Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

Multiply $\frac{1}{x - 1}$ by $\frac{x}{x}$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Rewrite $\frac{x + 2 (x - 1)}{x (x - 1)}$ in a factored form.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $2$ by $- 1$ .

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

Step 5.3

Add $x$ and $2 x$ .

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

Step 6

Set the numerator equal to zero.

$3 x - 2 = 0$

Step 7

Solve the equation for $x$ .

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Step 7.1

Add $2$ to both sides of the equation.

$3 x = 2$

Step 7.2

Divide each term in $3 x = 2$ by $3$ and simplify.

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Step 7.2.1

Divide each term in $3 x = 2$ by $3$ .

$\frac{3 x}{3} = \frac{2}{3}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $3$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{2}{3}$

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{3}$