Finite Math Examples

$\frac{3 x}{16} - \frac{1}{3} = 1 - \frac{4}{3} \cdot (x - \frac{1}{12})$

Step 1

Simplify $1 - \frac{4}{3} \cdot (x - \frac{1}{12})$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$\frac{3 x}{16} - \frac{1}{3} = 1 - \frac{4}{3} x - \frac{4}{3} (- \frac{1}{12})$

Step 1.1.2

Combine $x$ and $\frac{4}{3}$ .

$\frac{3 x}{16} - \frac{1}{3} = 1 - \frac{x \cdot 4}{3} - \frac{4}{3} (- \frac{1}{12})$

Step 1.1.3

Cancel the common factor of $4$ .

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Step 1.1.3.1

Move the leading negative in $- \frac{4}{3}$ into the numerator.

$\frac{3 x}{16} - \frac{1}{3} = 1 - \frac{x \cdot 4}{3} + \frac{- 4}{3} (- \frac{1}{12})$

Step 1.1.3.2

Move the leading negative in $- \frac{1}{12}$ into the numerator.

$\frac{3 x}{16} - \frac{1}{3} = 1 - \frac{x \cdot 4}{3} + \frac{- 4}{3} \cdot \frac{- 1}{12}$

Step 1.1.3.3

Factor $4$ out of $- 4$ .

$\frac{3 x}{16} - \frac{1}{3} = 1 - \frac{x \cdot 4}{3} + \frac{4 (- 1)}{3} \cdot \frac{- 1}{12}$

Step 1.1.3.4

Factor $4$ out of $12$ .

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

Step 1.1.3.5

Cancel the common factor.

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

Step 1.1.3.6

Rewrite the expression.

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

Step 1.1.4

Multiply $\frac{- 1}{3}$ by $\frac{- 1}{3}$ .

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

Step 1.1.5

Multiply $- 1$ by $- 1$ .

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

Step 1.1.6

Multiply $3$ by $3$ .

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

Step 1.1.7

Move $4$ to the left of $x$ .

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

Step 1.2

Simplify the expression.

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Step 1.2.1

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

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

Step 1.2.2

Combine the numerators over the common denominator.

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

Step 1.2.3

Add $9$ and $1$ .

$\frac{3 x}{16} - \frac{1}{3} = - \frac{4 x}{3} + \frac{10}{9}$

Step 2

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

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Step 2.1

Add $\frac{4 x}{3}$ to both sides of the equation.

$\frac{3 x}{16} - \frac{1}{3} + \frac{4 x}{3} = \frac{10}{9}$

Step 2.2

To write $\frac{3 x}{16}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{3 x}{16} \cdot \frac{3}{3} + \frac{4 x}{3} - \frac{1}{3} = \frac{10}{9}$

Step 2.3

To write $\frac{4 x}{3}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\frac{3 x}{16} \cdot \frac{3}{3} + \frac{4 x}{3} \cdot \frac{16}{16} - \frac{1}{3} = \frac{10}{9}$

Step 2.4

Write each expression with a common denominator of $48$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{3 x}{16}$ by $\frac{3}{3}$ .

$\frac{3 x \cdot 3}{16 \cdot 3} + \frac{4 x}{3} \cdot \frac{16}{16} - \frac{1}{3} = \frac{10}{9}$

Step 2.4.2

Multiply $16$ by $3$ .

$\frac{3 x \cdot 3}{48} + \frac{4 x}{3} \cdot \frac{16}{16} - \frac{1}{3} = \frac{10}{9}$

Step 2.4.3

Multiply $\frac{4 x}{3}$ by $\frac{16}{16}$ .

$\frac{3 x \cdot 3}{48} + \frac{4 x \cdot 16}{3 \cdot 16} - \frac{1}{3} = \frac{10}{9}$

Step 2.4.4

Multiply $3$ by $16$ .

$\frac{3 x \cdot 3}{48} + \frac{4 x \cdot 16}{48} - \frac{1}{3} = \frac{10}{9}$

Step 2.5

Combine the numerators over the common denominator.

$\frac{3 x \cdot 3 + 4 x \cdot 16}{48} - \frac{1}{3} = \frac{10}{9}$

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 $3 x \cdot 3 + 4 x \cdot 16$ .

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Step 2.6.1.1.1

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

$\frac{x (3 \cdot 3) + 4 x \cdot 16}{48} - \frac{1}{3} = \frac{10}{9}$

Step 2.6.1.1.2

Factor $x$ out of $4 x \cdot 16$ .

$\frac{x (3 \cdot 3) + x (4 \cdot 16)}{48} - \frac{1}{3} = \frac{10}{9}$

Step 2.6.1.1.3

Factor $x$ out of $x (3 \cdot 3) + x (4 \cdot 16)$ .

$\frac{x (3 \cdot 3 + 4 \cdot 16)}{48} - \frac{1}{3} = \frac{10}{9}$

Step 2.6.1.2

Multiply $3$ by $3$ .

$\frac{x (9 + 4 \cdot 16)}{48} - \frac{1}{3} = \frac{10}{9}$

Step 2.6.1.3

Multiply $4$ by $16$ .

$\frac{x (9 + 64)}{48} - \frac{1}{3} = \frac{10}{9}$

Step 2.6.1.4

Add $9$ and $64$ .

$\frac{x \cdot 73}{48} - \frac{1}{3} = \frac{10}{9}$

Step 2.6.2

Move $73$ to the left of $x$ .

$\frac{73 x}{48} - \frac{1}{3} = \frac{10}{9}$

Step 3

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

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Step 3.1

Add $\frac{1}{3}$ to both sides of the equation.

$\frac{73 x}{48} = \frac{10}{9} + \frac{1}{3}$

Step 3.2

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

$\frac{73 x}{48} = \frac{10}{9} + \frac{1}{3} \cdot \frac{3}{3}$

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 $\frac{1}{3}$ by $\frac{3}{3}$ .

$\frac{73 x}{48} = \frac{10}{9} + \frac{3}{3 \cdot 3}$

Step 3.3.2

Multiply $3$ by $3$ .

$\frac{73 x}{48} = \frac{10}{9} + \frac{3}{9}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{73 x}{48} = \frac{10 + 3}{9}$

Step 3.5

Add $10$ and $3$ .

$\frac{73 x}{48} = \frac{13}{9}$

Step 4

Multiply both sides of the equation by $\frac{48}{73}$ .

$\frac{48}{73} \cdot \frac{73 x}{48} = \frac{48}{73} \cdot \frac{13}{9}$

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 $\frac{48}{73} \cdot \frac{73 x}{48}$ .

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Step 5.1.1.1

Cancel the common factor of $48$ .

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Step 5.1.1.1.1

Cancel the common factor.

$\frac{48}{73} \cdot \frac{73 x}{48} = \frac{48}{73} \cdot \frac{13}{9}$

Step 5.1.1.1.2

Rewrite the expression.

$\frac{1}{73} (73 x) = \frac{48}{73} \cdot \frac{13}{9}$

Step 5.1.1.2

Cancel the common factor of $73$ .

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Step 5.1.1.2.1

Factor $73$ out of $73 x$ .

$\frac{1}{73} (73 (x)) = \frac{48}{73} \cdot \frac{13}{9}$

Step 5.1.1.2.2

Cancel the common factor.

$\frac{1}{73} (73 x) = \frac{48}{73} \cdot \frac{13}{9}$

Step 5.1.1.2.3

Rewrite the expression.

$x = \frac{48}{73} \cdot \frac{13}{9}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{48}{73} \cdot \frac{13}{9}$ .

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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

Factor $3$ out of $48$ .

$x = \frac{3 (16)}{73} \cdot \frac{13}{9}$

Step 5.2.1.1.2

Factor $3$ out of $9$ .

$x = \frac{3 \cdot 16}{73} \cdot \frac{13}{3 \cdot 3}$

Step 5.2.1.1.3

Cancel the common factor.

$x = \frac{3 \cdot 16}{73} \cdot \frac{13}{3 \cdot 3}$

Step 5.2.1.1.4

Rewrite the expression.

$x = \frac{16}{73} \cdot \frac{13}{3}$

Step 5.2.1.2

Multiply $\frac{16}{73}$ by $\frac{13}{3}$ .

$x = \frac{16 \cdot 13}{73 \cdot 3}$

Step 5.2.1.3

Multiply.

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Step 5.2.1.3.1

Multiply $16$ by $13$ .

$x = \frac{208}{73 \cdot 3}$

Step 5.2.1.3.2

Multiply $73$ by $3$ .

$x = \frac{208}{219}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{208}{219}$

Decimal Form:

$x = 0.94977168 \dots$