Algebra Examples

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

Step 1

Simplify $\frac{1}{4} x + \frac{2}{3} - x$ .

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Step 1.1

Combine $\frac{1}{4}$ and $x$ .

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

Step 1.2

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

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

Step 1.3

Simplify terms.

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Step 1.3.1

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

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

Step 1.3.2

Combine the numerators over the common denominator.

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

Step 1.4

Simplify each term.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Factor $x$ out of $x - x \cdot 4$ .

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Step 1.4.1.1.1

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

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

Step 1.4.1.1.2

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

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

Step 1.4.1.1.3

Factor $x$ out of $- x \cdot 4$ .

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

Step 1.4.1.1.4

Factor $x$ out of $x \cdot 1 + x (- 1 \cdot 4)$ .

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

Step 1.4.1.2

Multiply $- 1$ by $4$ .

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

Step 1.4.1.3

Subtract $4$ from $1$ .

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

Step 1.4.2

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

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

Step 1.4.3

Move the negative in front of the fraction.

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

Step 2

Combine $\frac{3}{5}$ and $x$ .

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

Step 3

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

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Step 3.1

Subtract $\frac{3 x}{5}$ from both sides of the equation.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

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

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

Step 3.4.2

Multiply $4$ by $5$ .

$- \frac{3 x \cdot 5}{20} - \frac{3 x}{5} \cdot \frac{4}{4} + \frac{2}{3} = - \frac{1}{2}$

Step 3.4.3

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

$- \frac{3 x \cdot 5}{20} - \frac{3 x \cdot 4}{5 \cdot 4} + \frac{2}{3} = - \frac{1}{2}$

Step 3.4.4

Multiply $5$ by $4$ .

$- \frac{3 x \cdot 5}{20} - \frac{3 x \cdot 4}{20} + \frac{2}{3} = - \frac{1}{2}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{- 3 x \cdot 5 - 3 x \cdot 4}{20} + \frac{2}{3} = - \frac{1}{2}$

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 $3 x$ out of $- 3 x \cdot 5 - 3 x \cdot 4$ .

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Step 3.6.1.1.1

Factor $3 x$ out of $- 3 x \cdot 5$ .

$\frac{3 x (- 1 \cdot 5) - 3 x \cdot 4}{20} + \frac{2}{3} = - \frac{1}{2}$

Step 3.6.1.1.2

Factor $3 x$ out of $- 3 x \cdot 4$ .

$\frac{3 x (- 1 \cdot 5) + 3 x (- 1 \cdot 4)}{20} + \frac{2}{3} = - \frac{1}{2}$

Step 3.6.1.1.3

Factor $3 x$ out of $3 x (- 1 \cdot 5) + 3 x (- 1 \cdot 4)$ .

$\frac{3 x (- 1 \cdot 5 - 1 \cdot 4)}{20} + \frac{2}{3} = - \frac{1}{2}$

Step 3.6.1.2

Combine exponents.

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Step 3.6.1.2.1

Multiply $- 1$ by $5$ .

$\frac{3 x (- 5 - 1 \cdot 4)}{20} + \frac{2}{3} = - \frac{1}{2}$

Step 3.6.1.2.2

Multiply $- 1$ by $4$ .

$\frac{3 x (- 5 - 4)}{20} + \frac{2}{3} = - \frac{1}{2}$

Step 3.6.1.3

Subtract $4$ from $- 5$ .

$\frac{3 x \cdot - 9}{20} + \frac{2}{3} = - \frac{1}{2}$

Step 3.6.1.4

Multiply $- 9$ by $3$ .

$\frac{- 27 x}{20} + \frac{2}{3} = - \frac{1}{2}$

Step 3.6.2

Move the negative in front of the fraction.

$- \frac{27 x}{20} + \frac{2}{3} = - \frac{1}{2}$

Step 4

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

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Step 4.1

Subtract $\frac{2}{3}$ from both sides of the equation.

$- \frac{27 x}{20} = - \frac{1}{2} - \frac{2}{3}$

Step 4.2

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

$- \frac{27 x}{20} = - \frac{1}{2} \cdot \frac{3}{3} - \frac{2}{3}$

Step 4.3

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

$- \frac{27 x}{20} = - \frac{1}{2} \cdot \frac{3}{3} - \frac{2}{3} \cdot \frac{2}{2}$

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 $\frac{3}{3}$ .

$- \frac{27 x}{20} = - \frac{3}{2 \cdot 3} - \frac{2}{3} \cdot \frac{2}{2}$

Step 4.4.2

Multiply $2$ by $3$ .

$- \frac{27 x}{20} = - \frac{3}{6} - \frac{2}{3} \cdot \frac{2}{2}$

Step 4.4.3

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

$- \frac{27 x}{20} = - \frac{3}{6} - \frac{2 \cdot 2}{3 \cdot 2}$

Step 4.4.4

Multiply $3$ by $2$ .

$- \frac{27 x}{20} = - \frac{3}{6} - \frac{2 \cdot 2}{6}$

Step 4.5

Combine the numerators over the common denominator.

$- \frac{27 x}{20} = \frac{- 3 - 2 \cdot 2}{6}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 2$ by $2$ .

$- \frac{27 x}{20} = \frac{- 3 - 4}{6}$

Step 4.6.2

Subtract $4$ from $- 3$ .

$- \frac{27 x}{20} = \frac{- 7}{6}$

Step 4.7

Move the negative in front of the fraction.

$- \frac{27 x}{20} = - \frac{7}{6}$

Step 5

Multiply both sides of the equation by $- \frac{20}{27}$ .

$- \frac{20}{27} (- \frac{27 x}{20}) = - \frac{20}{27} (- \frac{7}{6})$

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 $- \frac{20}{27} (- \frac{27 x}{20})$ .

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Step 6.1.1.1

Cancel the common factor of $20$ .

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Step 6.1.1.1.1

Move the leading negative in $- \frac{20}{27}$ into the numerator.

$\frac{- 20}{27} (- \frac{27 x}{20}) = - \frac{20}{27} (- \frac{7}{6})$

Step 6.1.1.1.2

Move the leading negative in $- \frac{27 x}{20}$ into the numerator.

$\frac{- 20}{27} \cdot \frac{- 27 x}{20} = - \frac{20}{27} (- \frac{7}{6})$

Step 6.1.1.1.3

Factor $20$ out of $- 20$ .

$\frac{20 (- 1)}{27} \cdot \frac{- 27 x}{20} = - \frac{20}{27} (- \frac{7}{6})$

Step 6.1.1.1.4

Cancel the common factor.

$\frac{20 \cdot - 1}{27} \cdot \frac{- 27 x}{20} = - \frac{20}{27} (- \frac{7}{6})$

Step 6.1.1.1.5

Rewrite the expression.

$\frac{- 1}{27} (- 27 x) = - \frac{20}{27} (- \frac{7}{6})$

Step 6.1.1.2

Cancel the common factor of $27$ .

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Step 6.1.1.2.1

Factor $27$ out of $- 27 x$ .

$\frac{- 1}{27} (27 (- x)) = - \frac{20}{27} (- \frac{7}{6})$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{- 1}{27} (27 (- x)) = - \frac{20}{27} (- \frac{7}{6})$

Step 6.1.1.2.3

Rewrite the expression.

$- - x = - \frac{20}{27} (- \frac{7}{6})$

Step 6.1.1.3

Multiply.

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Step 6.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{20}{27} (- \frac{7}{6})$

Step 6.1.1.3.2

Multiply $x$ by $1$ .

$x = - \frac{20}{27} (- \frac{7}{6})$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- \frac{20}{27} (- \frac{7}{6})$ .

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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

Move the leading negative in $- \frac{20}{27}$ into the numerator.

$x = \frac{- 20}{27} (- \frac{7}{6})$

Step 6.2.1.1.2

Move the leading negative in $- \frac{7}{6}$ into the numerator.

$x = \frac{- 20}{27} \cdot \frac{- 7}{6}$

Step 6.2.1.1.3

Factor $2$ out of $- 20$ .

$x = \frac{2 (- 10)}{27} \cdot \frac{- 7}{6}$

Step 6.2.1.1.4

Factor $2$ out of $6$ .

$x = \frac{2 \cdot - 10}{27} \cdot \frac{- 7}{2 \cdot 3}$

Step 6.2.1.1.5

Cancel the common factor.

$x = \frac{2 \cdot - 10}{27} \cdot \frac{- 7}{2 \cdot 3}$

Step 6.2.1.1.6

Rewrite the expression.

$x = \frac{- 10}{27} \cdot \frac{- 7}{3}$

Step 6.2.1.2

Multiply $\frac{- 10}{27}$ by $\frac{- 7}{3}$ .

$x = \frac{- 10 \cdot - 7}{27 \cdot 3}$

Step 6.2.1.3

Multiply.

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Step 6.2.1.3.1

Multiply $- 10$ by $- 7$ .

$x = \frac{70}{27 \cdot 3}$

Step 6.2.1.3.2

Multiply $27$ by $3$ .

$x = \frac{70}{81}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{70}{81}$

Decimal Form:

$x = 0.86419753 \dots$