Trigonometry Examples

$\frac{x + 8}{14} = \frac{1}{7} + \frac{x - 9}{8}$

Step 1

Multiply both sides by $14$ .

$\frac{x + 8}{14} \cdot 14 = (\frac{1}{7} + \frac{x - 9}{8}) \cdot 14$

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 $14$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{x + 8}{14} \cdot 14 = (\frac{1}{7} + \frac{x - 9}{8}) \cdot 14$

Step 2.1.1.2

Rewrite the expression.

$x + 8 = (\frac{1}{7} + \frac{x - 9}{8}) \cdot 14$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $(\frac{1}{7} + \frac{x - 9}{8}) \cdot 14$ .

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Step 2.2.1.1

To write $\frac{1}{7}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$x + 8 = (\frac{1}{7} \cdot \frac{8}{8} + \frac{x - 9}{8}) \cdot 14$

Step 2.2.1.2

To write $\frac{x - 9}{8}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$x + 8 = (\frac{1}{7} \cdot \frac{8}{8} + \frac{x - 9}{8} \cdot \frac{7}{7}) \cdot 14$

Step 2.2.1.3

Write each expression with a common denominator of $56$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.3.1

Multiply $\frac{1}{7}$ by $\frac{8}{8}$ .

$x + 8 = (\frac{8}{7 \cdot 8} + \frac{x - 9}{8} \cdot \frac{7}{7}) \cdot 14$

Step 2.2.1.3.2

Multiply $7$ by $8$ .

$x + 8 = (\frac{8}{56} + \frac{x - 9}{8} \cdot \frac{7}{7}) \cdot 14$

Step 2.2.1.3.3

Multiply $\frac{x - 9}{8}$ by $\frac{7}{7}$ .

$x + 8 = (\frac{8}{56} + \frac{(x - 9) \cdot 7}{8 \cdot 7}) \cdot 14$

Step 2.2.1.3.4

Multiply $8$ by $7$ .

$x + 8 = (\frac{8}{56} + \frac{(x - 9) \cdot 7}{56}) \cdot 14$

Step 2.2.1.4

Combine the numerators over the common denominator.

$x + 8 = \frac{8 + (x - 9) \cdot 7}{56} \cdot 14$

Step 2.2.1.5

Simplify the numerator.

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Step 2.2.1.5.1

Apply the distributive property.

$x + 8 = \frac{8 + x \cdot 7 - 9 \cdot 7}{56} \cdot 14$

Step 2.2.1.5.2

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

$x + 8 = \frac{8 + 7 \cdot x - 9 \cdot 7}{56} \cdot 14$

Step 2.2.1.5.3

Multiply $- 9$ by $7$ .

$x + 8 = \frac{8 + 7 \cdot x - 63}{56} \cdot 14$

Step 2.2.1.5.4

Subtract $63$ from $8$ .

$x + 8 = \frac{7 x - 55}{56} \cdot 14$

Step 2.2.1.6

Cancel the common factor of $14$ .

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Step 2.2.1.6.1

Factor $14$ out of $56$ .

$x + 8 = \frac{7 x - 55}{14 (4)} \cdot 14$

Step 2.2.1.6.2

Cancel the common factor.

$x + 8 = \frac{7 x - 55}{14 \cdot 4} \cdot 14$

Step 2.2.1.6.3

Rewrite the expression.

$x + 8 = \frac{7 x - 55}{4}$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides by $4$ .

$(x + 8) \cdot 4 = \frac{7 x - 55}{4} \cdot 4$

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 $(x + 8) \cdot 4$ .

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Step 3.2.1.1.1

Apply the distributive property.

$x \cdot 4 + 8 \cdot 4 = \frac{7 x - 55}{4} \cdot 4$

Step 3.2.1.1.2

Simplify the expression.

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Step 3.2.1.1.2.1

Move $4$ to the left of $x$ .

$4 \cdot x + 8 \cdot 4 = \frac{7 x - 55}{4} \cdot 4$

Step 3.2.1.1.2.2

Multiply $8$ by $4$ .

$4 x + 32 = \frac{7 x - 55}{4} \cdot 4$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.2.1.1

Cancel the common factor.

$4 x + 32 = \frac{7 x - 55}{4} \cdot 4$

Step 3.2.2.1.2

Rewrite the expression.

$4 x + 32 = 7 x - 55$

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 $7 x$ from both sides of the equation.

$4 x + 32 - 7 x = - 55$

Step 3.3.1.2

Subtract $7 x$ from $4 x$ .

$- 3 x + 32 = - 55$

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 $32$ from both sides of the equation.

$- 3 x = - 55 - 32$

Step 3.3.2.2

Subtract $32$ from $- 55$ .

$- 3 x = - 87$

Step 3.3.3

Divide each term in $- 3 x = - 87$ by $- 3$ and simplify.

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Step 3.3.3.1

Divide each term in $- 3 x = - 87$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 87}{- 3}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 3$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 87}{- 3}$

Step 3.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 87}{- 3}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Divide $- 87$ by $- 3$ .

$x = 29$