Algebra Examples

$\frac{1}{5} x + \frac{1}{4} = \frac{1}{2} (x + 5)$

Step 1

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

$\frac{x}{5} + \frac{1}{4} = \frac{1}{2} \cdot (x + 5)$

Step 2

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

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Step 2.1

Apply the distributive property.

$\frac{x}{5} + \frac{1}{4} = \frac{1}{2} x + \frac{1}{2} \cdot 5$

Step 2.2

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

$\frac{x}{5} + \frac{1}{4} = \frac{x}{2} + \frac{1}{2} \cdot 5$

Step 2.3

Combine $\frac{1}{2}$ and $5$ .

$\frac{x}{5} + \frac{1}{4} = \frac{x}{2} + \frac{5}{2}$

Step 3

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

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Step 3.1

Subtract $\frac{x}{2}$ from both sides of the equation.

$\frac{x}{5} + \frac{1}{4} - \frac{x}{2} = \frac{5}{2}$

Step 3.2

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

$\frac{x}{5} \cdot \frac{2}{2} - \frac{x}{2} + \frac{1}{4} = \frac{5}{2}$

Step 3.3

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

$\frac{x}{5} \cdot \frac{2}{2} - \frac{x}{2} \cdot \frac{5}{5} + \frac{1}{4} = \frac{5}{2}$

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

$\frac{x \cdot 2}{5 \cdot 2} - \frac{x}{2} \cdot \frac{5}{5} + \frac{1}{4} = \frac{5}{2}$

Step 3.4.2

Multiply $5$ by $2$ .

$\frac{x \cdot 2}{10} - \frac{x}{2} \cdot \frac{5}{5} + \frac{1}{4} = \frac{5}{2}$

Step 3.4.3

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

$\frac{x \cdot 2}{10} - \frac{x \cdot 5}{2 \cdot 5} + \frac{1}{4} = \frac{5}{2}$

Step 3.4.4

Multiply $2$ by $5$ .

$\frac{x \cdot 2}{10} - \frac{x \cdot 5}{10} + \frac{1}{4} = \frac{5}{2}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{x \cdot 2 - x \cdot 5}{10} + \frac{1}{4} = \frac{5}{2}$

Step 3.6

Simplify each term.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Factor $x$ out of $x \cdot 2 - x \cdot 5$ .

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Step 3.6.1.1.1

Factor $x$ out of $x \cdot 2$ .

$\frac{x (2) - x \cdot 5}{10} + \frac{1}{4} = \frac{5}{2}$

Step 3.6.1.1.2

Factor $x$ out of $- x \cdot 5$ .

$\frac{x (2) + x (- 1 \cdot 5)}{10} + \frac{1}{4} = \frac{5}{2}$

Step 3.6.1.1.3

Factor $x$ out of $x (2) + x (- 1 \cdot 5)$ .

$\frac{x (2 - 1 \cdot 5)}{10} + \frac{1}{4} = \frac{5}{2}$

Step 3.6.1.2

Multiply $- 1$ by $5$ .

$\frac{x (2 - 5)}{10} + \frac{1}{4} = \frac{5}{2}$

Step 3.6.1.3

Subtract $5$ from $2$ .

$\frac{x \cdot - 3}{10} + \frac{1}{4} = \frac{5}{2}$

Step 3.6.2

Move $- 3$ to the left of $x$ .

$\frac{- 3 \cdot x}{10} + \frac{1}{4} = \frac{5}{2}$

Step 3.6.3

Move the negative in front of the fraction.

$- \frac{3 x}{10} + \frac{1}{4} = \frac{5}{2}$

Step 4

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

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Step 4.1

Subtract $\frac{1}{4}$ from both sides of the equation.

$- \frac{3 x}{10} = \frac{5}{2} - \frac{1}{4}$

Step 4.2

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

$- \frac{3 x}{10} = \frac{5}{2} \cdot \frac{2}{2} - \frac{1}{4}$

Step 4.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.3.1

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

$- \frac{3 x}{10} = \frac{5 \cdot 2}{2 \cdot 2} - \frac{1}{4}$

Step 4.3.2

Multiply $2$ by $2$ .

$- \frac{3 x}{10} = \frac{5 \cdot 2}{4} - \frac{1}{4}$

Step 4.4

Combine the numerators over the common denominator.

$- \frac{3 x}{10} = \frac{5 \cdot 2 - 1}{4}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $5$ by $2$ .

$- \frac{3 x}{10} = \frac{10 - 1}{4}$

Step 4.5.2

Subtract $1$ from $10$ .

$- \frac{3 x}{10} = \frac{9}{4}$

Step 5

Multiply both sides of the equation by $- \frac{10}{3}$ .

$- \frac{10}{3} (- \frac{3 x}{10}) = - \frac{10}{3} \cdot \frac{9}{4}$

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 $- \frac{10}{3} (- \frac{3 x}{10})$ .

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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

Move the leading negative in $- \frac{10}{3}$ into the numerator.

$\frac{- 10}{3} (- \frac{3 x}{10}) = - \frac{10}{3} \cdot \frac{9}{4}$

Step 6.1.1.1.2

Move the leading negative in $- \frac{3 x}{10}$ into the numerator.

$\frac{- 10}{3} \cdot \frac{- 3 x}{10} = - \frac{10}{3} \cdot \frac{9}{4}$

Step 6.1.1.1.3

Factor $10$ out of $- 10$ .

$\frac{10 (- 1)}{3} \cdot \frac{- 3 x}{10} = - \frac{10}{3} \cdot \frac{9}{4}$

Step 6.1.1.1.4

Cancel the common factor.

$\frac{10 \cdot - 1}{3} \cdot \frac{- 3 x}{10} = - \frac{10}{3} \cdot \frac{9}{4}$

Step 6.1.1.1.5

Rewrite the expression.

$\frac{- 1}{3} (- 3 x) = - \frac{10}{3} \cdot \frac{9}{4}$

Step 6.1.1.2

Cancel the common factor of $3$ .

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Step 6.1.1.2.1

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

$\frac{- 1}{3} (3 (- x)) = - \frac{10}{3} \cdot \frac{9}{4}$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{- 1}{3} (3 (- x)) = - \frac{10}{3} \cdot \frac{9}{4}$

Step 6.1.1.2.3

Rewrite the expression.

$- - x = - \frac{10}{3} \cdot \frac{9}{4}$

Step 6.1.1.3

Multiply.

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Step 6.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{10}{3} \cdot \frac{9}{4}$

Step 6.1.1.3.2

Multiply $x$ by $1$ .

$x = - \frac{10}{3} \cdot \frac{9}{4}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- \frac{10}{3} \cdot \frac{9}{4}$ .

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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

Move the leading negative in $- \frac{10}{3}$ into the numerator.

$x = \frac{- 10}{3} \cdot \frac{9}{4}$

Step 6.2.1.1.2

Factor $2$ out of $- 10$ .

$x = \frac{2 (- 5)}{3} \cdot \frac{9}{4}$

Step 6.2.1.1.3

Factor $2$ out of $4$ .

$x = \frac{2 \cdot - 5}{3} \cdot \frac{9}{2 \cdot 2}$

Step 6.2.1.1.4

Cancel the common factor.

$x = \frac{2 \cdot - 5}{3} \cdot \frac{9}{2 \cdot 2}$

Step 6.2.1.1.5

Rewrite the expression.

$x = \frac{- 5}{3} \cdot \frac{9}{2}$

Step 6.2.1.2

Cancel the common factor of $3$ .

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Step 6.2.1.2.1

Factor $3$ out of $9$ .

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

Step 6.2.1.2.2

Cancel the common factor.

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

Step 6.2.1.2.3

Rewrite the expression.

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

Step 6.2.1.3

Combine $- 5$ and $\frac{3}{2}$ .

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

Step 6.2.1.4

Simplify the expression.

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Step 6.2.1.4.1

Multiply $- 5$ by $3$ .

$x = \frac{- 15}{2}$

Step 6.2.1.4.2

Move the negative in front of the fraction.

$x = - \frac{15}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{15}{2}$

Decimal Form:

$x = - 7.5$

Mixed Number Form:

$x = - 7 \frac{1}{2}$