Finite Math Examples

$\frac{1}{6} a = \frac{2}{3} \cdot (3 - a) + \frac{a - 8}{6}$

Step 1

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

$\frac{2}{3} \cdot (3 - a) + \frac{a - 8}{6} = \frac{1}{6} a$

Step 2

Simplify $\frac{2}{3} \cdot (3 - a) + \frac{a - 8}{6}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$\frac{2}{3} \cdot 3 + \frac{2}{3} (- a) + \frac{a - 8}{6} = \frac{1}{6} a$

Step 2.1.2

Cancel the common factor of $3$ .

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Step 2.1.2.1

Cancel the common factor.

$\frac{2}{3} \cdot 3 + \frac{2}{3} (- a) + \frac{a - 8}{6} = \frac{1}{6} a$

Step 2.1.2.2

Rewrite the expression.

$2 + \frac{2}{3} (- a) + \frac{a - 8}{6} = \frac{1}{6} a$

Step 2.1.3

Combine $\frac{2}{3}$ and $a$ .

$2 - \frac{2 a}{3} + \frac{a - 8}{6} = \frac{1}{6} a$

Step 2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{2 a}{3} + 2 \cdot \frac{6}{6} + \frac{a - 8}{6} = \frac{1}{6} a$

Step 2.3

Combine $2$ and $\frac{6}{6}$ .

$- \frac{2 a}{3} + \frac{2 \cdot 6}{6} + \frac{a - 8}{6} = \frac{1}{6} a$

Step 2.4

Combine the numerators over the common denominator.

$- \frac{2 a}{3} + \frac{2 \cdot 6 + a - 8}{6} = \frac{1}{6} a$

Step 2.5

Simplify the expression.

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Step 2.5.1

Multiply $2$ by $6$ .

$- \frac{2 a}{3} + \frac{12 + a - 8}{6} = \frac{1}{6} a$

Step 2.5.2

Subtract $8$ from $12$ .

$- \frac{2 a}{3} + \frac{a + 4}{6} = \frac{1}{6} a$

Step 2.6

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

$- \frac{2 a}{3} \cdot \frac{2}{2} + \frac{a + 4}{6} = \frac{1}{6} a$

Step 2.7

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.7.1

Multiply $\frac{2 a}{3}$ by $\frac{2}{2}$ .

$- \frac{2 a \cdot 2}{3 \cdot 2} + \frac{a + 4}{6} = \frac{1}{6} a$

Step 2.7.2

Multiply $3$ by $2$ .

$- \frac{2 a \cdot 2}{6} + \frac{a + 4}{6} = \frac{1}{6} a$

Step 2.8

Combine the numerators over the common denominator.

$\frac{- 2 a \cdot 2 + a + 4}{6} = \frac{1}{6} a$

Step 2.9

Simplify the numerator.

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Step 2.9.1

Multiply $2$ by $- 2$ .

$\frac{- 4 a + a + 4}{6} = \frac{1}{6} a$

Step 2.9.2

Add $- 4 a$ and $a$ .

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

Step 2.10

Simplify with factoring out.

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Step 2.10.1

Factor $- 1$ out of $- 3 a$ .

$\frac{- (3 a) + 4}{6} = \frac{1}{6} a$

Step 2.10.2

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{- (3 a) - 1 (- 4)}{6} = \frac{1}{6} a$

Step 2.10.3

Factor $- 1$ out of $- (3 a) - 1 (- 4)$ .

$\frac{- (3 a - 4)}{6} = \frac{1}{6} a$

Step 2.10.4

Simplify the expression.

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Step 2.10.4.1

Rewrite $- (3 a - 4)$ as $- 1 (3 a - 4)$ .

$\frac{- 1 (3 a - 4)}{6} = \frac{1}{6} a$

Step 2.10.4.2

Move the negative in front of the fraction.

$- \frac{3 a - 4}{6} = \frac{1}{6} a$

Step 3

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

$- \frac{3 a - 4}{6} = \frac{a}{6}$

Step 4

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

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Step 4.1

Subtract $\frac{a}{6}$ from both sides of the equation.

$- \frac{3 a - 4}{6} - \frac{a}{6} = 0$

Step 4.2

Combine the numerators over the common denominator.

$\frac{- (3 a - 4) - a}{6} = 0$

Step 4.3

Simplify each term.

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Step 4.3.1

Apply the distributive property.

$\frac{- (3 a) - - 4 - a}{6} = 0$

Step 4.3.2

Multiply $3$ by $- 1$ .

$\frac{- 3 a - - 4 - a}{6} = 0$

Step 4.3.3

Multiply $- 1$ by $- 4$ .

$\frac{- 3 a + 4 - a}{6} = 0$

Step 4.4

Subtract $a$ from $- 3 a$ .

$\frac{- 4 a + 4}{6} = 0$

Step 4.5

Cancel the common factor of $- 4 a + 4$ and $6$ .

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Step 4.5.1

Factor $2$ out of $- 4 a$ .

$\frac{2 (- 2 a) + 4}{6} = 0$

Step 4.5.2

Factor $2$ out of $4$ .

$\frac{2 (- 2 a) + 2 (2)}{6} = 0$

Step 4.5.3

Factor $2$ out of $2 (- 2 a) + 2 (2)$ .

$\frac{2 (- 2 a + 2)}{6} = 0$

Step 4.5.4

Cancel the common factors.

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Step 4.5.4.1

Factor $2$ out of $6$ .

$\frac{2 (- 2 a + 2)}{2 (3)} = 0$

Step 4.5.4.2

Cancel the common factor.

$\frac{2 (- 2 a + 2)}{2 \cdot 3} = 0$

Step 4.5.4.3

Rewrite the expression.

$\frac{- 2 a + 2}{3} = 0$

Step 4.6

Factor $2$ out of $- 2 a + 2$ .

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Step 4.6.1

Factor $2$ out of $- 2 a$ .

$\frac{2 (- a) + 2}{3} = 0$

Step 4.6.2

Factor $2$ out of $2$ .

$\frac{2 (- a) + 2 (1)}{3} = 0$

Step 4.6.3

Factor $2$ out of $2 (- a) + 2 (1)$ .

$\frac{2 (- a + 1)}{3} = 0$

Step 4.7

Factor $- 1$ out of $- a$ .

$\frac{2 (- (a) + 1)}{3} = 0$

Step 4.8

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

$\frac{2 (- (a) - 1 (- 1))}{3} = 0$

Step 4.9

Factor $- 1$ out of $- (a) - 1 (- 1)$ .

$\frac{2 (- (a - 1))}{3} = 0$

Step 4.10

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

$\frac{2 (- 1 (a - 1))}{3} = 0$

Step 4.11

Move the negative in front of the fraction.

$- \frac{2 (a - 1)}{3} = 0$

Step 5

Set the numerator equal to zero.

$2 (a - 1) = 0$

Step 6

Solve the equation for $a$ .

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Step 6.1

Divide each term in $2 (a - 1) = 0$ by $2$ and simplify.

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Step 6.1.1

Divide each term in $2 (a - 1) = 0$ by $2$ .

$\frac{2 (a - 1)}{2} = \frac{0}{2}$

Step 6.1.2

Simplify the left side.

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Step 6.1.2.1

Cancel the common factor of $2$ .

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Step 6.1.2.1.1

Cancel the common factor.

$\frac{2 (a - 1)}{2} = \frac{0}{2}$

Step 6.1.2.1.2

Divide $a - 1$ by $1$ .

$a - 1 = \frac{0}{2}$

Step 6.1.3

Simplify the right side.

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Step 6.1.3.1

Divide $0$ by $2$ .

$a - 1 = 0$

Step 6.2

Add $1$ to both sides of the equation.

$a = 1$

Step 7