Trigonometry Examples

$9 x + \frac{x}{2 y} \cdot (3 x) = 5 x y$

Step 1

Simplify each term.

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Step 1.1

Rewrite using the commutative property of multiplication.

$9 x + 3 \frac{x}{2 y} x = 5 x y$

Step 1.2

Combine $3$ and $\frac{x}{2 y}$ .

$9 x + \frac{3 x}{2 y} x = 5 x y$

Step 1.3

Multiply $\frac{3 x}{2 y} x$ .

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Step 1.3.1

Combine $\frac{3 x}{2 y}$ and $x$ .

$9 x + \frac{3 x \cdot x}{2 y} = 5 x y$

Step 1.3.2

Raise $x$ to the power of $1$ .

$9 x + \frac{3 (x^{1} x)}{2 y} = 5 x y$

Step 1.3.3

Raise $x$ to the power of $1$ .

$9 x + \frac{3 (x^{1} x^{1})}{2 y} = 5 x y$

Step 1.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$9 x + \frac{3 x^{1 + 1}}{2 y} = 5 x y$

Step 1.3.5

Add $1$ and $1$ .

$9 x + \frac{3 x^{2}}{2 y} = 5 x y$

Step 2

Subtract $5 x y$ from both sides of the equation.

$9 x + \frac{3 x^{2}}{2 y} - 5 x y = 0$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

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

$1, 2 y, 1, 1$

Step 3.2

The LCM of one and any expression is the expression.

$2 y$

Step 4

Multiply each term in $9 x + \frac{3 x^{2}}{2 y} - 5 x y = 0$ by $2 y$ to eliminate the fractions.

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Step 4.1

Multiply each term in $9 x + \frac{3 x^{2}}{2 y} - 5 x y = 0$ by $2 y$ .

$9 x (2 y) + \frac{3 x^{2}}{2 y} (2 y) - 5 x y (2 y) = 0 (2 y)$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Rewrite using the commutative property of multiplication.

$9 \cdot 2 x y + \frac{3 x^{2}}{2 y} (2 y) - 5 x y (2 y) = 0 (2 y)$

Step 4.2.1.2

Multiply $9$ by $2$ .

$18 x y + \frac{3 x^{2}}{2 y} (2 y) - 5 x y (2 y) = 0 (2 y)$

Step 4.2.1.3

Rewrite using the commutative property of multiplication.

$18 x y + 2 \frac{3 x^{2}}{2 y} y - 5 x y (2 y) = 0 (2 y)$

Step 4.2.1.4

Cancel the common factor of $2$ .

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Step 4.2.1.4.1

Factor $2$ out of $2 y$ .

$18 x y + 2 \frac{3 x^{2}}{2 (y)} y - 5 x y (2 y) = 0 (2 y)$

Step 4.2.1.4.2

Cancel the common factor.

$18 x y + 2 \frac{3 x^{2}}{2 y} y - 5 x y (2 y) = 0 (2 y)$

Step 4.2.1.4.3

Rewrite the expression.

$18 x y + \frac{3 x^{2}}{y} y - 5 x y (2 y) = 0 (2 y)$

Step 4.2.1.5

Cancel the common factor of $y$ .

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Step 4.2.1.5.1

Cancel the common factor.

$18 x y + \frac{3 x^{2}}{y} y - 5 x y (2 y) = 0 (2 y)$

Step 4.2.1.5.2

Rewrite the expression.

$18 x y + 3 x^{2} - 5 x y (2 y) = 0 (2 y)$

Step 4.2.1.6

Multiply $y$ by $y$ by adding the exponents.

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Step 4.2.1.6.1

Move $y$ .

$18 x y + 3 x^{2} - 5 x (y \cdot y) \cdot 2 = 0 (2 y)$

Step 4.2.1.6.2

Multiply $y$ by $y$ .

$18 x y + 3 x^{2} - 5 x y^{2} \cdot 2 = 0 (2 y)$

Step 4.2.1.7

Multiply $2$ by $- 5$ .

$18 x y + 3 x^{2} - 10 x y^{2} = 0 (2 y)$

Step 4.3

Simplify the right side.

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Step 4.3.1

Multiply $0 (2 y)$ .

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Step 4.3.1.1

Multiply $2$ by $0$ .

$18 x y + 3 x^{2} - 10 x y^{2} = 0 y$

Step 4.3.1.2

Multiply $0$ by $y$ .

$18 x y + 3 x^{2} - 10 x y^{2} = 0$

Step 5

Solve the equation.

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Step 5.1

Factor $x$ out of $18 x y + 3 x^{2} - 10 x y^{2}$ .

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Step 5.1.1

Factor $x$ out of $18 x y$ .

$x (18 y) + 3 x^{2} - 10 x y^{2} = 0$

Step 5.1.2

Factor $x$ out of $3 x^{2}$ .

$x (18 y) + x (3 x) - 10 x y^{2} = 0$

Step 5.1.3

Factor $x$ out of $- 10 x y^{2}$ .

$x (18 y) + x (3 x) + x (- 10 y^{2}) = 0$

Step 5.1.4

Factor $x$ out of $x (18 y) + x (3 x)$ .

$x (18 y + 3 x) + x (- 10 y^{2}) = 0$

Step 5.1.5

Factor $x$ out of $x (18 y + 3 x) + x (- 10 y^{2})$ .

$x (18 y + 3 x - 10 y^{2}) = 0$

Step 5.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$18 y + 3 x - 10 y^{2} = 0$

Step 5.3

Set $x$ equal to $0$ .

$x = 0$

Step 5.4

Set $18 y + 3 x - 10 y^{2}$ equal to $0$ and solve for $x$ .

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Step 5.4.1

Set $18 y + 3 x - 10 y^{2}$ equal to $0$ .

$18 y + 3 x - 10 y^{2} = 0$

Step 5.4.2

Solve $18 y + 3 x - 10 y^{2} = 0$ for $x$ .

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Step 5.4.2.1

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

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Step 5.4.2.1.1

Subtract $18 y$ from both sides of the equation.

$3 x - 10 y^{2} = - 18 y$

Step 5.4.2.1.2

Add $10 y^{2}$ to both sides of the equation.

$3 x = - 18 y + 10 y^{2}$

Step 5.4.2.2

Divide each term in $3 x = - 18 y + 10 y^{2}$ by $3$ and simplify.

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Step 5.4.2.2.1

Divide each term in $3 x = - 18 y + 10 y^{2}$ by $3$ .

$\frac{3 x}{3} = \frac{- 18 y}{3} + \frac{10 y^{2}}{3}$

Step 5.4.2.2.2

Simplify the left side.

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Step 5.4.2.2.2.1

Cancel the common factor of $3$ .

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Step 5.4.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 18 y}{3} + \frac{10 y^{2}}{3}$

Step 5.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 18 y}{3} + \frac{10 y^{2}}{3}$

Step 5.4.2.2.3

Simplify the right side.

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Step 5.4.2.2.3.1

Cancel the common factor of $- 18$ and $3$ .

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Step 5.4.2.2.3.1.1

Factor $3$ out of $- 18 y$ .

$x = \frac{3 (- 6 y)}{3} + \frac{10 y^{2}}{3}$

Step 5.4.2.2.3.1.2

Cancel the common factors.

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Step 5.4.2.2.3.1.2.1

Factor $3$ out of $3$ .

$x = \frac{3 (- 6 y)}{3 (1)} + \frac{10 y^{2}}{3}$

Step 5.4.2.2.3.1.2.2

Cancel the common factor.

$x = \frac{3 (- 6 y)}{3 \cdot 1} + \frac{10 y^{2}}{3}$

Step 5.4.2.2.3.1.2.3

Rewrite the expression.

$x = \frac{- 6 y}{1} + \frac{10 y^{2}}{3}$

Step 5.4.2.2.3.1.2.4

Divide $- 6 y$ by $1$ .

$x = - 6 y + \frac{10 y^{2}}{3}$

Step 5.5

The final solution is all the values that make $x (18 y + 3 x - 10 y^{2}) = 0$ true.

$x = 0$

$x = - 6 y + \frac{10 y^{2}}{3}$

$x = 0$

$x = - 6 y + \frac{10 y^{2}}{3}$