Solve by Factoring 4/(x-1)+6/(3x+1)=15/(3x+1)

$\frac{4}{x - 1} + \frac{6}{3 x + 1} = \frac{15}{3 x + 1}$

Step 1

Subtract $\frac{15}{3 x + 1}$ from both sides of the equation.

$\frac{4}{x - 1} + \frac{6}{3 x + 1} - \frac{15}{3 x + 1} = 0$

Step 2

Simplify $\frac{4}{x - 1} + \frac{6}{3 x + 1} - \frac{15}{3 x + 1}$ .

Combine the numerators over the common denominator.

$\frac{4}{x - 1} + \frac{6 - 15}{3 x + 1} = 0$

Subtract $15$ from $6$ .

$\frac{4}{x - 1} + \frac{- 9}{3 x + 1} = 0$

Move the negative in front of the fraction.

$\frac{4}{x - 1} - \frac{9}{3 x + 1} = 0$

To write $\frac{4}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{3 x + 1}{3 x + 1}$ .

$\frac{4}{x - 1} \cdot \frac{3 x + 1}{3 x + 1} - \frac{9}{3 x + 1} = 0$

To write $- \frac{9}{3 x + 1}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{4}{x - 1} \cdot \frac{3 x + 1}{3 x + 1} - \frac{9}{3 x + 1} \cdot \frac{x - 1}{x - 1} = 0$

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

Multiply $\frac{4}{x - 1}$ by $\frac{3 x + 1}{3 x + 1}$ .

$\frac{4 (3 x + 1)}{(x - 1) (3 x + 1)} - \frac{9}{3 x + 1} \cdot \frac{x - 1}{x - 1} = 0$

Multiply $\frac{9}{3 x + 1}$ by $\frac{x - 1}{x - 1}$ .

$\frac{4 (3 x + 1)}{(x - 1) (3 x + 1)} - \frac{9 (x - 1)}{(3 x + 1) (x - 1)} = 0$

Reorder the factors of $(x - 1) (3 x + 1)$ .

$\frac{4 (3 x + 1)}{(3 x + 1) (x - 1)} - \frac{9 (x - 1)}{(3 x + 1) (x - 1)} = 0$

Combine the numerators over the common denominator.

$\frac{4 (3 x + 1) - 9 (x - 1)}{(3 x + 1) (x - 1)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{4 (3 x) + 4 \cdot 1 - 9 (x - 1)}{(3 x + 1) (x - 1)} = 0$

Multiply $3$ by $4$ .

$\frac{12 x + 4 \cdot 1 - 9 (x - 1)}{(3 x + 1) (x - 1)} = 0$

Multiply $4$ by $1$ .

$\frac{12 x + 4 - 9 (x - 1)}{(3 x + 1) (x - 1)} = 0$

Apply the distributive property.

$\frac{12 x + 4 - 9 x - 9 \cdot - 1}{(3 x + 1) (x - 1)} = 0$

Multiply $- 9$ by $- 1$ .

$\frac{12 x + 4 - 9 x + 9}{(3 x + 1) (x - 1)} = 0$

Subtract $9 x$ from $12 x$ .

$\frac{3 x + 4 + 9}{(3 x + 1) (x - 1)} = 0$

Add $4$ and $9$ .

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

Step 3

Set the numerator equal to zero.

$3 x + 13 = 0$

Step 4

Solve the equation for $x$ .

Subtract $13$ from both sides of the equation.

$3 x = - 13$

Divide each term in $3 x = - 13$ by $3$ and simplify.

Divide each term in $3 x = - 13$ by $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Solve by Factoring 4/(x-1)+6/(3x+1)=15/(3x+1)

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