Algebra Examples

$y = 2 π - 3 π \cot (\frac{4 π}{k}) (x - 2)$

Step 1

Rewrite the equation as $2 π - 3 π \cot (\frac{4 π}{k}) (x - 2) = y$ .

$2 π - 3 π \cot (\frac{4 π}{k}) (x - 2) = y$

Step 2

Substitute $u$ for $\cot (\frac{4 π}{k})$ .

$2 π - 3 π u (x - 2) = y$

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

$2 π - 3 π u x - 3 π u \cdot - 2 = y$

Step 3.2

Multiply $- 2$ by $- 3$ .

$2 π - 3 π u x + 6 π u = y$

Step 4

Subtract $2 π$ from both sides of the equation.

$- 3 π u x + 6 π u = y - 2 π$

Step 5

Factor $3 π u$ out of $- 3 π u x + 6 π u$ .

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Step 5.1

Factor $3 π u$ out of $- 3 π u x$ .

$3 π u (- 1 x) + 6 π u = y - 2 π$

Step 5.2

Factor $3 π u$ out of $6 π u$ .

$3 π u (- 1 x) + 3 π u \cdot 2 = y - 2 π$

Step 5.3

Factor $3 π u$ out of $3 π u (- 1 x) + 3 π u \cdot 2$ .

$3 π u (- 1 x + 2) = y - 2 π$

Step 6

Rewrite $- 1 x$ as $- x$ .

$3 π u (- x + 2) = y - 2 π$

Step 7

Divide each term in $3 π u (- x + 2) = y - 2 π$ by $3 π (- x + 2)$ and simplify.

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Step 7.1

Divide each term in $3 π u (- x + 2) = y - 2 π$ by $3 π (- x + 2)$ .

$\frac{3 π u (- x + 2)}{3 π (- x + 2)} = \frac{y}{3 π (- x + 2)} + \frac{- 2 π}{3 π (- x + 2)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $3$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{3 π u (- x + 2)}{3 π (- x + 2)} = \frac{y}{3 π (- x + 2)} + \frac{- 2 π}{3 π (- x + 2)}$

Step 7.2.1.2

Rewrite the expression.

$\frac{π u (- x + 2)}{π (- x + 2)} = \frac{y}{3 π (- x + 2)} + \frac{- 2 π}{3 π (- x + 2)}$

Step 7.2.2

Cancel the common factor of $π$ .

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Step 7.2.2.1

Cancel the common factor.

$\frac{π u (- x + 2)}{π (- x + 2)} = \frac{y}{3 π (- x + 2)} + \frac{- 2 π}{3 π (- x + 2)}$

Step 7.2.2.2

Rewrite the expression.

$\frac{u (- x + 2)}{- x + 2} = \frac{y}{3 π (- x + 2)} + \frac{- 2 π}{3 π (- x + 2)}$

Step 7.2.3

Cancel the common factor of $- x + 2$ .

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Step 7.2.3.1

Cancel the common factor.

$\frac{u (- x + 2)}{- x + 2} = \frac{y}{3 π (- x + 2)} + \frac{- 2 π}{3 π (- x + 2)}$

Step 7.2.3.2

Divide $u$ by $1$ .

$u = \frac{y}{3 π (- x + 2)} + \frac{- 2 π}{3 π (- x + 2)}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

Cancel the common factor of $π$ .

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Step 7.3.1.1.1

Cancel the common factor.

$u = \frac{y}{3 π (- x + 2)} + \frac{- 2 π}{3 π (- x + 2)}$

Step 7.3.1.1.2

Rewrite the expression.

$u = \frac{y}{3 π (- x + 2)} + \frac{- 2}{3 (- x + 2)}$

Step 7.3.1.2

Move the negative in front of the fraction.

$u = \frac{y}{3 π (- x + 2)} - \frac{2}{3 (- x + 2)}$

Step 8

Substitute $\cot (\frac{4 π}{k})$ for $u$ .

$\cot (\frac{4 π}{k}) = \frac{y}{3 π (- x + 2)} - \frac{2}{3 (- x + 2)}$

Step 9

Take the inverse cotangent of both sides of the equation to extract $k$ from inside the cotangent.

$\frac{4 π}{k} = arccot (\frac{y}{3 π (- x + 2)} - \frac{2}{3 (- x + 2)})$

Step 10

Find the LCD of the terms in the equation.

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Step 10.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$k, 1$

Step 10.2

The LCM of one and any expression is the expression.

$k$

Step 11

Multiply each term in $\frac{4 π}{k} = arccot (\frac{y}{3 π (- x + 2)} - \frac{2}{3 (- x + 2)})$ by $k$ to eliminate the fractions.

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Step 11.1

Multiply each term in $\frac{4 π}{k} = arccot (\frac{y}{3 π (- x + 2)} - \frac{2}{3 (- x + 2)})$ by $k$ .

$\frac{4 π}{k} k = arccot (\frac{y}{3 π (- x + 2)} - \frac{2}{3 (- x + 2)}) k$

Step 11.2

Simplify the left side.

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Step 11.2.1

Cancel the common factor of $k$ .

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Step 11.2.1.1

Cancel the common factor.

$\frac{4 π}{k} k = arccot (\frac{y}{3 π (- x + 2)} - \frac{2}{3 (- x + 2)}) k$

Step 11.2.1.2

Rewrite the expression.

$4 π = arccot (\frac{y}{3 π (- x + 2)} - \frac{2}{3 (- x + 2)}) k$

Step 11.3

Simplify the right side.

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Step 11.3.1

Reorder factors in $arccot (\frac{y}{3 π (- x + 2)} - \frac{2}{3 (- x + 2)}) k$ .

$4 π = k arccot (\frac{y}{3 π (- x + 2)} - \frac{2}{3 (- x + 2)})$

Step 12

Solve the equation.

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Step 12.1

Rewrite the equation as $k arccot (\frac{y}{3 π (- x + 2)} - \frac{2}{3 (- x + 2)}) = 4 π$ .

$k arccot (\frac{y}{3 π (- x + 2)} - \frac{2}{3 (- x + 2)}) = 4 π$

Step 12.2

Divide each term in $k arccot (\frac{y}{3 π (- x + 2)} - \frac{2}{3 (- x + 2)}) = 4 π$ by $arccot (\frac{y}{3 π (- x + 2)} - \frac{2}{3 (- x + 2)})$ and simplify.

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Step 12.2.1

Divide each term in $k arccot (\frac{y}{3 π (- x + 2)} - \frac{2}{3 (- x + 2)}) = 4 π$ by $arccot (\frac{y}{3 π (- x + 2)} - \frac{2}{3 (- x + 2)})$ .

$\frac{k arccot (\frac{y}{3 π (- x + 2)} - \frac{2}{3 (- x + 2)})}{arccot (\frac{y}{3 π (- x + 2)} - \frac{2}{3 (- x + 2)})} = \frac{4 π}{arccot (\frac{y}{3 π (- x + 2)} - \frac{2}{3 (- x + 2)})}$

Step 12.2.2

Simplify the left side.

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Step 12.2.2.1

Cancel the common factor of $arccot (\frac{y}{3 π (- x + 2)} - \frac{2}{3 (- x + 2)})$ .

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Step 12.2.2.1.1

Cancel the common factor.

Step 12.2.2.1.2

Divide $k$ by $1$ .

$k = \frac{4 π}{arccot (\frac{y}{3 π (- x + 2)} - \frac{2}{3 (- x + 2)})}$

Step 12.2.3

Simplify the right side.

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Step 12.2.3.1

Simplify the denominator.

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Step 12.2.3.1.1

To write $- \frac{2}{3 (- x + 2)}$ as a fraction with a common denominator, multiply by $\frac{π}{π}$ .

$k = \frac{4 π}{arccot (\frac{y}{3 π (- x + 2)} - \frac{2}{3 (- x + 2)} \cdot \frac{π}{π})}$

Step 12.2.3.1.2

Write each expression with a common denominator of $3 π (- x + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 12.2.3.1.2.1

Multiply $\frac{2}{3 (- x + 2)}$ by $\frac{π}{π}$ .

$k = \frac{4 π}{arccot (\frac{y}{3 π (- x + 2)} - \frac{2 π}{3 (- x + 2) π})}$

Step 12.2.3.1.2.2

Reorder the factors of $3 (- x + 2) π$ .

$k = \frac{4 π}{arccot (\frac{y}{3 π (- x + 2)} - \frac{2 π}{3 π (- x + 2)})}$

Step 12.2.3.1.3

Combine the numerators over the common denominator.

$k = \frac{4 π}{arccot (\frac{y - 2 π}{3 π (- x + 2)})}$