Algebra Examples

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

Step 1

Subtract $\frac{6 x}{2 x - 6}$ from both sides of the equation.

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

Step 2

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

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Step 2.1

Cancel the common factor of $6$ and $2 x - 6$ .

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Step 2.1.1

Factor $2$ out of $6 x$ .

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

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $2$ out of $2 x$ .

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

Step 2.1.2.2

Factor $2$ out of $- 6$ .

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

Step 2.1.2.3

Factor $2$ out of $2 x + 2 \cdot - 3$ .

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

Step 2.1.2.4

Cancel the common factor.

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

Step 2.1.2.5

Rewrite the expression.

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 2.3

Subtract $3 x$ from $x$ .

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

Step 2.4

Move the negative in front of the fraction.

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Write each expression with a common denominator of $(x - 3) \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.7.1

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

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

Step 2.7.2

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

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

Step 2.7.3

Reorder the factors of $(x - 3) \cdot 2$ .

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

Step 2.8

Combine the numerators over the common denominator.

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

Step 2.9

Simplify the numerator.

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Step 2.9.1

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

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Step 2.9.1.1

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

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

Step 2.9.1.2

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

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

Step 2.9.2

Multiply $- 2$ by $2$ .

$\frac{x (- 4 + x - 3)}{2 (x - 3)} = 0$

Step 2.9.3

Subtract $3$ from $- 4$ .

$\frac{x (x - 7)}{2 (x - 3)} = 0$

Step 3

Set the numerator equal to zero.

$x (x - 7) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 7 = 0$

Step 4.2

Set $x$ equal to $0$ .

$x = 0$

Step 4.3

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 4.3.2

Add $7$ to both sides of the equation.

$x = 7$

Step 4.4

The final solution is all the values that make $x (x - 7) = 0$ true.

$x = 0, 7$