Finite Math Examples

$\frac{5 x}{9} - \frac{1}{5} = 1 - \frac{3}{5} \cdot (x - \frac{1}{15})$

Step 1

Simplify $1 - \frac{3}{5} \cdot (x - \frac{1}{15})$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$\frac{5 x}{9} - \frac{1}{5} = 1 - \frac{3}{5} x - \frac{3}{5} (- \frac{1}{15})$

Step 1.1.2

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

$\frac{5 x}{9} - \frac{1}{5} = 1 - \frac{x \cdot 3}{5} - \frac{3}{5} (- \frac{1}{15})$

Step 1.1.3

Cancel the common factor of $3$ .

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Step 1.1.3.1

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

$\frac{5 x}{9} - \frac{1}{5} = 1 - \frac{x \cdot 3}{5} + \frac{- 3}{5} (- \frac{1}{15})$

Step 1.1.3.2

Move the leading negative in $- \frac{1}{15}$ into the numerator.

$\frac{5 x}{9} - \frac{1}{5} = 1 - \frac{x \cdot 3}{5} + \frac{- 3}{5} \cdot \frac{- 1}{15}$

Step 1.1.3.3

Factor $3$ out of $- 3$ .

$\frac{5 x}{9} - \frac{1}{5} = 1 - \frac{x \cdot 3}{5} + \frac{3 (- 1)}{5} \cdot \frac{- 1}{15}$

Step 1.1.3.4

Factor $3$ out of $15$ .

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

Step 1.1.3.5

Cancel the common factor.

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

Step 1.1.3.6

Rewrite the expression.

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

Step 1.1.4

Multiply $\frac{- 1}{5}$ by $\frac{- 1}{5}$ .

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

Step 1.1.5

Multiply $- 1$ by $- 1$ .

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

Step 1.1.6

Multiply $5$ by $5$ .

$\frac{5 x}{9} - \frac{1}{5} = 1 - \frac{x \cdot 3}{5} + \frac{1}{25}$

Step 1.1.7

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

$\frac{5 x}{9} - \frac{1}{5} = 1 - \frac{3 x}{5} + \frac{1}{25}$

Step 1.2

Simplify the expression.

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Step 1.2.1

Write $1$ as a fraction with a common denominator.

$\frac{5 x}{9} - \frac{1}{5} = - \frac{3 x}{5} + \frac{25}{25} + \frac{1}{25}$

Step 1.2.2

Combine the numerators over the common denominator.

$\frac{5 x}{9} - \frac{1}{5} = - \frac{3 x}{5} + \frac{25 + 1}{25}$

Step 1.2.3

Add $25$ and $1$ .

$\frac{5 x}{9} - \frac{1}{5} = - \frac{3 x}{5} + \frac{26}{25}$

Step 2

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

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Step 2.1

Add $\frac{3 x}{5}$ to both sides of the equation.

$\frac{5 x}{9} - \frac{1}{5} + \frac{3 x}{5} = \frac{26}{25}$

Step 2.2

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

$\frac{5 x}{9} \cdot \frac{5}{5} + \frac{3 x}{5} - \frac{1}{5} = \frac{26}{25}$

Step 2.3

To write $\frac{3 x}{5}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{5 x}{9} \cdot \frac{5}{5} + \frac{3 x}{5} \cdot \frac{9}{9} - \frac{1}{5} = \frac{26}{25}$

Step 2.4

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

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Step 2.4.1

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

$\frac{5 x \cdot 5}{9 \cdot 5} + \frac{3 x}{5} \cdot \frac{9}{9} - \frac{1}{5} = \frac{26}{25}$

Step 2.4.2

Multiply $9$ by $5$ .

$\frac{5 x \cdot 5}{45} + \frac{3 x}{5} \cdot \frac{9}{9} - \frac{1}{5} = \frac{26}{25}$

Step 2.4.3

Multiply $\frac{3 x}{5}$ by $\frac{9}{9}$ .

$\frac{5 x \cdot 5}{45} + \frac{3 x \cdot 9}{5 \cdot 9} - \frac{1}{5} = \frac{26}{25}$

Step 2.4.4

Multiply $5$ by $9$ .

$\frac{5 x \cdot 5}{45} + \frac{3 x \cdot 9}{45} - \frac{1}{5} = \frac{26}{25}$

Step 2.5

Combine the numerators over the common denominator.

$\frac{5 x \cdot 5 + 3 x \cdot 9}{45} - \frac{1}{5} = \frac{26}{25}$

Step 2.6

Simplify each term.

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Factor $x$ out of $5 x \cdot 5 + 3 x \cdot 9$ .

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Step 2.6.1.1.1

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

$\frac{x (5 \cdot 5) + 3 x \cdot 9}{45} - \frac{1}{5} = \frac{26}{25}$

Step 2.6.1.1.2

Factor $x$ out of $3 x \cdot 9$ .

$\frac{x (5 \cdot 5) + x (3 \cdot 9)}{45} - \frac{1}{5} = \frac{26}{25}$

Step 2.6.1.1.3

Factor $x$ out of $x (5 \cdot 5) + x (3 \cdot 9)$ .

$\frac{x (5 \cdot 5 + 3 \cdot 9)}{45} - \frac{1}{5} = \frac{26}{25}$

Step 2.6.1.2

Multiply $5$ by $5$ .

$\frac{x (25 + 3 \cdot 9)}{45} - \frac{1}{5} = \frac{26}{25}$

Step 2.6.1.3

Multiply $3$ by $9$ .

$\frac{x (25 + 27)}{45} - \frac{1}{5} = \frac{26}{25}$

Step 2.6.1.4

Add $25$ and $27$ .

$\frac{x \cdot 52}{45} - \frac{1}{5} = \frac{26}{25}$

Step 2.6.2

Move $52$ to the left of $x$ .

$\frac{52 x}{45} - \frac{1}{5} = \frac{26}{25}$

Step 3

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

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Step 3.1

Add $\frac{1}{5}$ to both sides of the equation.

$\frac{52 x}{45} = \frac{26}{25} + \frac{1}{5}$

Step 3.2

To write $\frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{52 x}{45} = \frac{26}{25} + \frac{1}{5} \cdot \frac{5}{5}$

Step 3.3

Write each expression with a common denominator of $25$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

Multiply $\frac{1}{5}$ by $\frac{5}{5}$ .

$\frac{52 x}{45} = \frac{26}{25} + \frac{5}{5 \cdot 5}$

Step 3.3.2

Multiply $5$ by $5$ .

$\frac{52 x}{45} = \frac{26}{25} + \frac{5}{25}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{52 x}{45} = \frac{26 + 5}{25}$

Step 3.5

Add $26$ and $5$ .

$\frac{52 x}{45} = \frac{31}{25}$

Step 4

Multiply both sides of the equation by $\frac{45}{52}$ .

$\frac{45}{52} \cdot \frac{52 x}{45} = \frac{45}{52} \cdot \frac{31}{25}$

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 $\frac{45}{52} \cdot \frac{52 x}{45}$ .

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Step 5.1.1.1

Cancel the common factor of $45$ .

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Step 5.1.1.1.1

Cancel the common factor.

$\frac{45}{52} \cdot \frac{52 x}{45} = \frac{45}{52} \cdot \frac{31}{25}$

Step 5.1.1.1.2

Rewrite the expression.

$\frac{1}{52} (52 x) = \frac{45}{52} \cdot \frac{31}{25}$

Step 5.1.1.2

Cancel the common factor of $52$ .

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Step 5.1.1.2.1

Factor $52$ out of $52 x$ .

$\frac{1}{52} (52 (x)) = \frac{45}{52} \cdot \frac{31}{25}$

Step 5.1.1.2.2

Cancel the common factor.

$\frac{1}{52} (52 x) = \frac{45}{52} \cdot \frac{31}{25}$

Step 5.1.1.2.3

Rewrite the expression.

$x = \frac{45}{52} \cdot \frac{31}{25}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{45}{52} \cdot \frac{31}{25}$ .

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Step 5.2.1.1

Cancel the common factor of $5$ .

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Step 5.2.1.1.1

Factor $5$ out of $45$ .

$x = \frac{5 (9)}{52} \cdot \frac{31}{25}$

Step 5.2.1.1.2

Factor $5$ out of $25$ .

$x = \frac{5 \cdot 9}{52} \cdot \frac{31}{5 \cdot 5}$

Step 5.2.1.1.3

Cancel the common factor.

$x = \frac{5 \cdot 9}{52} \cdot \frac{31}{5 \cdot 5}$

Step 5.2.1.1.4

Rewrite the expression.

$x = \frac{9}{52} \cdot \frac{31}{5}$

Step 5.2.1.2

Multiply $\frac{9}{52}$ by $\frac{31}{5}$ .

$x = \frac{9 \cdot 31}{52 \cdot 5}$

Step 5.2.1.3

Multiply.

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Step 5.2.1.3.1

Multiply $9$ by $31$ .

$x = \frac{279}{52 \cdot 5}$

Step 5.2.1.3.2

Multiply $52$ by $5$ .

$x = \frac{279}{260}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{279}{260}$

Decimal Form:

$x = 1.07307692 \dots$

Mixed Number Form:

$x = 1 \frac{19}{260}$