Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

$\frac{b}{5} + \frac{1}{3} + \frac{b}{2} = \frac{1}{2}$

Step 3.2

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

$\frac{b}{5} \cdot \frac{2}{2} + \frac{b}{2} + \frac{1}{3} = \frac{1}{2}$

Step 3.3

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

$\frac{b}{5} \cdot \frac{2}{2} + \frac{b}{2} \cdot \frac{5}{5} + \frac{1}{3} = \frac{1}{2}$

Step 3.4

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

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Step 3.4.1

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

$\frac{b \cdot 2}{5 \cdot 2} + \frac{b}{2} \cdot \frac{5}{5} + \frac{1}{3} = \frac{1}{2}$

Step 3.4.2

Multiply $5$ by $2$ .

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

Step 3.4.3

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

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

Step 3.4.4

Multiply $2$ by $5$ .

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Reorder $b$ and $5$ .

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

Step 3.7

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

$\frac{7 b}{10} + \frac{1}{3} = \frac{1}{2}$

Step 4

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

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Step 4.1

Subtract $\frac{1}{3}$ from both sides of the equation.

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.4.1

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

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

Step 4.4.2

Multiply $2$ by $3$ .

$\frac{7 b}{10} = \frac{3}{6} - \frac{1}{3} \cdot \frac{2}{2}$

Step 4.4.3

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

$\frac{7 b}{10} = \frac{3}{6} - \frac{2}{3 \cdot 2}$

Step 4.4.4

Multiply $3$ by $2$ .

$\frac{7 b}{10} = \frac{3}{6} - \frac{2}{6}$

Step 4.5

Combine the numerators over the common denominator.

$\frac{7 b}{10} = \frac{3 - 2}{6}$

Step 4.6

Subtract $2$ from $3$ .

$\frac{7 b}{10} = \frac{1}{6}$

Step 5

Multiply both sides of the equation by $\frac{10}{7}$ .

$\frac{10}{7} \cdot \frac{7 b}{10} = \frac{10}{7} \cdot \frac{1}{6}$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify $\frac{10}{7} \cdot \frac{7 b}{10}$ .

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Step 6.1.1.1

Cancel the common factor of $10$ .

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Step 6.1.1.1.1

Cancel the common factor.

$\frac{10}{7} \cdot \frac{7 b}{10} = \frac{10}{7} \cdot \frac{1}{6}$

Step 6.1.1.1.2

Rewrite the expression.

$\frac{1}{7} (7 b) = \frac{10}{7} \cdot \frac{1}{6}$

Step 6.1.1.2

Cancel the common factor of $7$ .

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Step 6.1.1.2.1

Factor $7$ out of $7 b$ .

$\frac{1}{7} (7 (b)) = \frac{10}{7} \cdot \frac{1}{6}$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{1}{7} (7 b) = \frac{10}{7} \cdot \frac{1}{6}$

Step 6.1.1.2.3

Rewrite the expression.

$b = \frac{10}{7} \cdot \frac{1}{6}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{10}{7} \cdot \frac{1}{6}$ .

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Step 6.2.1.1

Cancel the common factor of $2$ .

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Step 6.2.1.1.1

Factor $2$ out of $10$ .

$b = \frac{2 (5)}{7} \cdot \frac{1}{6}$

Step 6.2.1.1.2

Factor $2$ out of $6$ .

$b = \frac{2 \cdot 5}{7} \cdot \frac{1}{2 \cdot 3}$

Step 6.2.1.1.3

Cancel the common factor.

$b = \frac{2 \cdot 5}{7} \cdot \frac{1}{2 \cdot 3}$

Step 6.2.1.1.4

Rewrite the expression.

$b = \frac{5}{7} \cdot \frac{1}{3}$

Step 6.2.1.2

Multiply $\frac{5}{7}$ by $\frac{1}{3}$ .

$b = \frac{5}{7 \cdot 3}$

Step 6.2.1.3

Multiply $7$ by $3$ .

$b = \frac{5}{21}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$b = \frac{5}{21}$

Decimal Form:

$b = 0.\overline{238095}$