Algebra Examples

$\frac{6}{x - 3} - \frac{2}{3} = \frac{x}{x - 3}$

Step 1

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

$\frac{6}{x - 3} - \frac{2}{3} - \frac{x}{x - 3} = 0$

Step 2

Simplify $\frac{6}{x - 3} - \frac{2}{3} - \frac{x}{x - 3}$ .

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Step 2.1

To write $\frac{6}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{6}{x - 3} \cdot \frac{3}{3} - \frac{2}{3} - \frac{x}{x - 3} = 0$

Step 2.2

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

$\frac{6}{x - 3} \cdot \frac{3}{3} - \frac{2}{3} \cdot \frac{x - 3}{x - 3} - \frac{x}{x - 3} = 0$

Step 2.3

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

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Step 2.3.1

Multiply $\frac{6}{x - 3}$ by $\frac{3}{3}$ .

$\frac{6 \cdot 3}{(x - 3) \cdot 3} - \frac{2}{3} \cdot \frac{x - 3}{x - 3} - \frac{x}{x - 3} = 0$

Step 2.3.2

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

$\frac{6 \cdot 3}{(x - 3) \cdot 3} - \frac{2 (x - 3)}{3 (x - 3)} - \frac{x}{x - 3} = 0$

Step 2.3.3

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

$\frac{6 \cdot 3}{3 (x - 3)} - \frac{2 (x - 3)}{3 (x - 3)} - \frac{x}{x - 3} = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{6 \cdot 3 - 2 (x - 3)}{3 (x - 3)} - \frac{x}{x - 3} = 0$

Step 2.5

Multiply $6$ by $3$ .

$\frac{18 - 2 (x - 3)}{3 (x - 3)} - \frac{x}{x - 3} = 0$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Factor $2$ out of $18 - 2 (x - 3)$ .

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Step 2.6.1.1

Factor $2$ out of $18$ .

$\frac{2 (9) - 2 (x - 3)}{3 (x - 3)} - \frac{x}{x - 3} = 0$

Step 2.6.1.2

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

$\frac{2 (9) + 2 (- (x - 3))}{3 (x - 3)} - \frac{x}{x - 3} = 0$

Step 2.6.1.3

Factor $2$ out of $2 (9) + 2 (- (x - 3))$ .

$\frac{2 (9 - (x - 3))}{3 (x - 3)} - \frac{x}{x - 3} = 0$

Step 2.6.2

Apply the distributive property.

$\frac{2 (9 - x - - 3)}{3 (x - 3)} - \frac{x}{x - 3} = 0$

Step 2.6.3

Multiply $- 1$ by $- 3$ .

$\frac{2 (9 - x + 3)}{3 (x - 3)} - \frac{x}{x - 3} = 0$

Step 2.6.4

Add $9$ and $3$ .

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

Step 2.7

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

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

Step 2.8

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

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Step 2.8.1

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

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

Step 2.8.2

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

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

Step 2.9

Combine the numerators over the common denominator.

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

Step 2.10

Simplify the numerator.

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Step 2.10.1

Apply the distributive property.

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

Step 2.10.2

Multiply $- 1$ by $2$ .

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

Step 2.10.3

Multiply $2$ by $12$ .

$\frac{- 2 x + 24 - x \cdot 3}{3 (x - 3)} = 0$

Step 2.10.4

Multiply $3$ by $- 1$ .

$\frac{- 2 x + 24 - 3 x}{3 (x - 3)} = 0$

Step 2.10.5

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

$\frac{- 5 x + 24}{3 (x - 3)} = 0$

Step 2.11

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

$\frac{- (5 x) + 24}{3 (x - 3)} = 0$

Step 2.12

Rewrite $24$ as $- 1 (- 24)$ .

$\frac{- (5 x) - 1 (- 24)}{3 (x - 3)} = 0$

Step 2.13

Factor $- 1$ out of $- (5 x) - 1 (- 24)$ .

$\frac{- (5 x - 24)}{3 (x - 3)} = 0$

Step 2.14

Rewrite $- (5 x - 24)$ as $- 1 (5 x - 24)$ .

$\frac{- 1 (5 x - 24)}{3 (x - 3)} = 0$

Step 2.15

Move the negative in front of the fraction.

$- \frac{5 x - 24}{3 (x - 3)} = 0$

Step 3

Set the numerator equal to zero.

$5 x - 24 = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Add $24$ to both sides of the equation.

$5 x = 24$

Step 4.2

Divide each term in $5 x = 24$ by $5$ and simplify.

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Step 4.2.1

Divide each term in $5 x = 24$ by $5$ .

$\frac{5 x}{5} = \frac{24}{5}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $5$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{24}{5}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{24}{5}$