Pre-Algebra Examples

$\frac{1}{3} k - (k + \frac{1}{3}) = \frac{1}{9} \cdot (k + 1)$

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$\frac{k}{3} - (k + \frac{1}{3}) = \frac{1}{9} \cdot (k + 1)$

Step 1.1.2

Apply the distributive property.

$\frac{k}{3} - k - \frac{1}{3} = \frac{1}{9} \cdot (k + 1)$

Step 1.2

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

$\frac{k}{3} - k \cdot \frac{3}{3} - \frac{1}{3} = \frac{1}{9} \cdot (k + 1)$

Step 1.3

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

$\frac{k}{3} + \frac{- k \cdot 3}{3} - \frac{1}{3} = \frac{1}{9} \cdot (k + 1)$

Step 1.4

Combine the numerators over the common denominator.

$\frac{k - k \cdot 3}{3} - \frac{1}{3} = \frac{1}{9} \cdot (k + 1)$

Step 1.5

Combine the numerators over the common denominator.

$\frac{k - k \cdot 3 - 1}{3} = \frac{1}{9} \cdot (k + 1)$

Step 1.6

Multiply $3$ by $- 1$ .

$\frac{k - 3 k - 1}{3} = \frac{1}{9} \cdot (k + 1)$

Step 1.7

Subtract $3 k$ from $k$ .

$\frac{- 2 k - 1}{3} = \frac{1}{9} \cdot (k + 1)$

Step 1.8

Factor $- 1$ out of $- 2 k$ .

$\frac{- (2 k) - 1}{3} = \frac{1}{9} \cdot (k + 1)$

Step 1.9

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

$\frac{- (2 k) - 1 (1)}{3} = \frac{1}{9} \cdot (k + 1)$

Step 1.10

Factor $- 1$ out of $- (2 k) - 1 (1)$ .

$\frac{- (2 k + 1)}{3} = \frac{1}{9} \cdot (k + 1)$

Step 1.11

Simplify the expression.

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Step 1.11.1

Rewrite $- (2 k + 1)$ as $- 1 (2 k + 1)$ .

$\frac{- 1 (2 k + 1)}{3} = \frac{1}{9} \cdot (k + 1)$

Step 1.11.2

Move the negative in front of the fraction.

$- \frac{2 k + 1}{3} = \frac{1}{9} \cdot (k + 1)$

Step 2

Simplify $\frac{1}{9} \cdot (k + 1)$ .

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Step 2.1

Apply the distributive property.

$- \frac{2 k + 1}{3} = \frac{1}{9} k + \frac{1}{9} \cdot 1$

Step 2.2

Combine $\frac{1}{9}$ and $k$ .

$- \frac{2 k + 1}{3} = \frac{k}{9} + \frac{1}{9} \cdot 1$

Step 2.3

Multiply $\frac{1}{9}$ by $1$ .

$- \frac{2 k + 1}{3} = \frac{k}{9} + \frac{1}{9}$

Step 3

Move all terms containing $k$ to the left side of the equation.

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Step 3.1

Subtract $\frac{k}{9}$ from both sides of the equation.

$- \frac{2 k + 1}{3} - \frac{k}{9} = \frac{1}{9}$

Step 3.2

To write $- \frac{2 k + 1}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{2 k + 1}{3} \cdot \frac{3}{3} - \frac{k}{9} = \frac{1}{9}$

Step 3.3

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

Multiply $\frac{2 k + 1}{3}$ by $\frac{3}{3}$ .

$- \frac{(2 k + 1) \cdot 3}{3 \cdot 3} - \frac{k}{9} = \frac{1}{9}$

Step 3.3.2

Multiply $3$ by $3$ .

$- \frac{(2 k + 1) \cdot 3}{9} - \frac{k}{9} = \frac{1}{9}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{- (2 k + 1) \cdot 3 - k}{9} = \frac{1}{9}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Apply the distributive property.

$\frac{(- (2 k) - 1 \cdot 1) \cdot 3 - k}{9} = \frac{1}{9}$

Step 3.5.2

Multiply $2$ by $- 1$ .

$\frac{(- 2 k - 1 \cdot 1) \cdot 3 - k}{9} = \frac{1}{9}$

Step 3.5.3

Multiply $- 1$ by $1$ .

$\frac{(- 2 k - 1) \cdot 3 - k}{9} = \frac{1}{9}$

Step 3.5.4

Apply the distributive property.

$\frac{- 2 k \cdot 3 - 1 \cdot 3 - k}{9} = \frac{1}{9}$

Step 3.5.5

Multiply $3$ by $- 2$ .

$\frac{- 6 k - 1 \cdot 3 - k}{9} = \frac{1}{9}$

Step 3.5.6

Multiply $- 1$ by $3$ .

$\frac{- 6 k - 3 - k}{9} = \frac{1}{9}$

Step 3.5.7

Subtract $k$ from $- 6 k$ .

$\frac{- 7 k - 3}{9} = \frac{1}{9}$

Step 3.6

Factor $- 1$ out of $- 7 k$ .

$\frac{- (7 k) - 3}{9} = \frac{1}{9}$

Step 3.7

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{- (7 k) - 1 (3)}{9} = \frac{1}{9}$

Step 3.8

Factor $- 1$ out of $- (7 k) - 1 (3)$ .

$\frac{- (7 k + 3)}{9} = \frac{1}{9}$

Step 3.9

Rewrite $- (7 k + 3)$ as $- 1 (7 k + 3)$ .

$\frac{- 1 (7 k + 3)}{9} = \frac{1}{9}$

Step 3.10

Move the negative in front of the fraction.

$- \frac{7 k + 3}{9} = \frac{1}{9}$

Step 4

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- (7 k + 3) = 1$

Step 5

Simplify $- (7 k + 3)$ .

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Step 5.1

Apply the distributive property.

$- (7 k) - 1 \cdot 3 = 1$

Step 5.2

Multiply.

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Step 5.2.1

Multiply $7$ by $- 1$ .

$- 7 k - 1 \cdot 3 = 1$

Step 5.2.2

Multiply $- 1$ by $3$ .

$- 7 k - 3 = 1$

Step 6

Move all terms not containing $k$ to the right side of the equation.

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Step 6.1

Add $3$ to both sides of the equation.

$- 7 k = 1 + 3$

Step 6.2

Add $1$ and $3$ .

$- 7 k = 4$

Step 7

Divide each term in $- 7 k = 4$ by $- 7$ and simplify.

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Step 7.1

Divide each term in $- 7 k = 4$ by $- 7$ .

$\frac{- 7 k}{- 7} = \frac{4}{- 7}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $- 7$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{- 7 k}{- 7} = \frac{4}{- 7}$

Step 7.2.1.2

Divide $k$ by $1$ .

$k = \frac{4}{- 7}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Move the negative in front of the fraction.

$k = - \frac{4}{7}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$k = - \frac{4}{7}$

Decimal Form:

$k = - 0.\overline{571428}$