Algebra Examples

$\frac{5 x}{x - 5} - \frac{4}{x} = \frac{20}{x^{2} - 5 x}$

Step 1

Subtract $\frac{20}{x^{2} - 5 x}$ from both sides of the equation.

$\frac{5 x}{x - 5} - \frac{4}{x} - \frac{20}{x^{2} - 5 x} = 0$

Step 2

Simplify $\frac{5 x}{x - 5} - \frac{4}{x} - \frac{20}{x^{2} - 5 x}$ .

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Step 2.1

Factor $x$ out of $x^{2} - 5 x$ .

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Step 2.1.1

Factor $x$ out of $x^{2}$ .

$\frac{5 x}{x - 5} - \frac{4}{x} - \frac{20}{x \cdot x - 5 x} = 0$

Step 2.1.2

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

$\frac{5 x}{x - 5} - \frac{4}{x} - \frac{20}{x \cdot x + x \cdot - 5} = 0$

Step 2.1.3

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

$\frac{5 x}{x - 5} - \frac{4}{x} - \frac{20}{x (x - 5)} = 0$

Step 2.2

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

$\frac{5 x}{x - 5} \cdot \frac{x}{x} - \frac{4}{x} - \frac{20}{x (x - 5)} = 0$

Step 2.3

To write $- \frac{4}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$\frac{5 x}{x - 5} \cdot \frac{x}{x} - \frac{4}{x} \cdot \frac{x - 5}{x - 5} - \frac{20}{x (x - 5)} = 0$

Step 2.4

Write each expression with a common denominator of $(x - 5) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{5 x}{x - 5}$ by $\frac{x}{x}$ .

$\frac{5 x \cdot x}{(x - 5) x} - \frac{4}{x} \cdot \frac{x - 5}{x - 5} - \frac{20}{x (x - 5)} = 0$

Step 2.4.2

Multiply $\frac{4}{x}$ by $\frac{x - 5}{x - 5}$ .

$\frac{5 x \cdot x}{(x - 5) x} - \frac{4 (x - 5)}{x (x - 5)} - \frac{20}{x (x - 5)} = 0$

Step 2.4.3

Reorder the factors of $(x - 5) x$ .

$\frac{5 x \cdot x}{x (x - 5)} - \frac{4 (x - 5)}{x (x - 5)} - \frac{20}{x (x - 5)} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{5 x \cdot x - 4 (x - 5)}{x (x - 5)} - \frac{20}{x (x - 5)} = 0$

Step 2.6

Combine the numerators over the common denominator.

$\frac{5 x \cdot x - 4 (x - 5) - 20}{x (x - 5)} = 0$

Step 2.7

Simplify each term.

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Step 2.7.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.7.1.1

Move $x$ .

$\frac{5 (x \cdot x) - 4 (x - 5) - 20}{x (x - 5)} = 0$

Step 2.7.1.2

Multiply $x$ by $x$ .

$\frac{5 x^{2} - 4 (x - 5) - 20}{x (x - 5)} = 0$

Step 2.7.2

Apply the distributive property.

$\frac{5 x^{2} - 4 x - 4 \cdot - 5 - 20}{x (x - 5)} = 0$

Step 2.7.3

Multiply $- 4$ by $- 5$ .

$\frac{5 x^{2} - 4 x + 20 - 20}{x (x - 5)} = 0$

Step 2.8

Combine the opposite terms in $5 x^{2} - 4 x + 20 - 20$ .

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Step 2.8.1

Subtract $20$ from $20$ .

$\frac{5 x^{2} - 4 x + 0}{x (x - 5)} = 0$

Step 2.8.2

Add $5 x^{2} - 4 x$ and $0$ .

$\frac{5 x^{2} - 4 x}{x (x - 5)} = 0$

Step 2.9

Factor $x$ out of $5 x^{2} - 4 x$ .

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Step 2.9.1

Factor $x$ out of $5 x^{2}$ .

$\frac{x (5 x) - 4 x}{x (x - 5)} = 0$

Step 2.9.2

Factor $x$ out of $- 4 x$ .

$\frac{x (5 x) + x \cdot - 4}{x (x - 5)} = 0$

Step 2.9.3

Factor $x$ out of $x (5 x) + x \cdot - 4$ .

$\frac{x (5 x - 4)}{x (x - 5)} = 0$

Step 2.10

Cancel the common factor of $x$ .

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Step 2.10.1

Cancel the common factor.

$\frac{x (5 x - 4)}{x (x - 5)} = 0$

Step 2.10.2

Rewrite the expression.

$\frac{5 x - 4}{x - 5} = 0$

Step 3

Set the numerator equal to zero.

$5 x - 4 = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Add $4$ to both sides of the equation.

$5 x = 4$

Step 4.2

Divide each term in $5 x = 4$ by $5$ and simplify.

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Step 4.2.1

Divide each term in $5 x = 4$ by $5$ .

$\frac{5 x}{5} = \frac{4}{5}$

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.

$\frac{5 x}{5} = \frac{4}{5}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{5}$