Algebra Examples

$- x + \frac{1}{2} x = \frac{67}{12} + x + \frac{5}{3} - 2$

Step 1

Simplify $- x + \frac{1}{2} x$ .

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Step 1.1

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

$- x + \frac{x}{2} = \frac{67}{12} + x + \frac{5}{3} - 2$

Step 1.2

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

$- x \cdot \frac{2}{2} + \frac{x}{2} = \frac{67}{12} + x + \frac{5}{3} - 2$

Step 1.3

Simplify terms.

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Step 1.3.1

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

$\frac{- x \cdot 2}{2} + \frac{x}{2} = \frac{67}{12} + x + \frac{5}{3} - 2$

Step 1.3.2

Combine the numerators over the common denominator.

$\frac{- x \cdot 2 + x}{2} = \frac{67}{12} + x + \frac{5}{3} - 2$

Step 1.4

Simplify the numerator.

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Step 1.4.1

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

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Step 1.4.1.1

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

$\frac{x (- 1 \cdot 2) + x}{2} = \frac{67}{12} + x + \frac{5}{3} - 2$

Step 1.4.1.2

Raise $x$ to the power of $1$ .

$\frac{x (- 1 \cdot 2) + x^{1}}{2} = \frac{67}{12} + x + \frac{5}{3} - 2$

Step 1.4.1.3

Factor $x$ out of $x^{1}$ .

$\frac{x (- 1 \cdot 2) + x \cdot 1}{2} = \frac{67}{12} + x + \frac{5}{3} - 2$

Step 1.4.1.4

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

$\frac{x (- 1 \cdot 2 + 1)}{2} = \frac{67}{12} + x + \frac{5}{3} - 2$

Step 1.4.2

Multiply $- 1$ by $2$ .

$\frac{x (- 2 + 1)}{2} = \frac{67}{12} + x + \frac{5}{3} - 2$

Step 1.4.3

Add $- 2$ and $1$ .

$\frac{x \cdot - 1}{2} = \frac{67}{12} + x + \frac{5}{3} - 2$

Step 1.5

Simplify the expression.

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Step 1.5.1

Move $- 1$ to the left of $x$ .

$\frac{- 1 \cdot x}{2} = \frac{67}{12} + x + \frac{5}{3} - 2$

Step 1.5.2

Move the negative in front of the fraction.

$- \frac{x}{2} = \frac{67}{12} + x + \frac{5}{3} - 2$

Step 2

Simplify $\frac{67}{12} + x + \frac{5}{3} - 2$ .

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Step 2.1

To write $\frac{5}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{x}{2} = x + \frac{67}{12} + \frac{5}{3} \cdot \frac{4}{4} - 2$

Step 2.2

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1

Multiply $\frac{5}{3}$ by $\frac{4}{4}$ .

$- \frac{x}{2} = x + \frac{67}{12} + \frac{5 \cdot 4}{3 \cdot 4} - 2$

Step 2.2.2

Multiply $3$ by $4$ .

$- \frac{x}{2} = x + \frac{67}{12} + \frac{5 \cdot 4}{12} - 2$

Step 2.3

Combine the numerators over the common denominator.

$- \frac{x}{2} = x + \frac{67 + 5 \cdot 4}{12} - 2$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Multiply $5$ by $4$ .

$- \frac{x}{2} = x + \frac{67 + 20}{12} - 2$

Step 2.4.2

Add $67$ and $20$ .

$- \frac{x}{2} = x + \frac{87}{12} - 2$

Step 2.5

Cancel the common factor of $87$ and $12$ .

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Step 2.5.1

Factor $3$ out of $87$ .

$- \frac{x}{2} = x + \frac{3 (29)}{12} - 2$

Step 2.5.2

Cancel the common factors.

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Step 2.5.2.1

Factor $3$ out of $12$ .

$- \frac{x}{2} = x + \frac{3 \cdot 29}{3 \cdot 4} - 2$

Step 2.5.2.2

Cancel the common factor.

$- \frac{x}{2} = x + \frac{3 \cdot 29}{3 \cdot 4} - 2$

Step 2.5.2.3

Rewrite the expression.

$- \frac{x}{2} = x + \frac{29}{4} - 2$

Step 2.6

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{x}{2} = x + \frac{29}{4} - 2 \cdot \frac{4}{4}$

Step 2.7

Combine $- 2$ and $\frac{4}{4}$ .

$- \frac{x}{2} = x + \frac{29}{4} + \frac{- 2 \cdot 4}{4}$

Step 2.8

Combine the numerators over the common denominator.

$- \frac{x}{2} = x + \frac{29 - 2 \cdot 4}{4}$

Step 2.9

Simplify the numerator.

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Step 2.9.1

Multiply $- 2$ by $4$ .

$- \frac{x}{2} = x + \frac{29 - 8}{4}$

Step 2.9.2

Subtract $8$ from $29$ .

$- \frac{x}{2} = x + \frac{21}{4}$

Step 3

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

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Step 3.1

Subtract $x$ from both sides of the equation.

$- \frac{x}{2} - x = \frac{21}{4}$

Step 3.2

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

$- \frac{x}{2} - x \cdot \frac{2}{2} = \frac{21}{4}$

Step 3.3

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

$- \frac{x}{2} + \frac{- x \cdot 2}{2} = \frac{21}{4}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{- x - x \cdot 2}{2} = \frac{21}{4}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

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

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Step 3.5.1.1

Factor $x$ out of $- x$ .

$\frac{x \cdot - 1 - x \cdot 2}{2} = \frac{21}{4}$

Step 3.5.1.2

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

$\frac{x \cdot - 1 + x (- 1 \cdot 2)}{2} = \frac{21}{4}$

Step 3.5.1.3

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

$\frac{x (- 1 - 1 \cdot 2)}{2} = \frac{21}{4}$

Step 3.5.2

Multiply $- 1$ by $2$ .

$\frac{x (- 1 - 2)}{2} = \frac{21}{4}$

Step 3.5.3

Subtract $2$ from $- 1$ .

$\frac{x \cdot - 3}{2} = \frac{21}{4}$

Step 3.6

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

$\frac{- 3 \cdot x}{2} = \frac{21}{4}$

Step 3.7

Move the negative in front of the fraction.

$- \frac{3 x}{2} = \frac{21}{4}$

Step 4

Multiply both sides of the equation by $- \frac{2}{3}$ .

$- \frac{2}{3} (- \frac{3 x}{2}) = - \frac{2}{3} \cdot \frac{21}{4}$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $- \frac{2}{3} (- \frac{3 x}{2})$ .

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Step 5.1.1.1

Cancel the common factor of $2$ .

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Step 5.1.1.1.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$\frac{- 2}{3} (- \frac{3 x}{2}) = - \frac{2}{3} \cdot \frac{21}{4}$

Step 5.1.1.1.2

Move the leading negative in $- \frac{3 x}{2}$ into the numerator.

$\frac{- 2}{3} \cdot \frac{- 3 x}{2} = - \frac{2}{3} \cdot \frac{21}{4}$

Step 5.1.1.1.3

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{3} \cdot \frac{- 3 x}{2} = - \frac{2}{3} \cdot \frac{21}{4}$

Step 5.1.1.1.4

Cancel the common factor.

$\frac{2 \cdot - 1}{3} \cdot \frac{- 3 x}{2} = - \frac{2}{3} \cdot \frac{21}{4}$

Step 5.1.1.1.5

Rewrite the expression.

$\frac{- 1}{3} (- 3 x) = - \frac{2}{3} \cdot \frac{21}{4}$

Step 5.1.1.2

Cancel the common factor of $3$ .

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Step 5.1.1.2.1

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

$\frac{- 1}{3} (3 (- x)) = - \frac{2}{3} \cdot \frac{21}{4}$

Step 5.1.1.2.2

Cancel the common factor.

$\frac{- 1}{3} (3 (- x)) = - \frac{2}{3} \cdot \frac{21}{4}$

Step 5.1.1.2.3

Rewrite the expression.

$- - x = - \frac{2}{3} \cdot \frac{21}{4}$

Step 5.1.1.3

Multiply.

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Step 5.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{2}{3} \cdot \frac{21}{4}$

Step 5.1.1.3.2

Multiply $x$ by $1$ .

$x = - \frac{2}{3} \cdot \frac{21}{4}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- \frac{2}{3} \cdot \frac{21}{4}$ .

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Step 5.2.1.1

Cancel the common factor of $2$ .

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Step 5.2.1.1.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$x = \frac{- 2}{3} \cdot \frac{21}{4}$

Step 5.2.1.1.2

Factor $2$ out of $- 2$ .

$x = \frac{2 (- 1)}{3} \cdot \frac{21}{4}$

Step 5.2.1.1.3

Factor $2$ out of $4$ .

$x = \frac{2 \cdot - 1}{3} \cdot \frac{21}{2 \cdot 2}$

Step 5.2.1.1.4

Cancel the common factor.

$x = \frac{2 \cdot - 1}{3} \cdot \frac{21}{2 \cdot 2}$

Step 5.2.1.1.5

Rewrite the expression.

$x = \frac{- 1}{3} \cdot \frac{21}{2}$

Step 5.2.1.2

Cancel the common factor of $3$ .

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Step 5.2.1.2.1

Factor $3$ out of $21$ .

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

Step 5.2.1.2.2

Cancel the common factor.

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

Step 5.2.1.2.3

Rewrite the expression.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 3.5$

Mixed Number Form:

$x = - 3 \frac{1}{2}$