Algebra Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

Simplify each term.

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Multiply $- 1$ by $- 1$ .

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

Step 2.3

Add $2$ and $1$ .

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

Step 2.4

Cancel the common factor of $- x + 3$ and $x - 3$ .

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Step 2.4.1

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

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

Step 2.4.2

Rewrite $3$ as $- 1 (- 3)$ .

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

Step 2.4.3

Factor $- 1$ out of $- (x) - 1 (- 3)$ .

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

Step 2.4.4

Rewrite $- (x - 3)$ as $- 1 (x - 3)$ .

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

Step 2.4.5

Cancel the common factor.

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

Step 2.4.6

Divide $- 1$ by $1$ .

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

Step 3

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

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

Step 4

Combine $- 1$ and $\frac{x}{x}$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Set the numerator equal to zero.

$- x + 1 = 0$

Step 7

Solve the equation for $x$ .

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Step 7.1

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 7.2

Divide each term in $- x = - 1$ by $- 1$ and simplify.

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Step 7.2.1

Divide each term in $- x = - 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 1}{- 1}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 1}{- 1}$

Step 7.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 1}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Divide $- 1$ by $- 1$ .

$x = 1$