Algebra Examples

$\frac{19}{x} = 4 - \frac{1}{x}$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $4$ from both sides of the equation.

$\frac{19}{x} - 4 = - \frac{1}{x}$

Step 1.2

Add $\frac{1}{x}$ to both sides of the equation.

$\frac{19}{x} - 4 + \frac{1}{x} = 0$

Step 2

Simplify $\frac{19}{x} - 4 + \frac{1}{x}$ .

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Step 2.1

Combine the numerators over the common denominator.

$- 4 + \frac{19 + 1}{x} = 0$

Step 2.2

Add $19$ and $1$ .

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

Step 3

Factor $4$ out of $- 4 + \frac{20}{x}$ .

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Step 3.1

Factor $4$ out of $- 4$ .

$4 (- 1) + \frac{20}{x} = 0$

Step 3.2

Factor $4$ out of $\frac{20}{x}$ .

$4 \cdot - 1 + 4 (\frac{5}{x}) = 0$

Step 3.3

Factor $4$ out of $4 \cdot - 1 + 4 \frac{5}{x}$ .

$4 (- 1 + \frac{5}{x}) = 0$

Step 4

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

$4 (- 1 \cdot \frac{x}{x} + \frac{5}{x}) = 0$

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify $4 (\frac{- x + 5}{x})$ .

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Step 7.1

Combine $4$ and $\frac{- x + 5}{x}$ .

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

Step 7.2

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

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

Step 7.3

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

$\frac{4 (- (x) - 1 (- 5))}{x} = 0$

Step 7.4

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

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

Step 7.5

Simplify the expression.

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Step 7.5.1

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

$\frac{4 (- 1 (x - 5))}{x} = 0$

Step 7.5.2

Move the negative in front of the fraction.

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

Step 8

Set the numerator equal to zero.

$4 (x - 5) = 0$

Step 9

Solve the equation for $x$ .

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Step 9.1

Divide each term in $4 (x - 5) = 0$ by $4$ and simplify.

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Step 9.1.1

Divide each term in $4 (x - 5) = 0$ by $4$ .

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

Step 9.1.2

Simplify the left side.

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Step 9.1.2.1

Cancel the common factor of $4$ .

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Step 9.1.2.1.1

Cancel the common factor.

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

Step 9.1.2.1.2

Divide $x - 5$ by $1$ .

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

Step 9.1.3

Simplify the right side.

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Step 9.1.3.1

Divide $0$ by $4$ .

$x - 5 = 0$

Step 9.2

Add $5$ to both sides of the equation.

$x = 5$