Algebra Examples

$\frac{a}{6} - 10 = \frac{a}{3}$

Step 1

Multiply both sides by $3$ .

$(\frac{a}{6} - 10) \cdot 3 = \frac{a}{3} \cdot 3$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $(\frac{a}{6} - 10) \cdot 3$ .

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Step 2.1.1.1

Apply the distributive property.

$\frac{a}{6} \cdot 3 - 10 \cdot 3 = \frac{a}{3} \cdot 3$

Step 2.1.1.2

Cancel the common factor of $3$ .

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Step 2.1.1.2.1

Factor $3$ out of $6$ .

$\frac{a}{3 (2)} \cdot 3 - 10 \cdot 3 = \frac{a}{3} \cdot 3$

Step 2.1.1.2.2

Cancel the common factor.

$\frac{a}{3 \cdot 2} \cdot 3 - 10 \cdot 3 = \frac{a}{3} \cdot 3$

Step 2.1.1.2.3

Rewrite the expression.

$\frac{a}{2} - 10 \cdot 3 = \frac{a}{3} \cdot 3$

Step 2.1.1.3

Multiply $- 10$ by $3$ .

$\frac{a}{2} - 30 = \frac{a}{3} \cdot 3$

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{a}{2} - 30 = \frac{a}{3} \cdot 3$

Step 2.2.1.2

Rewrite the expression.

$\frac{a}{2} - 30 = a$

Step 3

Solve for $a$ .

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Step 3.1

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

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Step 3.1.1

Subtract $a$ from both sides of the equation.

$\frac{a}{2} - 30 - a = 0$

Step 3.1.2

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

$\frac{a}{2} - a \cdot \frac{2}{2} - 30 = 0$

Step 3.1.3

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

$\frac{a}{2} + \frac{- a \cdot 2}{2} - 30 = 0$

Step 3.1.4

Combine the numerators over the common denominator.

$\frac{a - a \cdot 2}{2} - 30 = 0$

Step 3.1.5

Simplify each term.

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Step 3.1.5.1

Simplify the numerator.

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Step 3.1.5.1.1

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

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Step 3.1.5.1.1.1

Raise $a$ to the power of $1$ .

$\frac{a^{1} - a \cdot 2}{2} - 30 = 0$

Step 3.1.5.1.1.2

Factor $a$ out of $a^{1}$ .

$\frac{a \cdot 1 - a \cdot 2}{2} - 30 = 0$

Step 3.1.5.1.1.3

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

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

Step 3.1.5.1.1.4

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

$\frac{a (1 - 1 \cdot 2)}{2} - 30 = 0$

Step 3.1.5.1.2

Multiply $- 1$ by $2$ .

$\frac{a (1 - 2)}{2} - 30 = 0$

Step 3.1.5.1.3

Subtract $2$ from $1$ .

$\frac{a \cdot - 1}{2} - 30 = 0$

Step 3.1.5.2

Move $- 1$ to the left of $a$ .

$\frac{- 1 \cdot a}{2} - 30 = 0$

Step 3.1.5.3

Move the negative in front of the fraction.

$- \frac{a}{2} - 30 = 0$

Step 3.2

Add $30$ to both sides of the equation.

$- \frac{a}{2} = 30$

Step 3.3

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{a}{2}) = - 2 \cdot 30$

Step 3.4

Simplify both sides of the equation.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

Simplify $- 2 (- \frac{a}{2})$ .

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Step 3.4.1.1.1

Cancel the common factor of $2$ .

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Step 3.4.1.1.1.1

Move the leading negative in $- \frac{a}{2}$ into the numerator.

$- 2 \frac{- a}{2} = - 2 \cdot 30$

Step 3.4.1.1.1.2

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- a}{2} = - 2 \cdot 30$

Step 3.4.1.1.1.3

Cancel the common factor.

$2 \cdot - 1 \frac{- a}{2} = - 2 \cdot 30$

Step 3.4.1.1.1.4

Rewrite the expression.

$- - a = - 2 \cdot 30$

Step 3.4.1.1.2

Multiply.

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Step 3.4.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 a = - 2 \cdot 30$

Step 3.4.1.1.2.2

Multiply $a$ by $1$ .

$a = - 2 \cdot 30$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Multiply $- 2$ by $30$ .

$a = - 60$