Algebra Examples

$\frac{3}{4} (\frac{1}{6} x + 16) = \frac{3}{7} (4 \frac{3}{8} x - 21)$

Step 1

Simplify the left side.

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Step 1.1

Simplify $\frac{3}{4} \cdot (\frac{1}{6} x + 16)$ .

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Step 1.1.1

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

$\frac{3}{4} \cdot (\frac{x}{6} + 16) = \frac{3}{7} \cdot (4 \frac{3}{8} x - 21)$

Step 1.1.2

Apply the distributive property.

$\frac{3}{4} \cdot \frac{x}{6} + \frac{3}{4} \cdot 16 = \frac{3}{7} \cdot (4 \frac{3}{8} x - 21)$

Step 1.1.3

Cancel the common factor of $3$ .

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Step 1.1.3.1

Factor $3$ out of $6$ .

$\frac{3}{4} \cdot \frac{x}{3 (2)} + \frac{3}{4} \cdot 16 = \frac{3}{7} \cdot (4 \frac{3}{8} x - 21)$

Step 1.1.3.2

Cancel the common factor.

$\frac{3}{4} \cdot \frac{x}{3 \cdot 2} + \frac{3}{4} \cdot 16 = \frac{3}{7} \cdot (4 \frac{3}{8} x - 21)$

Step 1.1.3.3

Rewrite the expression.

$\frac{1}{4} \cdot \frac{x}{2} + \frac{3}{4} \cdot 16 = \frac{3}{7} \cdot (4 \frac{3}{8} x - 21)$

Step 1.1.4

Multiply $\frac{1}{4}$ by $\frac{x}{2}$ .

$\frac{x}{4 \cdot 2} + \frac{3}{4} \cdot 16 = \frac{3}{7} \cdot (4 \frac{3}{8} x - 21)$

Step 1.1.5

Multiply $4$ by $2$ .

$\frac{x}{8} + \frac{3}{4} \cdot 16 = \frac{3}{7} \cdot (4 \frac{3}{8} x - 21)$

Step 1.1.6

Cancel the common factor of $4$ .

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Step 1.1.6.1

Factor $4$ out of $16$ .

$\frac{x}{8} + \frac{3}{4} \cdot (4 (4)) = \frac{3}{7} \cdot (4 \frac{3}{8} x - 21)$

Step 1.1.6.2

Cancel the common factor.

$\frac{x}{8} + \frac{3}{4} \cdot (4 \cdot 4) = \frac{3}{7} \cdot (4 \frac{3}{8} x - 21)$

Step 1.1.6.3

Rewrite the expression.

$\frac{x}{8} + 3 \cdot 4 = \frac{3}{7} \cdot (4 \frac{3}{8} x - 21)$

Step 1.1.7

Multiply $3$ by $4$ .

$\frac{x}{8} + 12 = \frac{3}{7} \cdot (4 \frac{3}{8} x - 21)$

Step 2

Simplify the right side.

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Step 2.1

Simplify $\frac{3}{7} \cdot (4 \frac{3}{8} x - 21)$ .

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Step 2.1.1

Convert $4 \frac{3}{8}$ to an improper fraction.

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Step 2.1.1.1

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

$\frac{x}{8} + 12 = \frac{3}{7} \cdot ((4 + \frac{3}{8}) x - 21)$

Step 2.1.1.2

Add $4$ and $\frac{3}{8}$ .

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Step 2.1.1.2.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{x}{8} + 12 = \frac{3}{7} \cdot ((4 \cdot \frac{8}{8} + \frac{3}{8}) x - 21)$

Step 2.1.1.2.2

Combine $4$ and $\frac{8}{8}$ .

$\frac{x}{8} + 12 = \frac{3}{7} \cdot ((\frac{4 \cdot 8}{8} + \frac{3}{8}) x - 21)$

Step 2.1.1.2.3

Combine the numerators over the common denominator.

$\frac{x}{8} + 12 = \frac{3}{7} \cdot (\frac{4 \cdot 8 + 3}{8} x - 21)$

Step 2.1.1.2.4

Simplify the numerator.

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Step 2.1.1.2.4.1

Multiply $4$ by $8$ .

$\frac{x}{8} + 12 = \frac{3}{7} \cdot (\frac{32 + 3}{8} x - 21)$

Step 2.1.1.2.4.2

Add $32$ and $3$ .

$\frac{x}{8} + 12 = \frac{3}{7} \cdot (\frac{35}{8} x - 21)$

Step 2.1.2

Combine $\frac{35}{8}$ and $x$ .

$\frac{x}{8} + 12 = \frac{3}{7} \cdot (\frac{35 x}{8} - 21)$

Step 2.1.3

Apply the distributive property.

$\frac{x}{8} + 12 = \frac{3}{7} \cdot \frac{35 x}{8} + \frac{3}{7} \cdot - 21$

Step 2.1.4

Cancel the common factor of $7$ .

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Step 2.1.4.1

Factor $7$ out of $35 x$ .

$\frac{x}{8} + 12 = \frac{3}{7} \cdot \frac{7 (5 x)}{8} + \frac{3}{7} \cdot - 21$

Step 2.1.4.2

Cancel the common factor.

$\frac{x}{8} + 12 = \frac{3}{7} \cdot \frac{7 (5 x)}{8} + \frac{3}{7} \cdot - 21$

Step 2.1.4.3

Rewrite the expression.

$\frac{x}{8} + 12 = 3 \frac{5 x}{8} + \frac{3}{7} \cdot - 21$

Step 2.1.5

Combine $3$ and $\frac{5 x}{8}$ .

$\frac{x}{8} + 12 = \frac{3 (5 x)}{8} + \frac{3}{7} \cdot - 21$

Step 2.1.6

Multiply $5$ by $3$ .

$\frac{x}{8} + 12 = \frac{15 x}{8} + \frac{3}{7} \cdot - 21$

Step 2.1.7

Cancel the common factor of $7$ .

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Step 2.1.7.1

Factor $7$ out of $- 21$ .

$\frac{x}{8} + 12 = \frac{15 x}{8} + \frac{3}{7} \cdot (7 (- 3))$

Step 2.1.7.2

Cancel the common factor.

$\frac{x}{8} + 12 = \frac{15 x}{8} + \frac{3}{7} \cdot (7 \cdot - 3)$

Step 2.1.7.3

Rewrite the expression.

$\frac{x}{8} + 12 = \frac{15 x}{8} + 3 \cdot - 3$

Step 2.1.8

Multiply $3$ by $- 3$ .

$\frac{x}{8} + 12 = \frac{15 x}{8} - 9$

Step 3

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

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Step 3.1

Subtract $\frac{15 x}{8}$ from both sides of the equation.

$\frac{x}{8} + 12 - \frac{15 x}{8} = - 9$

Step 3.2

Combine the numerators over the common denominator.

$12 + \frac{x - 15 x}{8} = - 9$

Step 3.3

Subtract $15 x$ from $x$ .

$12 + \frac{- 14 x}{8} = - 9$

Step 3.4

Simplify each term.

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Step 3.4.1

Cancel the common factor of $- 14$ and $8$ .

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Step 3.4.1.1

Factor $2$ out of $- 14 x$ .

$12 + \frac{2 (- 7 x)}{8} = - 9$

Step 3.4.1.2

Cancel the common factors.

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Step 3.4.1.2.1

Factor $2$ out of $8$ .

$12 + \frac{2 (- 7 x)}{2 (4)} = - 9$

Step 3.4.1.2.2

Cancel the common factor.

$12 + \frac{2 (- 7 x)}{2 \cdot 4} = - 9$

Step 3.4.1.2.3

Rewrite the expression.

$12 + \frac{- 7 x}{4} = - 9$

Step 3.4.2

Move the negative in front of the fraction.

$12 - \frac{7 x}{4} = - 9$

Step 4

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

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Step 4.1

Subtract $12$ from both sides of the equation.

$- \frac{7 x}{4} = - 9 - 12$

Step 4.2

Subtract $12$ from $- 9$ .

$- \frac{7 x}{4} = - 21$

Step 5

Multiply both sides of the equation by $- \frac{4}{7}$ .

$- \frac{4}{7} (- \frac{7 x}{4}) = - \frac{4}{7} \cdot - 21$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify $- \frac{4}{7} (- \frac{7 x}{4})$ .

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Step 6.1.1.1

Cancel the common factor of $4$ .

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Step 6.1.1.1.1

Move the leading negative in $- \frac{4}{7}$ into the numerator.

$\frac{- 4}{7} (- \frac{7 x}{4}) = - \frac{4}{7} \cdot - 21$

Step 6.1.1.1.2

Move the leading negative in $- \frac{7 x}{4}$ into the numerator.

$\frac{- 4}{7} \cdot \frac{- 7 x}{4} = - \frac{4}{7} \cdot - 21$

Step 6.1.1.1.3

Factor $4$ out of $- 4$ .

$\frac{4 (- 1)}{7} \cdot \frac{- 7 x}{4} = - \frac{4}{7} \cdot - 21$

Step 6.1.1.1.4

Cancel the common factor.

$\frac{4 \cdot - 1}{7} \cdot \frac{- 7 x}{4} = - \frac{4}{7} \cdot - 21$

Step 6.1.1.1.5

Rewrite the expression.

$\frac{- 1}{7} (- 7 x) = - \frac{4}{7} \cdot - 21$

Step 6.1.1.2

Cancel the common factor of $7$ .

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Step 6.1.1.2.1

Factor $7$ out of $- 7 x$ .

$\frac{- 1}{7} (7 (- x)) = - \frac{4}{7} \cdot - 21$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{- 1}{7} (7 (- x)) = - \frac{4}{7} \cdot - 21$

Step 6.1.1.2.3

Rewrite the expression.

$- - x = - \frac{4}{7} \cdot - 21$

Step 6.1.1.3

Multiply.

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Step 6.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{4}{7} \cdot - 21$

Step 6.1.1.3.2

Multiply $x$ by $1$ .

$x = - \frac{4}{7} \cdot - 21$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- \frac{4}{7} \cdot - 21$ .

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Step 6.2.1.1

Cancel the common factor of $7$ .

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Step 6.2.1.1.1

Move the leading negative in $- \frac{4}{7}$ into the numerator.

$x = \frac{- 4}{7} \cdot - 21$

Step 6.2.1.1.2

Factor $7$ out of $- 21$ .

$x = \frac{- 4}{7} \cdot (7 (- 3))$

Step 6.2.1.1.3

Cancel the common factor.

$x = \frac{- 4}{7} \cdot (7 \cdot - 3)$

Step 6.2.1.1.4

Rewrite the expression.

$x = - 4 \cdot - 3$

Step 6.2.1.2

Multiply $- 4$ by $- 3$ .

$x = 12$