Solve by Factoring (n-6)/(n-7)=9/2

$\frac{n - 6}{n - 7} = \frac{9}{2}$

Step 1

Subtract $\frac{9}{2}$ from both sides of the equation.

$\frac{n - 6}{n - 7} - \frac{9}{2} = 0$

Step 2

Simplify $\frac{n - 6}{n - 7} - \frac{9}{2}$ .

To write $\frac{n - 6}{n - 7}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{n - 6}{n - 7} \cdot \frac{2}{2} - \frac{9}{2} = 0$

To write $- \frac{9}{2}$ as a fraction with a common denominator, multiply by $\frac{n - 7}{n - 7}$ .

$\frac{n - 6}{n - 7} \cdot \frac{2}{2} - \frac{9}{2} \cdot \frac{n - 7}{n - 7} = 0$

Write each expression with a common denominator of $(n - 7) \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{n - 6}{n - 7}$ by $\frac{2}{2}$ .

$\frac{(n - 6) \cdot 2}{(n - 7) \cdot 2} - \frac{9}{2} \cdot \frac{n - 7}{n - 7} = 0$

Multiply $\frac{9}{2}$ by $\frac{n - 7}{n - 7}$ .

$\frac{(n - 6) \cdot 2}{(n - 7) \cdot 2} - \frac{9 (n - 7)}{2 (n - 7)} = 0$

Reorder the factors of $(n - 7) \cdot 2$ .

$\frac{(n - 6) \cdot 2}{2 (n - 7)} - \frac{9 (n - 7)}{2 (n - 7)} = 0$

Combine the numerators over the common denominator.

$\frac{(n - 6) \cdot 2 - 9 (n - 7)}{2 (n - 7)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{n \cdot 2 - 6 \cdot 2 - 9 (n - 7)}{2 (n - 7)} = 0$

Move $2$ to the left of $n$ .

$\frac{2 \cdot n - 6 \cdot 2 - 9 (n - 7)}{2 (n - 7)} = 0$

Multiply $- 6$ by $2$ .

$\frac{2 \cdot n - 12 - 9 (n - 7)}{2 (n - 7)} = 0$

Apply the distributive property.

$\frac{2 n - 12 - 9 n - 9 \cdot - 7}{2 (n - 7)} = 0$

Multiply $- 9$ by $- 7$ .

$\frac{2 n - 12 - 9 n + 63}{2 (n - 7)} = 0$

Subtract $9 n$ from $2 n$ .

$\frac{- 7 n - 12 + 63}{2 (n - 7)} = 0$

Add $- 12$ and $63$ .

$\frac{- 7 n + 51}{2 (n - 7)} = 0$

Factor $- 1$ out of $- 7 n$ .

$\frac{- (7 n) + 51}{2 (n - 7)} = 0$

Rewrite $51$ as $- 1 (- 51)$ .

$\frac{- (7 n) - 1 (- 51)}{2 (n - 7)} = 0$

Factor $- 1$ out of $- (7 n) - 1 (- 51)$ .

$\frac{- (7 n - 51)}{2 (n - 7)} = 0$

Rewrite $- (7 n - 51)$ as $- 1 (7 n - 51)$ .

$\frac{- 1 (7 n - 51)}{2 (n - 7)} = 0$

Move the negative in front of the fraction.

$- \frac{7 n - 51}{2 (n - 7)} = 0$

Step 3

Set the numerator equal to zero.

$7 n - 51 = 0$

Step 4

Solve the equation for $n$ .

Add $51$ to both sides of the equation.

$7 n = 51$

Divide each term in $7 n = 51$ by $7$ and simplify.

Divide each term in $7 n = 51$ by $7$ .

$\frac{7 n}{7} = \frac{51}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 n}{7} = \frac{51}{7}$

Divide $n$ by $1$ .

$n = \frac{51}{7}$

Solve by Factoring (n-6)/(n-7)=9/2

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