Algebra Examples

$\frac{1}{5} x + \frac{1}{2} x - \frac{1}{3} \cdot (x + 2) = \frac{1}{10}$

Step 1

Simplify $\frac{1}{5} x + \frac{1}{2} x - \frac{1}{3} \cdot (x + 2)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $\frac{1}{5}$ and $x$ .

$\frac{x}{5} + \frac{1}{2} x - \frac{1}{3} \cdot (x + 2) = \frac{1}{10}$

Step 1.1.2

Combine $\frac{1}{2}$ and $x$ .

$\frac{x}{5} + \frac{x}{2} - \frac{1}{3} \cdot (x + 2) = \frac{1}{10}$

Step 1.1.3

Apply the distributive property.

$\frac{x}{5} + \frac{x}{2} - \frac{1}{3} x - \frac{1}{3} \cdot 2 = \frac{1}{10}$

Step 1.1.4

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

$\frac{x}{5} + \frac{x}{2} - \frac{x}{3} - \frac{1}{3} \cdot 2 = \frac{1}{10}$

Step 1.1.5

Multiply $- \frac{1}{3} \cdot 2$ .

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Step 1.1.5.1

Multiply $2$ by $- 1$ .

$\frac{x}{5} + \frac{x}{2} - \frac{x}{3} - 2 (\frac{1}{3}) = \frac{1}{10}$

Step 1.1.5.2

Combine $- 2$ and $\frac{1}{3}$ .

$\frac{x}{5} + \frac{x}{2} - \frac{x}{3} + \frac{- 2}{3} = \frac{1}{10}$

Step 1.1.6

Move the negative in front of the fraction.

$\frac{x}{5} + \frac{x}{2} - \frac{x}{3} - \frac{2}{3} = \frac{1}{10}$

Step 1.2

To write $\frac{x}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x}{5} \cdot \frac{2}{2} + \frac{x}{2} - \frac{x}{3} - \frac{2}{3} = \frac{1}{10}$

Step 1.3

To write $\frac{x}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{x}{5} \cdot \frac{2}{2} + \frac{x}{2} \cdot \frac{5}{5} - \frac{x}{3} - \frac{2}{3} = \frac{1}{10}$

Step 1.4

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{x}{5}$ by $\frac{2}{2}$ .

$\frac{x \cdot 2}{5 \cdot 2} + \frac{x}{2} \cdot \frac{5}{5} - \frac{x}{3} - \frac{2}{3} = \frac{1}{10}$

Step 1.4.2

Multiply $5$ by $2$ .

$\frac{x \cdot 2}{10} + \frac{x}{2} \cdot \frac{5}{5} - \frac{x}{3} - \frac{2}{3} = \frac{1}{10}$

Step 1.4.3

Multiply $\frac{x}{2}$ by $\frac{5}{5}$ .

$\frac{x \cdot 2}{10} + \frac{x \cdot 5}{2 \cdot 5} - \frac{x}{3} - \frac{2}{3} = \frac{1}{10}$

Step 1.4.4

Multiply $2$ by $5$ .

$\frac{x \cdot 2}{10} + \frac{x \cdot 5}{10} - \frac{x}{3} - \frac{2}{3} = \frac{1}{10}$

Step 1.5

Combine the numerators over the common denominator.

$\frac{x \cdot 2 + x \cdot 5}{10} - \frac{x}{3} - \frac{2}{3} = \frac{1}{10}$

Step 1.6

Reorder $x$ and $5$ .

$\frac{2 \cdot x + 5 \cdot x}{10} - \frac{x}{3} - \frac{2}{3} = \frac{1}{10}$

Step 1.7

Add $2 \cdot x$ and $5 \cdot x$ .

$\frac{7 \cdot x}{10} - \frac{x}{3} - \frac{2}{3} = \frac{1}{10}$

Step 1.8

To write $\frac{7 x}{10}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{7 x}{10} \cdot \frac{3}{3} - \frac{x}{3} - \frac{2}{3} = \frac{1}{10}$

Step 1.9

To write $- \frac{x}{3}$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$\frac{7 x}{10} \cdot \frac{3}{3} - \frac{x}{3} \cdot \frac{10}{10} - \frac{2}{3} = \frac{1}{10}$

Step 1.10

Write each expression with a common denominator of $30$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.10.1

Multiply $\frac{7 x}{10}$ by $\frac{3}{3}$ .

$\frac{7 x \cdot 3}{10 \cdot 3} - \frac{x}{3} \cdot \frac{10}{10} - \frac{2}{3} = \frac{1}{10}$

Step 1.10.2

Multiply $10$ by $3$ .

$\frac{7 x \cdot 3}{30} - \frac{x}{3} \cdot \frac{10}{10} - \frac{2}{3} = \frac{1}{10}$

Step 1.10.3

Multiply $\frac{x}{3}$ by $\frac{10}{10}$ .

$\frac{7 x \cdot 3}{30} - \frac{x \cdot 10}{3 \cdot 10} - \frac{2}{3} = \frac{1}{10}$

Step 1.10.4

Multiply $3$ by $10$ .

$\frac{7 x \cdot 3}{30} - \frac{x \cdot 10}{30} - \frac{2}{3} = \frac{1}{10}$

Step 1.11

Combine the numerators over the common denominator.

$\frac{7 x \cdot 3 - x \cdot 10}{30} - \frac{2}{3} = \frac{1}{10}$

Step 1.12

Simplify each term.

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Step 1.12.1

Simplify the numerator.

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Step 1.12.1.1

Factor $x$ out of $7 x \cdot 3 - x \cdot 10$ .

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Step 1.12.1.1.1

Factor $x$ out of $7 x \cdot 3$ .

$\frac{x (7 \cdot 3) - x \cdot 10}{30} - \frac{2}{3} = \frac{1}{10}$

Step 1.12.1.1.2

Factor $x$ out of $- x \cdot 10$ .

$\frac{x (7 \cdot 3) + x (- 1 \cdot 10)}{30} - \frac{2}{3} = \frac{1}{10}$

Step 1.12.1.1.3

Factor $x$ out of $x (7 \cdot 3) + x (- 1 \cdot 10)$ .

$\frac{x (7 \cdot 3 - 1 \cdot 10)}{30} - \frac{2}{3} = \frac{1}{10}$

Step 1.12.1.2

Multiply $7$ by $3$ .

$\frac{x (21 - 1 \cdot 10)}{30} - \frac{2}{3} = \frac{1}{10}$

Step 1.12.1.3

Multiply $- 1$ by $10$ .

$\frac{x (21 - 10)}{30} - \frac{2}{3} = \frac{1}{10}$

Step 1.12.1.4

Subtract $10$ from $21$ .

$\frac{x \cdot 11}{30} - \frac{2}{3} = \frac{1}{10}$

Step 1.12.2

Move $11$ to the left of $x$ .

$\frac{11 x}{30} - \frac{2}{3} = \frac{1}{10}$

Step 2

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

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Step 2.1

Add $\frac{2}{3}$ to both sides of the equation.

$\frac{11 x}{30} = \frac{1}{10} + \frac{2}{3}$

Step 2.2

To write $\frac{1}{10}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{11 x}{30} = \frac{1}{10} \cdot \frac{3}{3} + \frac{2}{3}$

Step 2.3

To write $\frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$\frac{11 x}{30} = \frac{1}{10} \cdot \frac{3}{3} + \frac{2}{3} \cdot \frac{10}{10}$

Step 2.4

Write each expression with a common denominator of $30$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{1}{10}$ by $\frac{3}{3}$ .

$\frac{11 x}{30} = \frac{3}{10 \cdot 3} + \frac{2}{3} \cdot \frac{10}{10}$

Step 2.4.2

Multiply $10$ by $3$ .

$\frac{11 x}{30} = \frac{3}{30} + \frac{2}{3} \cdot \frac{10}{10}$

Step 2.4.3

Multiply $\frac{2}{3}$ by $\frac{10}{10}$ .

$\frac{11 x}{30} = \frac{3}{30} + \frac{2 \cdot 10}{3 \cdot 10}$

Step 2.4.4

Multiply $3$ by $10$ .

$\frac{11 x}{30} = \frac{3}{30} + \frac{2 \cdot 10}{30}$

Step 2.5

Combine the numerators over the common denominator.

$\frac{11 x}{30} = \frac{3 + 2 \cdot 10}{30}$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $2$ by $10$ .

$\frac{11 x}{30} = \frac{3 + 20}{30}$

Step 2.6.2

Add $3$ and $20$ .

$\frac{11 x}{30} = \frac{23}{30}$

Step 3

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

$11 x = 23$

Step 4

Divide each term in $11 x = 23$ by $11$ and simplify.

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Step 4.1

Divide each term in $11 x = 23$ by $11$ .

$\frac{11 x}{11} = \frac{23}{11}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $11$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{11 x}{11} = \frac{23}{11}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{23}{11}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{23}{11}$

Decimal Form:

$x = 2.\overline{09}$

Mixed Number Form:

$x = 2 \frac{1}{11}$