Solve by Factoring (x-3)/(x+1)=(x-6)/(x+5)

$\frac{x - 3}{x + 1} = \frac{x - 6}{x + 5}$

Step 1

Subtract $\frac{x - 6}{x + 5}$ from both sides of the equation.

$\frac{x - 3}{x + 1} - \frac{x - 6}{x + 5} = 0$

Step 2

Simplify $\frac{x - 3}{x + 1} - \frac{x - 6}{x + 5}$ .

To write $\frac{x - 3}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{x - 3}{x + 1} \cdot \frac{x + 5}{x + 5} - \frac{x - 6}{x + 5} = 0$

To write $- \frac{x - 6}{x + 5}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$\frac{x - 3}{x + 1} \cdot \frac{x + 5}{x + 5} - \frac{x - 6}{x + 5} \cdot \frac{x + 1}{x + 1} = 0$

Write each expression with a common denominator of $(x + 1) (x + 5)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x - 3}{x + 1}$ by $\frac{x + 5}{x + 5}$ .

$\frac{(x - 3) (x + 5)}{(x + 1) (x + 5)} - \frac{x - 6}{x + 5} \cdot \frac{x + 1}{x + 1} = 0$

Multiply $\frac{x - 6}{x + 5}$ by $\frac{x + 1}{x + 1}$ .

$\frac{(x - 3) (x + 5)}{(x + 1) (x + 5)} - \frac{(x - 6) (x + 1)}{(x + 5) (x + 1)} = 0$

Reorder the factors of $(x + 5) (x + 1)$ .

$\frac{(x - 3) (x + 5)}{(x + 1) (x + 5)} - \frac{(x - 6) (x + 1)}{(x + 1) (x + 5)} = 0$

Combine the numerators over the common denominator.

$\frac{(x - 3) (x + 5) - (x - 6) (x + 1)}{(x + 1) (x + 5)} = 0$

Simplify the numerator.

Expand $(x - 3) (x + 5)$ using the FOIL Method.

Apply the distributive property.

$\frac{x (x + 5) - 3 (x + 5) - (x - 6) (x + 1)}{(x + 1) (x + 5)} = 0$

Apply the distributive property.

$\frac{x \cdot x + x \cdot 5 - 3 (x + 5) - (x - 6) (x + 1)}{(x + 1) (x + 5)} = 0$

Apply the distributive property.

$\frac{x \cdot x + x \cdot 5 - 3 x - 3 \cdot 5 - (x - 6) (x + 1)}{(x + 1) (x + 5)} = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 5 - 3 x - 3 \cdot 5 - (x - 6) (x + 1)}{(x + 1) (x + 5)} = 0$

Move $5$ to the left of $x$ .

$\frac{x^{2} + 5 \cdot x - 3 x - 3 \cdot 5 - (x - 6) (x + 1)}{(x + 1) (x + 5)} = 0$

Multiply $- 3$ by $5$ .

$\frac{x^{2} + 5 x - 3 x - 15 - (x - 6) (x + 1)}{(x + 1) (x + 5)} = 0$

Subtract $3 x$ from $5 x$ .

$\frac{x^{2} + 2 x - 15 - (x - 6) (x + 1)}{(x + 1) (x + 5)} = 0$

Apply the distributive property.

$\frac{x^{2} + 2 x - 15 + (- x - - 6) (x + 1)}{(x + 1) (x + 5)} = 0$

Multiply $- 1$ by $- 6$ .

$\frac{x^{2} + 2 x - 15 + (- x + 6) (x + 1)}{(x + 1) (x + 5)} = 0$

Expand $(- x + 6) (x + 1)$ using the FOIL Method.

Apply the distributive property.

$\frac{x^{2} + 2 x - 15 - x (x + 1) + 6 (x + 1)}{(x + 1) (x + 5)} = 0$

Apply the distributive property.

$\frac{x^{2} + 2 x - 15 - x \cdot x - x \cdot 1 + 6 (x + 1)}{(x + 1) (x + 5)} = 0$

Apply the distributive property.

$\frac{x^{2} + 2 x - 15 - x \cdot x - x \cdot 1 + 6 x + 6 \cdot 1}{(x + 1) (x + 5)} = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{x^{2} + 2 x - 15 - (x \cdot x) - x \cdot 1 + 6 x + 6 \cdot 1}{(x + 1) (x + 5)} = 0$

Multiply $x$ by $x$ .

$\frac{x^{2} + 2 x - 15 - x^{2} - x \cdot 1 + 6 x + 6 \cdot 1}{(x + 1) (x + 5)} = 0$

Multiply $- 1$ by $1$ .

$\frac{x^{2} + 2 x - 15 - x^{2} - x + 6 x + 6 \cdot 1}{(x + 1) (x + 5)} = 0$

Multiply $6$ by $1$ .

$\frac{x^{2} + 2 x - 15 - x^{2} - x + 6 x + 6}{(x + 1) (x + 5)} = 0$

Add $- x$ and $6 x$ .

$\frac{x^{2} + 2 x - 15 - x^{2} + 5 x + 6}{(x + 1) (x + 5)} = 0$

Subtract $x^{2}$ from $x^{2}$ .

$\frac{2 x - 15 + 0 + 5 x + 6}{(x + 1) (x + 5)} = 0$

Add $2 x$ and $0$ .

$\frac{2 x - 15 + 5 x + 6}{(x + 1) (x + 5)} = 0$

Add $2 x$ and $5 x$ .

$\frac{7 x - 15 + 6}{(x + 1) (x + 5)} = 0$

Add $- 15$ and $6$ .

$\frac{7 x - 9}{(x + 1) (x + 5)} = 0$

Step 3

Set the numerator equal to zero.

$7 x - 9 = 0$

Step 4

Solve the equation for $x$ .

Add $9$ to both sides of the equation.

$7 x = 9$

Divide each term in $7 x = 9$ by $7$ and simplify.

Divide each term in $7 x = 9$ by $7$ .

$\frac{7 x}{7} = \frac{9}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x}{7} = \frac{9}{7}$

Divide $x$ by $1$ .

$x = \frac{9}{7}$

Solve by Factoring (x-3)/(x+1)=(x-6)/(x+5)

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