Solve for z (z+9)/(z^2-81)=1

$\frac{z + 9}{z^{2} - 81} = 1$

Factor each term.

Rewrite $81$ as $9^{2}$ .

$\frac{z + 9}{z^{2} - 9^{2}} = 1$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = z$ and $b = 9$ .

$\frac{z + 9}{(z + 9) (z - 9)} = 1$

Reduce the expression $\frac{z + 9}{(z + 9) (z - 9)}$ by cancelling the common factors.

Cancel the common factor.

$\frac{z + 9}{(z + 9) (z - 9)} = 1$

Rewrite the expression.

$\frac{1}{z - 9} = 1$

Find the LCD of the terms in the equation.

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

$z - 9, 1$

Remove parentheses.

$z - 9, 1$

The LCM of one and any expression is the expression.

$z - 9$

Multiply each term in $\frac{1}{z - 9} = 1$ by $z - 9$ to eliminate the fractions.

Multiply each term in $\frac{1}{z - 9} = 1$ by $z - 9$ .

$\frac{1}{z - 9} (z - 9) = 1 (z - 9)$

Simplify the left side.

Cancel the common factor of $z - 9$ .

Cancel the common factor.

$\frac{1}{z - 9} (z - 9) = 1 (z - 9)$

Rewrite the expression.

$1 = 1 (z - 9)$

Simplify the right side.

Multiply $z - 9$ by $1$ .

$1 = z - 9$

Solve the equation.

Rewrite the equation as $z - 9 = 1$ .

$z - 9 = 1$

Move all terms not containing $z$ to the right side of the equation.

Add $9$ to both sides of the equation.

$z = 1 + 9$

Add $1$ and $9$ .

$z = 10$