Basic Math Examples

$\frac{- 3 a + 9}{15} - \frac{2 a - 5}{5} - \frac{a + 2}{10} = 0$

Step 1

Simplify $\frac{- 3 a + 9}{15} - \frac{2 a - 5}{5} - \frac{a + 2}{10}$ .

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Step 1.1

Find the common denominator.

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Step 1.1.1

Multiply $\frac{- 3 a + 9}{15}$ by $\frac{2}{2}$ .

$\frac{- 3 a + 9}{15} \cdot \frac{2}{2} - \frac{2 a - 5}{5} - \frac{a + 2}{10} = 0$

Step 1.1.2

Multiply $\frac{- 3 a + 9}{15}$ by $\frac{2}{2}$ .

$\frac{(- 3 a + 9) \cdot 2}{15 \cdot 2} - \frac{2 a - 5}{5} - \frac{a + 2}{10} = 0$

Step 1.1.3

Multiply $\frac{2 a - 5}{5}$ by $\frac{6}{6}$ .

$\frac{(- 3 a + 9) \cdot 2}{15 \cdot 2} - (\frac{2 a - 5}{5} \cdot \frac{6}{6}) - \frac{a + 2}{10} = 0$

Step 1.1.4

Multiply $\frac{2 a - 5}{5}$ by $\frac{6}{6}$ .

$\frac{(- 3 a + 9) \cdot 2}{15 \cdot 2} - \frac{(2 a - 5) \cdot 6}{5 \cdot 6} - \frac{a + 2}{10} = 0$

Step 1.1.5

Multiply $\frac{a + 2}{10}$ by $\frac{3}{3}$ .

$\frac{(- 3 a + 9) \cdot 2}{15 \cdot 2} - \frac{(2 a - 5) \cdot 6}{5 \cdot 6} - (\frac{a + 2}{10} \cdot \frac{3}{3}) = 0$

Step 1.1.6

Multiply $\frac{a + 2}{10}$ by $\frac{3}{3}$ .

$\frac{(- 3 a + 9) \cdot 2}{15 \cdot 2} - \frac{(2 a - 5) \cdot 6}{5 \cdot 6} - \frac{(a + 2) \cdot 3}{10 \cdot 3} = 0$

Step 1.1.7

Reorder the factors of $15 \cdot 2$ .

$\frac{(- 3 a + 9) \cdot 2}{2 \cdot 15} - \frac{(2 a - 5) \cdot 6}{5 \cdot 6} - \frac{(a + 2) \cdot 3}{10 \cdot 3} = 0$

Step 1.1.8

Multiply $2$ by $15$ .

$\frac{(- 3 a + 9) \cdot 2}{30} - \frac{(2 a - 5) \cdot 6}{5 \cdot 6} - \frac{(a + 2) \cdot 3}{10 \cdot 3} = 0$

Step 1.1.9

Multiply $5$ by $6$ .

$\frac{(- 3 a + 9) \cdot 2}{30} - \frac{(2 a - 5) \cdot 6}{30} - \frac{(a + 2) \cdot 3}{10 \cdot 3} = 0$

Step 1.1.10

Reorder the factors of $10 \cdot 3$ .

$\frac{(- 3 a + 9) \cdot 2}{30} - \frac{(2 a - 5) \cdot 6}{30} - \frac{(a + 2) \cdot 3}{3 \cdot 10} = 0$

Step 1.1.11

Multiply $3$ by $10$ .

$\frac{(- 3 a + 9) \cdot 2}{30} - \frac{(2 a - 5) \cdot 6}{30} - \frac{(a + 2) \cdot 3}{30} = 0$

Step 1.2

Combine the numerators over the common denominator.

$\frac{(- 3 a + 9) \cdot 2 - (2 a - 5) \cdot 6 - (a + 2) \cdot 3}{30} = 0$

Step 1.3

Simplify each term.

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Step 1.3.1

Apply the distributive property.

$\frac{- 3 a \cdot 2 + 9 \cdot 2 - (2 a - 5) \cdot 6 - (a + 2) \cdot 3}{30} = 0$

Step 1.3.2

Multiply $2$ by $- 3$ .

$\frac{- 6 a + 9 \cdot 2 - (2 a - 5) \cdot 6 - (a + 2) \cdot 3}{30} = 0$

Step 1.3.3

Multiply $9$ by $2$ .

$\frac{- 6 a + 18 - (2 a - 5) \cdot 6 - (a + 2) \cdot 3}{30} = 0$

Step 1.3.4

Apply the distributive property.

$\frac{- 6 a + 18 + (- (2 a) - - 5) \cdot 6 - (a + 2) \cdot 3}{30} = 0$

Step 1.3.5

Multiply $2$ by $- 1$ .

$\frac{- 6 a + 18 + (- 2 a - - 5) \cdot 6 - (a + 2) \cdot 3}{30} = 0$

Step 1.3.6

Multiply $- 1$ by $- 5$ .

$\frac{- 6 a + 18 + (- 2 a + 5) \cdot 6 - (a + 2) \cdot 3}{30} = 0$

Step 1.3.7

Apply the distributive property.

$\frac{- 6 a + 18 - 2 a \cdot 6 + 5 \cdot 6 - (a + 2) \cdot 3}{30} = 0$

Step 1.3.8

Multiply $6$ by $- 2$ .

$\frac{- 6 a + 18 - 12 a + 5 \cdot 6 - (a + 2) \cdot 3}{30} = 0$

Step 1.3.9

Multiply $5$ by $6$ .

$\frac{- 6 a + 18 - 12 a + 30 - (a + 2) \cdot 3}{30} = 0$

Step 1.3.10

Apply the distributive property.

$\frac{- 6 a + 18 - 12 a + 30 + (- a - 1 \cdot 2) \cdot 3}{30} = 0$

Step 1.3.11

Multiply $- 1$ by $2$ .

$\frac{- 6 a + 18 - 12 a + 30 + (- a - 2) \cdot 3}{30} = 0$

Step 1.3.12

Apply the distributive property.

$\frac{- 6 a + 18 - 12 a + 30 - a \cdot 3 - 2 \cdot 3}{30} = 0$

Step 1.3.13

Multiply $3$ by $- 1$ .

$\frac{- 6 a + 18 - 12 a + 30 - 3 a - 2 \cdot 3}{30} = 0$

Step 1.3.14

Multiply $- 2$ by $3$ .

$\frac{- 6 a + 18 - 12 a + 30 - 3 a - 6}{30} = 0$

Step 1.4

Simplify terms.

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Step 1.4.1

Subtract $12 a$ from $- 6 a$ .

$\frac{- 18 a + 18 + 30 - 3 a - 6}{30} = 0$

Step 1.4.2

Subtract $3 a$ from $- 18 a$ .

$\frac{- 21 a + 18 + 30 - 6}{30} = 0$

Step 1.4.3

Simplify by adding and subtracting.

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Step 1.4.3.1

Add $18$ and $30$ .

$\frac{- 21 a + 48 - 6}{30} = 0$

Step 1.4.3.2

Subtract $6$ from $48$ .

$\frac{- 21 a + 42}{30} = 0$

Step 1.4.4

Cancel the common factor of $- 21 a + 42$ and $30$ .

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Step 1.4.4.1

Factor $3$ out of $- 21 a$ .

$\frac{3 (- 7 a) + 42}{30} = 0$

Step 1.4.4.2

Factor $3$ out of $42$ .

$\frac{3 (- 7 a) + 3 (14)}{30} = 0$

Step 1.4.4.3

Factor $3$ out of $3 (- 7 a) + 3 (14)$ .

$\frac{3 (- 7 a + 14)}{30} = 0$

Step 1.4.4.4

Cancel the common factors.

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Step 1.4.4.4.1

Factor $3$ out of $30$ .

$\frac{3 (- 7 a + 14)}{3 (10)} = 0$

Step 1.4.4.4.2

Cancel the common factor.

$\frac{3 (- 7 a + 14)}{3 \cdot 10} = 0$

Step 1.4.4.4.3

Rewrite the expression.

$\frac{- 7 a + 14}{10} = 0$

Step 1.4.5

Factor $7$ out of $- 7 a + 14$ .

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Step 1.4.5.1

Factor $7$ out of $- 7 a$ .

$\frac{7 (- a) + 14}{10} = 0$

Step 1.4.5.2

Factor $7$ out of $14$ .

$\frac{7 (- a) + 7 (2)}{10} = 0$

Step 1.4.5.3

Factor $7$ out of $7 (- a) + 7 (2)$ .

$\frac{7 (- a + 2)}{10} = 0$

Step 1.4.6

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

$\frac{7 (- (a) + 2)}{10} = 0$

Step 1.4.7

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

$\frac{7 (- (a) - 1 (- 2))}{10} = 0$

Step 1.4.8

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

$\frac{7 (- (a - 2))}{10} = 0$

Step 1.4.9

Simplify the expression.

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Step 1.4.9.1

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

$\frac{7 (- 1 (a - 2))}{10} = 0$

Step 1.4.9.2

Move the negative in front of the fraction.

$- \frac{7 (a - 2)}{10} = 0$

Step 2

Set the numerator equal to zero.

$7 (a - 2) = 0$

Step 3

Solve the equation for $a$ .

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Step 3.1

Divide each term in $7 (a - 2) = 0$ by $7$ and simplify.

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Step 3.1.1

Divide each term in $7 (a - 2) = 0$ by $7$ .

$\frac{7 (a - 2)}{7} = \frac{0}{7}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $7$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{7 (a - 2)}{7} = \frac{0}{7}$

Step 3.1.2.1.2

Divide $a - 2$ by $1$ .

$a - 2 = \frac{0}{7}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Divide $0$ by $7$ .

$a - 2 = 0$

Step 3.2

Add $2$ to both sides of the equation.

$a = 2$