Algebra Examples

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

Step 1

Simplify $\frac{1}{3} \cdot (x - k)$ .

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Step 1.1

Rewrite.

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

Step 1.2

Simplify by adding zeros.

$\frac{1}{3} \cdot (x - k) = k x$

Step 1.3

Apply the distributive property.

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

Step 1.4

Combine $\frac{1}{3}$ and $x$ .

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

Step 1.5

Combine $\frac{1}{3}$ and $k$ .

$\frac{x}{3} - \frac{k}{3} = k x$

Step 2

Subtract $k x$ from both sides of the equation.

$\frac{x}{3} - \frac{k}{3} - k x = 0$

Step 3

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

$- \frac{k}{3} - k x = - \frac{x}{3}$

Step 4

Factor $k$ out of $- \frac{k}{3} - k x$ .

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Step 4.1

Factor $k$ out of $- \frac{k}{3}$ .

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

Step 4.2

Factor $k$ out of $- k x$ .

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

Step 4.3

Factor $k$ out of $k (- \frac{1}{3}) + k (- 1 x)$ .

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

Step 5

Rewrite $- 1 x$ as $- x$ .

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

Step 6

Divide each term in $k (- \frac{1}{3} - x) = - \frac{x}{3}$ by $- \frac{1}{3} - x$ and simplify.

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Step 6.1

Divide each term in $k (- \frac{1}{3} - x) = - \frac{x}{3}$ by $- \frac{1}{3} - x$ .

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

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $- \frac{1}{3} - x$ .

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Step 6.2.1.1

Cancel the common factor.

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

Step 6.2.1.2

Divide $k$ by $1$ .

$k = \frac{- \frac{x}{3}}{- \frac{1}{3} - x}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Multiply the numerator by the reciprocal of the denominator.

$k = - \frac{x}{3} \cdot \frac{1}{- \frac{1}{3} - x}$

Step 6.3.2

Simplify the denominator.

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Step 6.3.2.1

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

$k = - \frac{x}{3} \cdot \frac{1}{- \frac{1}{3} - x \cdot \frac{3}{3}}$

Step 6.3.2.2

Combine $- x$ and $\frac{3}{3}$ .

$k = - \frac{x}{3} \cdot \frac{1}{- \frac{1}{3} + \frac{- x \cdot 3}{3}}$

Step 6.3.2.3

Combine the numerators over the common denominator.

$k = - \frac{x}{3} \cdot \frac{1}{\frac{- 1 - x \cdot 3}{3}}$

Step 6.3.2.4

Multiply $3$ by $- 1$ .

$k = - \frac{x}{3} \cdot \frac{1}{\frac{- 1 - 3 x}{3}}$

Step 6.3.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.3.4

Multiply $\frac{3}{- 1 - 3 x}$ by $1$ .

$k = - \frac{x}{3} \cdot \frac{3}{- 1 - 3 x}$

Step 6.3.5

Cancel the common factor of $3$ .

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Step 6.3.5.1

Move the leading negative in $- \frac{x}{3}$ into the numerator.

$k = \frac{- x}{3} \cdot \frac{3}{- 1 - 3 x}$

Step 6.3.5.2

Cancel the common factor.

$k = \frac{- x}{3} \cdot \frac{3}{- 1 - 3 x}$

Step 6.3.5.3

Rewrite the expression.

$k = - x \frac{1}{- 1 - 3 x}$

Step 6.3.6

Combine $\frac{1}{- 1 - 3 x}$ and $x$ .

$k = - \frac{x}{- 1 - 3 x}$

Step 6.3.7

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

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

Step 6.3.8

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

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

Step 6.3.9

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

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

Step 6.3.10

Simplify the expression.

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Step 6.3.10.1

Move the negative in front of the fraction.

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

Step 6.3.10.2

Multiply $- 1$ by $- 1$ .

$k = 1 \frac{x}{1 + 3 x}$

Step 6.3.10.3

Multiply $\frac{x}{1 + 3 x}$ by $1$ .

$k = \frac{x}{1 + 3 x}$