Solve by Factoring 14/(n+7)=10/(n-4)

$\frac{14}{n + 7} = \frac{10}{n - 4}$

Step 1

Subtract $\frac{10}{n - 4}$ from both sides of the equation.

$\frac{14}{n + 7} - \frac{10}{n - 4} = 0$

Step 2

Simplify $\frac{14}{n + 7} - \frac{10}{n - 4}$ .

To write $\frac{14}{n + 7}$ as a fraction with a common denominator, multiply by $\frac{n - 4}{n - 4}$ .

$\frac{14}{n + 7} \cdot \frac{n - 4}{n - 4} - \frac{10}{n - 4} = 0$

To write $- \frac{10}{n - 4}$ as a fraction with a common denominator, multiply by $\frac{n + 7}{n + 7}$ .

$\frac{14}{n + 7} \cdot \frac{n - 4}{n - 4} - \frac{10}{n - 4} \cdot \frac{n + 7}{n + 7} = 0$

Write each expression with a common denominator of $(n + 7) (n - 4)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{14}{n + 7}$ by $\frac{n - 4}{n - 4}$ .

$\frac{14 (n - 4)}{(n + 7) (n - 4)} - \frac{10}{n - 4} \cdot \frac{n + 7}{n + 7} = 0$

Multiply $\frac{10}{n - 4}$ by $\frac{n + 7}{n + 7}$ .

$\frac{14 (n - 4)}{(n + 7) (n - 4)} - \frac{10 (n + 7)}{(n - 4) (n + 7)} = 0$

Reorder the factors of $(n - 4) (n + 7)$ .

$\frac{14 (n - 4)}{(n + 7) (n - 4)} - \frac{10 (n + 7)}{(n + 7) (n - 4)} = 0$

Combine the numerators over the common denominator.

$\frac{14 (n - 4) - 10 (n + 7)}{(n + 7) (n - 4)} = 0$

Simplify the numerator.

Factor $2$ out of $14 (n - 4) - 10 (n + 7)$ .

Factor $2$ out of $14 (n - 4)$ .

$\frac{2 (7 (n - 4)) - 10 (n + 7)}{(n + 7) (n - 4)} = 0$

Factor $2$ out of $- 10 (n + 7)$ .

$\frac{2 (7 (n - 4)) + 2 (- 5 (n + 7))}{(n + 7) (n - 4)} = 0$

Factor $2$ out of $2 (7 (n - 4)) + 2 (- 5 (n + 7))$ .

$\frac{2 (7 (n - 4) - 5 (n + 7))}{(n + 7) (n - 4)} = 0$

Apply the distributive property.

$\frac{2 (7 n + 7 \cdot - 4 - 5 (n + 7))}{(n + 7) (n - 4)} = 0$

Multiply $7$ by $- 4$ .

$\frac{2 (7 n - 28 - 5 (n + 7))}{(n + 7) (n - 4)} = 0$

Apply the distributive property.

$\frac{2 (7 n - 28 - 5 n - 5 \cdot 7)}{(n + 7) (n - 4)} = 0$

Multiply $- 5$ by $7$ .

$\frac{2 (7 n - 28 - 5 n - 35)}{(n + 7) (n - 4)} = 0$

Subtract $5 n$ from $7 n$ .

$\frac{2 (2 n - 28 - 35)}{(n + 7) (n - 4)} = 0$

Subtract $35$ from $- 28$ .

$\frac{2 (2 n - 63)}{(n + 7) (n - 4)} = 0$

Step 3

Set the numerator equal to zero.

$2 (2 n - 63) = 0$

Step 4

Solve the equation for $n$ .

Divide each term in $2 (2 n - 63) = 0$ by $2$ and simplify.

Divide each term in $2 (2 n - 63) = 0$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $2 n - 63$ by $1$ .

$2 n - 63 = \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$2 n - 63 = 0$

Add $63$ to both sides of the equation.

$2 n = 63$

Divide each term in $2 n = 63$ by $2$ and simplify.

Divide each term in $2 n = 63$ by $2$ .

$\frac{2 n}{2} = \frac{63}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 n}{2} = \frac{63}{2}$

Divide $n$ by $1$ .

$n = \frac{63}{2}$

Solve by Factoring 14/(n+7)=10/(n-4)

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