Linear Algebra Examples

$\frac{x + 6}{12} = \frac{1}{6} + \frac{x - 5}{7}$

Step 1

Multiply both sides by $12$ .

$\frac{x + 6}{12} \cdot 12 = (\frac{1}{6} + \frac{x - 5}{7}) \cdot 12$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $12$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{x + 6}{12} \cdot 12 = (\frac{1}{6} + \frac{x - 5}{7}) \cdot 12$

Step 2.1.1.2

Rewrite the expression.

$x + 6 = (\frac{1}{6} + \frac{x - 5}{7}) \cdot 12$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $(\frac{1}{6} + \frac{x - 5}{7}) \cdot 12$ .

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Step 2.2.1.1

To write $\frac{1}{6}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$x + 6 = (\frac{1}{6} \cdot \frac{7}{7} + \frac{x - 5}{7}) \cdot 12$

Step 2.2.1.2

To write $\frac{x - 5}{7}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x + 6 = (\frac{1}{6} \cdot \frac{7}{7} + \frac{x - 5}{7} \cdot \frac{6}{6}) \cdot 12$

Step 2.2.1.3

Write each expression with a common denominator of $42$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.3.1

Multiply $\frac{1}{6}$ by $\frac{7}{7}$ .

$x + 6 = (\frac{7}{6 \cdot 7} + \frac{x - 5}{7} \cdot \frac{6}{6}) \cdot 12$

Step 2.2.1.3.2

Multiply $6$ by $7$ .

$x + 6 = (\frac{7}{42} + \frac{x - 5}{7} \cdot \frac{6}{6}) \cdot 12$

Step 2.2.1.3.3

Multiply $\frac{x - 5}{7}$ by $\frac{6}{6}$ .

$x + 6 = (\frac{7}{42} + \frac{(x - 5) \cdot 6}{7 \cdot 6}) \cdot 12$

Step 2.2.1.3.4

Multiply $7$ by $6$ .

$x + 6 = (\frac{7}{42} + \frac{(x - 5) \cdot 6}{42}) \cdot 12$

Step 2.2.1.4

Combine the numerators over the common denominator.

$x + 6 = \frac{7 + (x - 5) \cdot 6}{42} \cdot 12$

Step 2.2.1.5

Simplify the numerator.

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Step 2.2.1.5.1

Apply the distributive property.

$x + 6 = \frac{7 + x \cdot 6 - 5 \cdot 6}{42} \cdot 12$

Step 2.2.1.5.2

Move $6$ to the left of $x$ .

$x + 6 = \frac{7 + 6 \cdot x - 5 \cdot 6}{42} \cdot 12$

Step 2.2.1.5.3

Multiply $- 5$ by $6$ .

$x + 6 = \frac{7 + 6 \cdot x - 30}{42} \cdot 12$

Step 2.2.1.5.4

Subtract $30$ from $7$ .

$x + 6 = \frac{6 x - 23}{42} \cdot 12$

Step 2.2.1.6

Simplify terms.

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Step 2.2.1.6.1

Cancel the common factor of $6$ .

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Step 2.2.1.6.1.1

Factor $6$ out of $42$ .

$x + 6 = \frac{6 x - 23}{6 (7)} \cdot 12$

Step 2.2.1.6.1.2

Factor $6$ out of $12$ .

$x + 6 = \frac{6 x - 23}{6 \cdot 7} \cdot (6 \cdot 2)$

Step 2.2.1.6.1.3

Cancel the common factor.

$x + 6 = \frac{6 x - 23}{6 \cdot 7} \cdot (6 \cdot 2)$

Step 2.2.1.6.1.4

Rewrite the expression.

$x + 6 = \frac{6 x - 23}{7} \cdot 2$

Step 2.2.1.6.2

Combine $\frac{6 x - 23}{7}$ and $2$ .

$x + 6 = \frac{(6 x - 23) \cdot 2}{7}$

Step 2.2.1.6.3

Move $2$ to the left of $6 x - 23$ .

$x + 6 = \frac{2 (6 x - 23)}{7}$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides by $7$ .

$(x + 6) \cdot 7 = \frac{2 (6 x - 23)}{7} \cdot 7$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $(x + 6) \cdot 7$ .

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Step 3.2.1.1.1

Apply the distributive property.

$x \cdot 7 + 6 \cdot 7 = \frac{2 (6 x - 23)}{7} \cdot 7$

Step 3.2.1.1.2

Simplify the expression.

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Step 3.2.1.1.2.1

Move $7$ to the left of $x$ .

$7 \cdot x + 6 \cdot 7 = \frac{2 (6 x - 23)}{7} \cdot 7$

Step 3.2.1.1.2.2

Multiply $6$ by $7$ .

$7 x + 42 = \frac{2 (6 x - 23)}{7} \cdot 7$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $\frac{2 (6 x - 23)}{7} \cdot 7$ .

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Step 3.2.2.1.1

Cancel the common factor of $7$ .

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Step 3.2.2.1.1.1

Cancel the common factor.

$7 x + 42 = \frac{2 (6 x - 23)}{7} \cdot 7$

Step 3.2.2.1.1.2

Rewrite the expression.

$7 x + 42 = 2 (6 x - 23)$

Step 3.2.2.1.2

Apply the distributive property.

$7 x + 42 = 2 (6 x) + 2 \cdot - 23$

Step 3.2.2.1.3

Multiply.

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Step 3.2.2.1.3.1

Multiply $6$ by $2$ .

$7 x + 42 = 12 x + 2 \cdot - 23$

Step 3.2.2.1.3.2

Multiply $2$ by $- 23$ .

$7 x + 42 = 12 x - 46$

Step 3.3

Solve for $x$ .

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Step 3.3.1

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

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Step 3.3.1.1

Subtract $12 x$ from both sides of the equation.

$7 x + 42 - 12 x = - 46$

Step 3.3.1.2

Subtract $12 x$ from $7 x$ .

$- 5 x + 42 = - 46$

Step 3.3.2

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

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Step 3.3.2.1

Subtract $42$ from both sides of the equation.

$- 5 x = - 46 - 42$

Step 3.3.2.2

Subtract $42$ from $- 46$ .

$- 5 x = - 88$

Step 3.3.3

Divide each term in $- 5 x = - 88$ by $- 5$ and simplify.

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Step 3.3.3.1

Divide each term in $- 5 x = - 88$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{- 88}{- 5}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 5$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{- 88}{- 5}$

Step 3.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 88}{- 5}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Dividing two negative values results in a positive value.

$x = \frac{88}{5}$

Step 4

The domain is the set of all valid $x$ values.

${x | x = \frac{88}{5}}$

Step 5