Algebra Examples

$2 - \frac{3 x}{1 - x} = \frac{8}{x - 1}$

Step 1

Subtract $\frac{8}{x - 1}$ from both sides of the equation.

$2 - \frac{3 x}{1 - x} - \frac{8}{x - 1} = 0$

Step 2

Simplify $2 - \frac{3 x}{1 - x} - \frac{8}{x - 1}$ .

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Step 2.1

To write $2$ as a fraction with a common denominator, multiply by $\frac{1 - x}{1 - x}$ .

$\frac{2 (1 - x)}{1 - x} - \frac{3 x}{1 - x} - \frac{8}{x - 1} = 0$

Step 2.2

Combine the numerators over the common denominator.

$\frac{2 (1 - x) - 3 x}{1 - x} - \frac{8}{x - 1} = 0$

Step 2.3

Simplify the numerator.

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Step 2.3.1

Apply the distributive property.

$\frac{2 \cdot 1 + 2 (- x) - 3 x}{1 - x} - \frac{8}{x - 1} = 0$

Step 2.3.2

Multiply $2$ by $1$ .

$\frac{2 + 2 (- x) - 3 x}{1 - x} - \frac{8}{x - 1} = 0$

Step 2.3.3

Multiply $- 1$ by $2$ .

$\frac{2 - 2 x - 3 x}{1 - x} - \frac{8}{x - 1} = 0$

Step 2.3.4

Subtract $3 x$ from $- 2 x$ .

$\frac{2 - 5 x}{1 - x} - \frac{8}{x - 1} = 0$

Step 2.4

Factor $- 1$ out of $x$ .

$\frac{2 - 5 x}{1 - x} - \frac{8}{- 1 (- x) - 1} = 0$

Step 2.5

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

$\frac{2 - 5 x}{1 - x} - \frac{8}{- 1 (- x) - 1 (1)} = 0$

Step 2.6

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

$\frac{2 - 5 x}{1 - x} - \frac{8}{- 1 (- x + 1)} = 0$

Step 2.7

Reorder terms.

$\frac{2 - 5 x}{- x + 1} - \frac{8}{- 1 (- x + 1)} = 0$

Step 2.8

To write $\frac{2 - 5 x}{- x + 1}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

$\frac{2 - 5 x}{- x + 1} \cdot \frac{- 1}{- 1} - \frac{8}{- 1 (- x + 1)} = 0$

Step 2.9

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

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Step 2.9.1

Multiply $\frac{2 - 5 x}{- x + 1}$ by $\frac{- 1}{- 1}$ .

$\frac{(2 - 5 x) \cdot - 1}{(- x + 1) \cdot - 1} - \frac{8}{- 1 (- x + 1)} = 0$

Step 2.9.2

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

$\frac{(2 - 5 x) \cdot - 1}{- (- x + 1)} - \frac{8}{- 1 (- x + 1)} = 0$

Step 2.10

Combine the numerators over the common denominator.

$\frac{(2 - 5 x) \cdot - 1 - 8}{- (- x + 1)} = 0$

Step 2.11

Simplify the numerator.

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Step 2.11.1

Apply the distributive property.

$\frac{2 \cdot - 1 - 5 x \cdot - 1 - 8}{- (- x + 1)} = 0$

Step 2.11.2

Multiply $2$ by $- 1$ .

$\frac{- 2 - 5 x \cdot - 1 - 8}{- (- x + 1)} = 0$

Step 2.11.3

Multiply $- 1$ by $- 5$ .

$\frac{- 2 + 5 x - 8}{- (- x + 1)} = 0$

Step 2.11.4

Subtract $8$ from $- 2$ .

$\frac{5 x - 10}{- (- x + 1)} = 0$

Step 2.11.5

Factor $5$ out of $5 x - 10$ .

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Step 2.11.5.1

Factor $5$ out of $5 x$ .

$\frac{5 (x) - 10}{- (- x + 1)} = 0$

Step 2.11.5.2

Factor $5$ out of $- 10$ .

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

Step 2.11.5.3

Factor $5$ out of $5 x + 5 \cdot - 2$ .

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

Step 2.12

Move the negative in front of the fraction.

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

Step 2.13

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

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

Step 2.14

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

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

Step 2.15

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

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

Step 2.16

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

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

Step 2.17

Move the negative in front of the fraction.

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

Step 2.18

Multiply $- 1$ by $- 1$ .

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

Step 2.19

Multiply $\frac{5 (x - 2)}{x - 1}$ by $1$ .

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

Step 3

Set the numerator equal to zero.

$5 (x - 2) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Divide each term in $5 (x - 2) = 0$ by $5$ and simplify.

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Step 4.1.1

Divide each term in $5 (x - 2) = 0$ by $5$ .

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

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $5$ .

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Step 4.1.2.1.1

Cancel the common factor.

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

Step 4.1.2.1.2

Divide $x - 2$ by $1$ .

$x - 2 = \frac{0}{5}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $5$ .

$x - 2 = 0$

Step 4.2

Add $2$ to both sides of the equation.

$x = 2$