.7.1

Set $y - 2$ equal to $0$ .

$y - 2 = 0$

Step 4.7.2

Add $2$ to both sides of the equation.

$y = 2$

Step 4.8

The final solution is all the values that make $5 (y - 7) (y - 2) = 0$ true.

$y = 7, 2$

Step 5

Exclude the solutions that do not make $\frac{- 7}{y + 8} + \frac{6}{7 - y} = \frac{(- 8) y - 9}{y^{2} + y - 56}$ true.

$y = 2$