Solve by Factoring 36/(3x-6)+1/(x-2)=13/3

$\frac{36}{3 x - 6} + \frac{1}{x - 2} = \frac{13}{3}$

Step 1

Subtract $\frac{13}{3}$ from both sides of the equation.

$\frac{36}{3 x - 6} + \frac{1}{x - 2} - \frac{13}{3} = 0$

Step 2

Simplify $\frac{36}{3 x - 6} + \frac{1}{x - 2} - \frac{13}{3}$ .

Cancel the common factor of $36$ and $3 x - 6$ .

Factor $3$ out of $36$ .

$\frac{3 \cdot 12}{3 x - 6} + \frac{1}{x - 2} - \frac{13}{3} = 0$

Cancel the common factors.

Factor $3$ out of $3 x$ .

$\frac{3 \cdot 12}{3 (x) - 6} + \frac{1}{x - 2} - \frac{13}{3} = 0$

Factor $3$ out of $- 6$ .

$\frac{3 \cdot 12}{3 (x) + 3 (- 2)} + \frac{1}{x - 2} - \frac{13}{3} = 0$

Factor $3$ out of $3 (x) + 3 (- 2)$ .

$\frac{3 \cdot 12}{3 (x - 2)} + \frac{1}{x - 2} - \frac{13}{3} = 0$

Cancel the common factor.

$\frac{3 \cdot 12}{3 (x - 2)} + \frac{1}{x - 2} - \frac{13}{3} = 0$

Rewrite the expression.

$\frac{12}{x - 2} + \frac{1}{x - 2} - \frac{13}{3} = 0$

Combine the numerators over the common denominator.

$\frac{12 + 1}{x - 2} + \frac{- 13}{3} = 0$

Add $12$ and $1$ .

$\frac{13}{x - 2} + \frac{- 13}{3} = 0$

Move the negative in front of the fraction.

$\frac{13}{x - 2} - \frac{13}{3} = 0$

To write $\frac{13}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{13}{x - 2} \cdot \frac{3}{3} - \frac{13}{3} = 0$

To write $- \frac{13}{3}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{13}{x - 2} \cdot \frac{3}{3} - \frac{13}{3} \cdot \frac{x - 2}{x - 2} = 0$

Write each expression with a common denominator of $(x - 2) \cdot 3$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{13}{x - 2}$ by $\frac{3}{3}$ .

$\frac{13 \cdot 3}{(x - 2) \cdot 3} - \frac{13}{3} \cdot \frac{x - 2}{x - 2} = 0$

Multiply $\frac{13}{3}$ by $\frac{x - 2}{x - 2}$ .

$\frac{13 \cdot 3}{(x - 2) \cdot 3} - \frac{13 (x - 2)}{3 (x - 2)} = 0$

Reorder the factors of $(x - 2) \cdot 3$ .

$\frac{13 \cdot 3}{3 (x - 2)} - \frac{13 (x - 2)}{3 (x - 2)} = 0$

Combine the numerators over the common denominator.

$\frac{13 \cdot 3 - 13 (x - 2)}{3 (x - 2)} = 0$

Simplify the numerator.

Factor $13$ out of $13 \cdot 3 - 13 (x - 2)$ .

Factor $13$ out of $13 \cdot 3$ .

$\frac{13 (3) - 13 (x - 2)}{3 (x - 2)} = 0$

Factor $13$ out of $- 13 (x - 2)$ .

$\frac{13 (3) + 13 (- (x - 2))}{3 (x - 2)} = 0$

Factor $13$ out of $13 (3) + 13 (- (x - 2))$ .

$\frac{13 (3 - (x - 2))}{3 (x - 2)} = 0$

Apply the distributive property.

$\frac{13 (3 - x - - 2)}{3 (x - 2)} = 0$

Multiply $- 1$ by $- 2$ .

$\frac{13 (3 - x + 2)}{3 (x - 2)} = 0$

Add $3$ and $2$ .

$\frac{13 (- x + 5)}{3 (x - 2)} = 0$

Factor $- 1$ out of $- x$ .

$\frac{13 (- (x) + 5)}{3 (x - 2)} = 0$

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{13 (- (x) - 1 (- 5))}{3 (x - 2)} = 0$

Factor $- 1$ out of $- (x) - 1 (- 5)$ .

$\frac{13 (- (x - 5))}{3 (x - 2)} = 0$

Rewrite $- (x - 5)$ as $- 1 (x - 5)$ .

$\frac{13 (- 1 (x - 5))}{3 (x - 2)} = 0$

Move the negative in front of the fraction.

$- \frac{13 (x - 5)}{3 (x - 2)} = 0$

Step 3

Set the numerator equal to zero.

$13 (x - 5) = 0$

Step 4

Solve the equation for $x$ .

Divide each term in $13 (x - 5) = 0$ by $13$ and simplify.

Divide each term in $13 (x - 5) = 0$ by $13$ .

$\frac{13 (x - 5)}{13} = \frac{0}{13}$

Simplify the left side.

Cancel the common factor of $13$ .

Cancel the common factor.

$\frac{13 (x - 5)}{13} = \frac{0}{13}$

Divide $x - 5$ by $1$ .

$x - 5 = \frac{0}{13}$

Simplify the right side.

Divide $0$ by $13$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Solve by Factoring 36/(3x-6)+1/(x-2)=13/3

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