Finite Math Examples

$- \frac{3}{4} w - \frac{4}{3} = - \frac{1}{6} w + \frac{1}{12} + \frac{5}{6} w$

Step 1

Since $w$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- \frac{1}{6} w + \frac{1}{12} + \frac{5}{6} w = - \frac{3}{4} w - \frac{4}{3}$

Step 2

Simplify $- \frac{1}{6} w + \frac{1}{12} + \frac{5}{6} w$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine $w$ and $\frac{1}{6}$ .

$- \frac{w}{6} + \frac{1}{12} + \frac{5}{6} w = - \frac{3}{4} w - \frac{4}{3}$

Step 2.1.2

Combine $\frac{5}{6}$ and $w$ .

$- \frac{w}{6} + \frac{1}{12} + \frac{5 w}{6} = - \frac{3}{4} w - \frac{4}{3}$

Step 2.2

Simplify terms.

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Step 2.2.1

Combine the numerators over the common denominator.

$\frac{- w + 5 w}{6} + \frac{1}{12} = - \frac{3}{4} w - \frac{4}{3}$

Step 2.2.2

Add $- w$ and $5 w$ .

$\frac{4 w}{6} + \frac{1}{12} = - \frac{3}{4} w - \frac{4}{3}$

Step 2.2.3

Cancel the common factor of $4$ and $6$ .

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Step 2.2.3.1

Factor $2$ out of $4 w$ .

$\frac{2 (2 w)}{6} + \frac{1}{12} = - \frac{3}{4} w - \frac{4}{3}$

Step 2.2.3.2

Cancel the common factors.

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Step 2.2.3.2.1

Factor $2$ out of $6$ .

$\frac{2 (2 w)}{2 (3)} + \frac{1}{12} = - \frac{3}{4} w - \frac{4}{3}$

Step 2.2.3.2.2

Cancel the common factor.

$\frac{2 (2 w)}{2 \cdot 3} + \frac{1}{12} = - \frac{3}{4} w - \frac{4}{3}$

Step 2.2.3.2.3

Rewrite the expression.

$\frac{2 w}{3} + \frac{1}{12} = - \frac{3}{4} w - \frac{4}{3}$

Step 3

Simplify each term.

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Step 3.1

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

$\frac{2 w}{3} + \frac{1}{12} = - \frac{w \cdot 3}{4} - \frac{4}{3}$

Step 3.2

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

$\frac{2 w}{3} + \frac{1}{12} = - \frac{3 w}{4} - \frac{4}{3}$

Step 4

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

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Step 4.1

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

$\frac{2 w}{3} + \frac{1}{12} + \frac{3 w}{4} = - \frac{4}{3}$

Step 4.2

To write $\frac{2 w}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{2 w}{3} \cdot \frac{4}{4} + \frac{3 w}{4} + \frac{1}{12} = - \frac{4}{3}$

Step 4.3

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

$\frac{2 w}{3} \cdot \frac{4}{4} + \frac{3 w}{4} \cdot \frac{3}{3} + \frac{1}{12} = - \frac{4}{3}$

Step 4.4

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.4.1

Multiply $\frac{2 w}{3}$ by $\frac{4}{4}$ .

$\frac{2 w \cdot 4}{3 \cdot 4} + \frac{3 w}{4} \cdot \frac{3}{3} + \frac{1}{12} = - \frac{4}{3}$

Step 4.4.2

Multiply $3$ by $4$ .

$\frac{2 w \cdot 4}{12} + \frac{3 w}{4} \cdot \frac{3}{3} + \frac{1}{12} = - \frac{4}{3}$

Step 4.4.3

Multiply $\frac{3 w}{4}$ by $\frac{3}{3}$ .

$\frac{2 w \cdot 4}{12} + \frac{3 w \cdot 3}{4 \cdot 3} + \frac{1}{12} = - \frac{4}{3}$

Step 4.4.4

Multiply $4$ by $3$ .

$\frac{2 w \cdot 4}{12} + \frac{3 w \cdot 3}{12} + \frac{1}{12} = - \frac{4}{3}$

Step 4.5

Combine the numerators over the common denominator.

$\frac{2 w \cdot 4 + 3 w \cdot 3}{12} + \frac{1}{12} = - \frac{4}{3}$

Step 4.6

Combine the numerators over the common denominator.

$\frac{2 w \cdot 4 + 3 w \cdot 3 + 1}{12} = - \frac{4}{3}$

Step 4.7

Simplify each term.

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Step 4.7.1

Multiply $4$ by $2$ .

$\frac{8 w + 3 w \cdot 3 + 1}{12} = - \frac{4}{3}$

Step 4.7.2

Multiply $3$ by $3$ .

$\frac{8 w + 9 w + 1}{12} = - \frac{4}{3}$

Step 4.8

Add $8 w$ and $9 w$ .

$\frac{17 w + 1}{12} = - \frac{4}{3}$

Step 5

Multiply both sides by $12$ .

$\frac{17 w + 1}{12} \cdot 12 = - \frac{4}{3} \cdot 12$

Step 6

Simplify.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Cancel the common factor of $12$ .

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Step 6.1.1.1

Cancel the common factor.

$\frac{17 w + 1}{12} \cdot 12 = - \frac{4}{3} \cdot 12$

Step 6.1.1.2

Rewrite the expression.

$17 w + 1 = - \frac{4}{3} \cdot 12$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- \frac{4}{3} \cdot 12$ .

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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 $- \frac{4}{3}$ into the numerator.

$17 w + 1 = \frac{- 4}{3} \cdot 12$

Step 6.2.1.1.2

Factor $3$ out of $12$ .

$17 w + 1 = \frac{- 4}{3} \cdot (3 (4))$

Step 6.2.1.1.3

Cancel the common factor.

$17 w + 1 = \frac{- 4}{3} \cdot (3 \cdot 4)$

Step 6.2.1.1.4

Rewrite the expression.

$17 w + 1 = - 4 \cdot 4$

Step 6.2.1.2

Multiply $- 4$ by $4$ .

$17 w + 1 = - 16$

Step 7

Solve for $w$ .

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Step 7.1

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

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Step 7.1.1

Subtract $1$ from both sides of the equation.

$17 w = - 16 - 1$

Step 7.1.2

Subtract $1$ from $- 16$ .

$17 w = - 17$

Step 7.2

Divide each term in $17 w = - 17$ by $17$ and simplify.

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Step 7.2.1

Divide each term in $17 w = - 17$ by $17$ .

$\frac{17 w}{17} = \frac{- 17}{17}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $17$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{17 w}{17} = \frac{- 17}{17}$

Step 7.2.2.1.2

Divide $w$ by $1$ .

$w = \frac{- 17}{17}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Divide $- 17$ by $17$ .

$w = - 1$