Finite Math Examples

$- \frac{5}{9} x + \frac{16}{9} = \frac{2}{6} x + \frac{4}{3}$

Step 1

Simplify each term.

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Step 1.1

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

$- \frac{x \cdot 5}{9} + \frac{16}{9} = \frac{2}{6} x + \frac{4}{3}$

Step 1.2

Move $5$ to the left of $x$ .

$- \frac{5 x}{9} + \frac{16}{9} = \frac{2}{6} x + \frac{4}{3}$

Step 2

Simplify each term.

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Step 2.1

Cancel the common factor of $2$ and $6$ .

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Step 2.1.1

Factor $2$ out of $2$ .

$- \frac{5 x}{9} + \frac{16}{9} = \frac{2 (1)}{6} x + \frac{4}{3}$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $2$ out of $6$ .

$- \frac{5 x}{9} + \frac{16}{9} = \frac{2 \cdot 1}{2 \cdot 3} x + \frac{4}{3}$

Step 2.1.2.2

Cancel the common factor.

$- \frac{5 x}{9} + \frac{16}{9} = \frac{2 \cdot 1}{2 \cdot 3} x + \frac{4}{3}$

Step 2.1.2.3

Rewrite the expression.

$- \frac{5 x}{9} + \frac{16}{9} = \frac{1}{3} x + \frac{4}{3}$

Step 2.2

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

$- \frac{5 x}{9} + \frac{16}{9} = \frac{x}{3} + \frac{4}{3}$

Step 3

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

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Step 3.1

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

$- \frac{5 x}{9} + \frac{16}{9} - \frac{x}{3} = \frac{4}{3}$

Step 3.2

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

$- \frac{5 x}{9} - \frac{x}{3} \cdot \frac{3}{3} + \frac{16}{9} = \frac{4}{3}$

Step 3.3

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

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

$- \frac{5 x}{9} - \frac{x \cdot 3}{3 \cdot 3} + \frac{16}{9} = \frac{4}{3}$

Step 3.3.2

Multiply $3$ by $3$ .

$- \frac{5 x}{9} - \frac{x \cdot 3}{9} + \frac{16}{9} = \frac{4}{3}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{- 5 x - x \cdot 3}{9} + \frac{16}{9} = \frac{4}{3}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{- 5 x - x \cdot 3 + 16}{9} = \frac{4}{3}$

Step 3.6

Multiply $3$ by $- 1$ .

$\frac{- 5 x - 3 x + 16}{9} = \frac{4}{3}$

Step 3.7

Subtract $3 x$ from $- 5 x$ .

$\frac{- 8 x + 16}{9} = \frac{4}{3}$

Step 3.8

Factor $8$ out of $- 8 x + 16$ .

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Step 3.8.1

Factor $8$ out of $- 8 x$ .

$\frac{8 (- x) + 16}{9} = \frac{4}{3}$

Step 3.8.2

Factor $8$ out of $16$ .

$\frac{8 (- x) + 8 (2)}{9} = \frac{4}{3}$

Step 3.8.3

Factor $8$ out of $8 (- x) + 8 (2)$ .

$\frac{8 (- x + 2)}{9} = \frac{4}{3}$

Step 3.9

Factor $- 1$ out of $- x$ .

$\frac{8 (- (x) + 2)}{9} = \frac{4}{3}$

Step 3.10

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{8 (- (x) - 1 (- 2))}{9} = \frac{4}{3}$

Step 3.11

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

$\frac{8 (- (x - 2))}{9} = \frac{4}{3}$

Step 3.12

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

$\frac{8 (- 1 (x - 2))}{9} = \frac{4}{3}$

Step 3.13

Move the negative in front of the fraction.

$- \frac{8 (x - 2)}{9} = \frac{4}{3}$

Step 4

Multiply both sides of the equation by $- \frac{9}{8}$ .

$- \frac{9}{8} (- \frac{8 (x - 2)}{9}) = - \frac{9}{8} \cdot \frac{4}{3}$

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 $- \frac{9}{8} (- \frac{8 (x - 2)}{9})$ .

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Step 5.1.1.1

Cancel the common factor of $9$ .

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Step 5.1.1.1.1

Move the leading negative in $- \frac{9}{8}$ into the numerator.

$\frac{- 9}{8} (- \frac{8 (x - 2)}{9}) = - \frac{9}{8} \cdot \frac{4}{3}$

Step 5.1.1.1.2

Move the leading negative in $- \frac{8 (x - 2)}{9}$ into the numerator.

$\frac{- 9}{8} \cdot \frac{- 8 (x - 2)}{9} = - \frac{9}{8} \cdot \frac{4}{3}$

Step 5.1.1.1.3

Factor $9$ out of $- 9$ .

$\frac{9 (- 1)}{8} \cdot \frac{- 8 (x - 2)}{9} = - \frac{9}{8} \cdot \frac{4}{3}$

Step 5.1.1.1.4

Cancel the common factor.

$\frac{9 \cdot - 1}{8} \cdot \frac{- 8 (x - 2)}{9} = - \frac{9}{8} \cdot \frac{4}{3}$

Step 5.1.1.1.5

Rewrite the expression.

$\frac{- 1}{8} (- 8 (x - 2)) = - \frac{9}{8} \cdot \frac{4}{3}$

Step 5.1.1.2

Cancel the common factor of $8$ .

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Step 5.1.1.2.1

Factor $8$ out of $- 8 (x - 2)$ .

$\frac{- 1}{8} (8 (- (x - 2))) = - \frac{9}{8} \cdot \frac{4}{3}$

Step 5.1.1.2.2

Cancel the common factor.

$\frac{- 1}{8} (8 (- (x - 2))) = - \frac{9}{8} \cdot \frac{4}{3}$

Step 5.1.1.2.3

Rewrite the expression.

$- - (x - 2) = - \frac{9}{8} \cdot \frac{4}{3}$

Step 5.1.1.3

Multiply.

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Step 5.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 (x - 2) = - \frac{9}{8} \cdot \frac{4}{3}$

Step 5.1.1.3.2

Multiply $x - 2$ by $1$ .

$x - 2 = - \frac{9}{8} \cdot \frac{4}{3}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- \frac{9}{8} \cdot \frac{4}{3}$ .

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Step 5.2.1.1

Cancel the common factor of $3$ .

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Step 5.2.1.1.1

Move the leading negative in $- \frac{9}{8}$ into the numerator.

$x - 2 = \frac{- 9}{8} \cdot \frac{4}{3}$

Step 5.2.1.1.2

Factor $3$ out of $- 9$ .

$x - 2 = \frac{3 (- 3)}{8} \cdot \frac{4}{3}$

Step 5.2.1.1.3

Cancel the common factor.

$x - 2 = \frac{3 \cdot - 3}{8} \cdot \frac{4}{3}$

Step 5.2.1.1.4

Rewrite the expression.

$x - 2 = \frac{- 3}{8} \cdot 4$

Step 5.2.1.2

Cancel the common factor of $4$ .

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Step 5.2.1.2.1

Factor $4$ out of $8$ .

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

Step 5.2.1.2.2

Cancel the common factor.

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

Step 5.2.1.2.3

Rewrite the expression.

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

Step 5.2.1.3

Move the negative in front of the fraction.

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

Step 6

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

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Step 6.1

Add $2$ to both sides of the equation.

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

Combine the numerators over the common denominator.

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

Step 6.5

Simplify the numerator.

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Step 6.5.1

Multiply $2$ by $2$ .

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

Step 6.5.2

Add $- 3$ and $4$ .

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.5$