Pre-Algebra Examples

$\frac{5}{9} x + \frac{2}{3} = \frac{4}{5} x - 1$

Step 1

Combine $\frac{5}{9}$ and $x$ .

$\frac{5 x}{9} + \frac{2}{3} = \frac{4}{5} x - 1$

Step 2

Combine $\frac{4}{5}$ and $x$ .

$\frac{5 x}{9} + \frac{2}{3} = \frac{4 x}{5} - 1$

Step 3

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

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Step 3.1

Subtract $\frac{4 x}{5}$ from both sides of the equation.

$\frac{5 x}{9} + \frac{2}{3} - \frac{4 x}{5} = - 1$

Step 3.2

To write $\frac{5 x}{9}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{5 x}{9} \cdot \frac{5}{5} - \frac{4 x}{5} + \frac{2}{3} = - 1$

Step 3.3

To write $- \frac{4 x}{5}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{5 x}{9} \cdot \frac{5}{5} - \frac{4 x}{5} \cdot \frac{9}{9} + \frac{2}{3} = - 1$

Step 3.4

Write each expression with a common denominator of $45$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{5 x}{9}$ by $\frac{5}{5}$ .

$\frac{5 x \cdot 5}{9 \cdot 5} - \frac{4 x}{5} \cdot \frac{9}{9} + \frac{2}{3} = - 1$

Step 3.4.2

Multiply $9$ by $5$ .

$\frac{5 x \cdot 5}{45} - \frac{4 x}{5} \cdot \frac{9}{9} + \frac{2}{3} = - 1$

Step 3.4.3

Multiply $\frac{4 x}{5}$ by $\frac{9}{9}$ .

$\frac{5 x \cdot 5}{45} - \frac{4 x \cdot 9}{5 \cdot 9} + \frac{2}{3} = - 1$

Step 3.4.4

Multiply $5$ by $9$ .

$\frac{5 x \cdot 5}{45} - \frac{4 x \cdot 9}{45} + \frac{2}{3} = - 1$

Step 3.5

Combine the numerators over the common denominator.

$\frac{5 x \cdot 5 - 4 x \cdot 9}{45} + \frac{2}{3} = - 1$

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 $5 x \cdot 5 - 4 x \cdot 9$ .

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Step 3.6.1.1.1

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

$\frac{x (5 \cdot 5) - 4 x \cdot 9}{45} + \frac{2}{3} = - 1$

Step 3.6.1.1.2

Factor $x$ out of $- 4 x \cdot 9$ .

$\frac{x (5 \cdot 5) + x (- 4 \cdot 9)}{45} + \frac{2}{3} = - 1$

Step 3.6.1.1.3

Factor $x$ out of $x (5 \cdot 5) + x (- 4 \cdot 9)$ .

$\frac{x (5 \cdot 5 - 4 \cdot 9)}{45} + \frac{2}{3} = - 1$

Step 3.6.1.2

Multiply $5$ by $5$ .

$\frac{x (25 - 4 \cdot 9)}{45} + \frac{2}{3} = - 1$

Step 3.6.1.3

Multiply $- 4$ by $9$ .

$\frac{x (25 - 36)}{45} + \frac{2}{3} = - 1$

Step 3.6.1.4

Subtract $36$ from $25$ .

$\frac{x \cdot - 11}{45} + \frac{2}{3} = - 1$

Step 3.6.2

Move $- 11$ to the left of $x$ .

$\frac{- 11 \cdot x}{45} + \frac{2}{3} = - 1$

Step 3.6.3

Move the negative in front of the fraction.

$- \frac{11 x}{45} + \frac{2}{3} = - 1$

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.

$- \frac{11 x}{45} = - 1 - \frac{2}{3}$

Step 4.2

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

$- \frac{11 x}{45} = - 1 \cdot \frac{3}{3} - \frac{2}{3}$

Step 4.3

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

$- \frac{11 x}{45} = \frac{- 1 \cdot 3}{3} - \frac{2}{3}$

Step 4.4

Combine the numerators over the common denominator.

$- \frac{11 x}{45} = \frac{- 1 \cdot 3 - 2}{3}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 1$ by $3$ .

$- \frac{11 x}{45} = \frac{- 3 - 2}{3}$

Step 4.5.2

Subtract $2$ from $- 3$ .

$- \frac{11 x}{45} = \frac{- 5}{3}$

Step 4.6

Move the negative in front of the fraction.

$- \frac{11 x}{45} = - \frac{5}{3}$

Step 5

Multiply both sides of the equation by $- \frac{45}{11}$ .

$- \frac{45}{11} (- \frac{11 x}{45}) = - \frac{45}{11} (- \frac{5}{3})$

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 $- \frac{45}{11} (- \frac{11 x}{45})$ .

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Step 6.1.1.1

Cancel the common factor of $45$ .

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Step 6.1.1.1.1

Move the leading negative in $- \frac{45}{11}$ into the numerator.

$\frac{- 45}{11} (- \frac{11 x}{45}) = - \frac{45}{11} (- \frac{5}{3})$

Step 6.1.1.1.2

Move the leading negative in $- \frac{11 x}{45}$ into the numerator.

$\frac{- 45}{11} \cdot \frac{- 11 x}{45} = - \frac{45}{11} (- \frac{5}{3})$

Step 6.1.1.1.3

Factor $45$ out of $- 45$ .

$\frac{45 (- 1)}{11} \cdot \frac{- 11 x}{45} = - \frac{45}{11} (- \frac{5}{3})$

Step 6.1.1.1.4

Cancel the common factor.

$\frac{45 \cdot - 1}{11} \cdot \frac{- 11 x}{45} = - \frac{45}{11} (- \frac{5}{3})$

Step 6.1.1.1.5

Rewrite the expression.

$\frac{- 1}{11} (- 11 x) = - \frac{45}{11} (- \frac{5}{3})$

Step 6.1.1.2

Cancel the common factor of $11$ .

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Step 6.1.1.2.1

Factor $11$ out of $- 11 x$ .

$\frac{- 1}{11} (11 (- x)) = - \frac{45}{11} (- \frac{5}{3})$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{- 1}{11} (11 (- x)) = - \frac{45}{11} (- \frac{5}{3})$

Step 6.1.1.2.3

Rewrite the expression.

$- - x = - \frac{45}{11} (- \frac{5}{3})$

Step 6.1.1.3

Multiply.

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Step 6.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{45}{11} (- \frac{5}{3})$

Step 6.1.1.3.2

Multiply $x$ by $1$ .

$x = - \frac{45}{11} (- \frac{5}{3})$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- \frac{45}{11} (- \frac{5}{3})$ .

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Step 6.2.1.1

Cancel the common factor of $3$ .

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Step 6.2.1.1.1

Move the leading negative in $- \frac{45}{11}$ into the numerator.

$x = \frac{- 45}{11} (- \frac{5}{3})$

Step 6.2.1.1.2

Move the leading negative in $- \frac{5}{3}$ into the numerator.

$x = \frac{- 45}{11} \cdot \frac{- 5}{3}$

Step 6.2.1.1.3

Factor $3$ out of $- 45$ .

$x = \frac{3 (- 15)}{11} \cdot \frac{- 5}{3}$

Step 6.2.1.1.4

Cancel the common factor.

$x = \frac{3 \cdot - 15}{11} \cdot \frac{- 5}{3}$

Step 6.2.1.1.5

Rewrite the expression.

$x = \frac{- 15}{11} \cdot - 5$

Step 6.2.1.2

Combine $\frac{- 15}{11}$ and $- 5$ .

$x = \frac{- 15 \cdot - 5}{11}$

Step 6.2.1.3

Multiply $- 15$ by $- 5$ .

$x = \frac{75}{11}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{75}{11}$

Decimal Form:

$x = 6.\overline{81}$

Mixed Number Form:

$x = 6 \frac{9}{11}$