Pre-Algebra Examples

$\frac{5 a}{6} - \frac{7}{12} + \frac{3 a}{4} = - 2 \frac{1}{6}$

Step 1

Simplify the left side.

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Step 1.1

Simplify $\frac{5 a}{6} - \frac{7}{12} + \frac{3 a}{4}$ .

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Step 1.1.1

To write $\frac{5 a}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{5 a}{6} \cdot \frac{2}{2} + \frac{3 a}{4} - \frac{7}{12} = - 2 \frac{1}{6}$

Step 1.1.2

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

$\frac{5 a}{6} \cdot \frac{2}{2} + \frac{3 a}{4} \cdot \frac{3}{3} - \frac{7}{12} = - 2 \frac{1}{6}$

Step 1.1.3

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

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Step 1.1.3.1

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

$\frac{5 a \cdot 2}{6 \cdot 2} + \frac{3 a}{4} \cdot \frac{3}{3} - \frac{7}{12} = - 2 \frac{1}{6}$

Step 1.1.3.2

Multiply $6$ by $2$ .

$\frac{5 a \cdot 2}{12} + \frac{3 a}{4} \cdot \frac{3}{3} - \frac{7}{12} = - 2 \frac{1}{6}$

Step 1.1.3.3

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

$\frac{5 a \cdot 2}{12} + \frac{3 a \cdot 3}{4 \cdot 3} - \frac{7}{12} = - 2 \frac{1}{6}$

Step 1.1.3.4

Multiply $4$ by $3$ .

$\frac{5 a \cdot 2}{12} + \frac{3 a \cdot 3}{12} - \frac{7}{12} = - 2 \frac{1}{6}$

Step 1.1.4

Simplify terms.

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Step 1.1.4.1

Combine the numerators over the common denominator.

$\frac{5 a \cdot 2 + 3 a \cdot 3}{12} - \frac{7}{12} = - 2 \frac{1}{6}$

Step 1.1.4.2

Combine the numerators over the common denominator.

$\frac{5 a \cdot 2 + 3 a \cdot 3 - 7}{12} = - 2 \frac{1}{6}$

Step 1.1.5

Simplify each term.

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Step 1.1.5.1

Multiply $2$ by $5$ .

$\frac{10 a + 3 a \cdot 3 - 7}{12} = - 2 \frac{1}{6}$

Step 1.1.5.2

Multiply $3$ by $3$ .

$\frac{10 a + 9 a - 7}{12} = - 2 \frac{1}{6}$

Step 1.1.6

Add $10 a$ and $9 a$ .

$\frac{19 a - 7}{12} = - 2 \frac{1}{6}$

Step 2

Simplify the right side.

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Step 2.1

Convert $2 \frac{1}{6}$ to an improper fraction.

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Step 2.1.1

A mixed number is an addition of its whole and fractional parts.

$\frac{19 a - 7}{12} = - (2 + \frac{1}{6})$

Step 2.1.2

Add $2$ and $\frac{1}{6}$ .

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Step 2.1.2.1

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

$\frac{19 a - 7}{12} = - (2 \cdot \frac{6}{6} + \frac{1}{6})$

Step 2.1.2.2

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

$\frac{19 a - 7}{12} = - (\frac{2 \cdot 6}{6} + \frac{1}{6})$

Step 2.1.2.3

Combine the numerators over the common denominator.

$\frac{19 a - 7}{12} = - \frac{2 \cdot 6 + 1}{6}$

Step 2.1.2.4

Simplify the numerator.

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Step 2.1.2.4.1

Multiply $2$ by $6$ .

$\frac{19 a - 7}{12} = - \frac{12 + 1}{6}$

Step 2.1.2.4.2

Add $12$ and $1$ .

$\frac{19 a - 7}{12} = - \frac{13}{6}$

Step 3

Multiply both sides by $12$ .

$\frac{19 a - 7}{12} \cdot 12 = - \frac{13}{6} \cdot 12$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $12$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{19 a - 7}{12} \cdot 12 = - \frac{13}{6} \cdot 12$

Step 4.1.1.2

Rewrite the expression.

$19 a - 7 = - \frac{13}{6} \cdot 12$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{13}{6} \cdot 12$ .

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Step 4.2.1.1

Cancel the common factor of $6$ .

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Step 4.2.1.1.1

Move the leading negative in $- \frac{13}{6}$ into the numerator.

$19 a - 7 = \frac{- 13}{6} \cdot 12$

Step 4.2.1.1.2

Factor $6$ out of $12$ .

$19 a - 7 = \frac{- 13}{6} \cdot (6 (2))$

Step 4.2.1.1.3

Cancel the common factor.

$19 a - 7 = \frac{- 13}{6} \cdot (6 \cdot 2)$

Step 4.2.1.1.4

Rewrite the expression.

$19 a - 7 = - 13 \cdot 2$

Step 4.2.1.2

Multiply $- 13$ by $2$ .

$19 a - 7 = - 26$

Step 5

Solve for $a$ .

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Step 5.1

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

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Step 5.1.1

Add $7$ to both sides of the equation.

$19 a = - 26 + 7$

Step 5.1.2

Add $- 26$ and $7$ .

$19 a = - 19$

Step 5.2

Divide each term in $19 a = - 19$ by $19$ and simplify.

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Step 5.2.1

Divide each term in $19 a = - 19$ by $19$ .

$\frac{19 a}{19} = \frac{- 19}{19}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $19$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{19 a}{19} = \frac{- 19}{19}$

Step 5.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 19}{19}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $- 19$ by $19$ .

$a = - 1$