Trigonometry Examples

$\frac{x - 9}{21} = \frac{1}{7} - \frac{x - 2}{14}$

Step 1

Multiply both sides by $21$ .

$\frac{x - 9}{21} \cdot 21 = (\frac{1}{7} - \frac{x - 2}{14}) \cdot 21$

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 $21$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{x - 9}{21} \cdot 21 = (\frac{1}{7} - \frac{x - 2}{14}) \cdot 21$

Step 2.1.1.2

Rewrite the expression.

$x - 9 = (\frac{1}{7} - \frac{x - 2}{14}) \cdot 21$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $(\frac{1}{7} - \frac{x - 2}{14}) \cdot 21$ .

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Step 2.2.1.1

To write $\frac{1}{7}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x - 9 = (\frac{1}{7} \cdot \frac{2}{2} - \frac{x - 2}{14}) \cdot 21$

Step 2.2.1.2

Write each expression with a common denominator of $14$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.2.1

Multiply $\frac{1}{7}$ by $\frac{2}{2}$ .

$x - 9 = (\frac{2}{7 \cdot 2} - \frac{x - 2}{14}) \cdot 21$

Step 2.2.1.2.2

Multiply $7$ by $2$ .

$x - 9 = (\frac{2}{14} - \frac{x - 2}{14}) \cdot 21$

Step 2.2.1.3

Combine the numerators over the common denominator.

$x - 9 = \frac{2 - (x - 2)}{14} \cdot 21$

Step 2.2.1.4

Simplify the numerator.

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Step 2.2.1.4.1

Apply the distributive property.

$x - 9 = \frac{2 - x - - 2}{14} \cdot 21$

Step 2.2.1.4.2

Multiply $- 1$ by $- 2$ .

$x - 9 = \frac{2 - x + 2}{14} \cdot 21$

Step 2.2.1.4.3

Add $2$ and $2$ .

$x - 9 = \frac{- x + 4}{14} \cdot 21$

Step 2.2.1.5

Simplify terms.

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Step 2.2.1.5.1

Cancel the common factor of $7$ .

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Step 2.2.1.5.1.1

Factor $7$ out of $14$ .

$x - 9 = \frac{- x + 4}{7 (2)} \cdot 21$

Step 2.2.1.5.1.2

Factor $7$ out of $21$ .

$x - 9 = \frac{- x + 4}{7 \cdot 2} \cdot (7 \cdot 3)$

Step 2.2.1.5.1.3

Cancel the common factor.

$x - 9 = \frac{- x + 4}{7 \cdot 2} \cdot (7 \cdot 3)$

Step 2.2.1.5.1.4

Rewrite the expression.

$x - 9 = \frac{- x + 4}{2} \cdot 3$

Step 2.2.1.5.2

Combine $\frac{- x + 4}{2}$ and $3$ .

$x - 9 = \frac{(- x + 4) \cdot 3}{2}$

Step 2.2.1.5.3

Move $3$ to the left of $- x + 4$ .

$x - 9 = \frac{3 \cdot (- x + 4)}{2}$

Step 2.2.1.5.4

Factor $- 1$ out of $- x$ .

$x - 9 = \frac{3 (- (x) + 4)}{2}$

Step 2.2.1.5.5

Rewrite $4$ as $- 1 (- 4)$ .

$x - 9 = \frac{3 (- (x) - 1 (- 4))}{2}$

Step 2.2.1.5.6

Factor $- 1$ out of $- (x) - 1 (- 4)$ .

$x - 9 = \frac{3 (- (x - 4))}{2}$

Step 2.2.1.5.7

Simplify the expression.

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Step 2.2.1.5.7.1

Rewrite $- (x - 4)$ as $- 1 (x - 4)$ .

$x - 9 = \frac{3 (- 1 (x - 4))}{2}$

Step 2.2.1.5.7.2

Move the negative in front of the fraction.

$x - 9 = - \frac{3 (x - 4)}{2}$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides by $2$ .

$(x - 9) \cdot 2 = - \frac{3 (x - 4)}{2} \cdot 2$

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 $(x - 9) \cdot 2$ .

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Step 3.2.1.1.1

Apply the distributive property.

$x \cdot 2 - 9 \cdot 2 = - \frac{3 (x - 4)}{2} \cdot 2$

Step 3.2.1.1.2

Simplify the expression.

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Step 3.2.1.1.2.1

Move $2$ to the left of $x$ .

$2 \cdot x - 9 \cdot 2 = - \frac{3 (x - 4)}{2} \cdot 2$

Step 3.2.1.1.2.2

Multiply $- 9$ by $2$ .

$2 x - 18 = - \frac{3 (x - 4)}{2} \cdot 2$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $- \frac{3 (x - 4)}{2} \cdot 2$ .

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Step 3.2.2.1.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1.1

Move the leading negative in $- \frac{3 (x - 4)}{2}$ into the numerator.

$2 x - 18 = \frac{- 3 (x - 4)}{2} \cdot 2$

Step 3.2.2.1.1.2

Cancel the common factor.

$2 x - 18 = \frac{- 3 (x - 4)}{2} \cdot 2$

Step 3.2.2.1.1.3

Rewrite the expression.

$2 x - 18 = - 3 (x - 4)$

Step 3.2.2.1.2

Apply the distributive property.

$2 x - 18 = - 3 x - 3 \cdot - 4$

Step 3.2.2.1.3

Multiply $- 3$ by $- 4$ .

$2 x - 18 = - 3 x + 12$

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

Add $3 x$ to both sides of the equation.

$2 x - 18 + 3 x = 12$

Step 3.3.1.2

Add $2 x$ and $3 x$ .

$5 x - 18 = 12$

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

Add $18$ to both sides of the equation.

$5 x = 12 + 18$

Step 3.3.2.2

Add $12$ and $18$ .

$5 x = 30$

Step 3.3.3

Divide each term in $5 x = 30$ by $5$ and simplify.

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Step 3.3.3.1

Divide each term in $5 x = 30$ by $5$ .

$\frac{5 x}{5} = \frac{30}{5}$

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.

$\frac{5 x}{5} = \frac{30}{5}$

Step 3.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{30}{5}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Divide $30$ by $5$ .

$x = 6$