Algebra Examples

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

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 2

Simplify $- x + \frac{2}{3} x - 1$ .

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Step 2.1

Combine $\frac{2}{3}$ and $x$ .

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

Step 2.2

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

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

Step 2.3

Simplify terms.

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Step 2.3.1

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

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

Step 2.3.2

Combine the numerators over the common denominator.

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

Step 2.4

Simplify each term.

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Factor $x$ out of $- x \cdot 3 + 2 x$ .

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Step 2.4.1.1.1

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

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

Step 2.4.1.1.2

Factor $x$ out of $2 x$ .

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

Step 2.4.1.1.3

Factor $x$ out of $x (- 1 \cdot 3) + x \cdot 2$ .

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

Step 2.4.1.2

Multiply $- 1$ by $3$ .

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

Step 2.4.1.3

Add $- 3$ and $2$ .

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

Step 2.4.2

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

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

Step 2.4.3

Move the negative in front of the fraction.

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

Step 3

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

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

Step 4

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

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Step 4.1

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

$- \frac{x}{3} - 1 - \frac{x}{3} = 0$

Step 4.2

Combine the numerators over the common denominator.

$- 1 + \frac{- x - x}{3} = 0$

Step 4.3

Subtract $x$ from $- x$ .

$- 1 + \frac{- 2 x}{3} = 0$

Step 4.4

Move the negative in front of the fraction.

$- 1 - \frac{2 x}{3} = 0$

Step 5

Add $1$ to both sides of the equation.

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

Step 6

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

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

Step 7

Simplify both sides of the equation.

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Step 7.1

Simplify the left side.

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Step 7.1.1

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

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Step 7.1.1.1

Cancel the common factor of $3$ .

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Step 7.1.1.1.1

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

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

Step 7.1.1.1.2

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

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

Step 7.1.1.1.3

Factor $3$ out of $- 3$ .

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

Step 7.1.1.1.4

Cancel the common factor.

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

Step 7.1.1.1.5

Rewrite the expression.

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

Step 7.1.1.2

Cancel the common factor of $2$ .

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Step 7.1.1.2.1

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

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

Step 7.1.1.2.2

Cancel the common factor.

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

Step 7.1.1.2.3

Rewrite the expression.

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

Step 7.1.1.3

Multiply.

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Step 7.1.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 7.1.1.3.2

Multiply $x$ by $1$ .

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

Step 7.2

Simplify the right side.

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Step 7.2.1

Multiply $- 1$ by $1$ .

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 1.5$

Mixed Number Form:

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