Solve 2:3:5=x:y:8

Solve the equation

Solve for x

Solve for y

x = \frac{16 y}{15}

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Testing for symmetry

Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Symmetry with respect to the origin

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Rewrite the equation

r = 0 θ = arccot (\frac{16}{15}) + kπ, k \in Z

Evaluate

2 \div 3 \div 5 = x \div y \div 8

Evaluate

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\frac{2}{15} = x \div y \div 8

Evaluate

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\frac{2}{15} = \frac{x}{8 y}

Multiply both sides of the equation by LCD

\frac{2}{15} \times 120 y = \frac{x}{8 y} \times 120 y

Simplify the equation

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16 y = \frac{x}{8 y} \times 120 y

Simplify the equation

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16 y = 15 x

Move the expression to the left side

16 y - 15 x = 0

To convert the equation to polar coordinates,substitute x for r cos (θ) and y for r sin (θ)

16 sin (θ) \times r - 15 cos (θ) \times r = 0

Factor the expression

(16 sin (θ) - 15 cos (θ)) r = 0

Separate into possible cases

r = 0 16 sin (θ) - 15 cos (θ) = 0

Solution

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r = 0 θ = arccot (\frac{16}{15}) + kπ, k \in Z

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Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = \frac{y}{x}

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Find the second derivative

Find the second derivative with respect to x

Find the second derivative with respect to y

\frac{d ^{2} y}{d x ^{2}} = 0

Calculate

2 : 3 : 5 = x : y : 8

Simplify the expression

\frac{2}{15} = \frac{x}{8 y}

Take the derivative of both sides

\frac{d}{d x} (\frac{2}{15}) = \frac{d}{d x} (\frac{x}{8 y})

Calculate the derivative

0 = \frac{d}{d x} (\frac{x}{8 y})

Calculate the derivative

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0 = \frac{y - x \frac{d y}{d x}}{8 y ^{2}}

Swap the sides of the equation

\frac{y - x \frac{d y}{d x}}{8 y ^{2}} = 0

Simplify

y - x \frac{d y}{d x} = 0

Move the constant to the right side

- x \frac{d y}{d x} = 0 - y

Removing 0 doesn't change the value,so remove it from the expression

- x \frac{d y}{d x} = - y

Divide both sides

\frac{- x \frac{d y}{d x}}{- x} = \frac{- y}{- x}

Divide the numbers

\frac{d y}{d x} = \frac{- y}{- x}

Divide the numbers

\frac{d y}{d x} = \frac{y}{x}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{y}{x})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{y}{x})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( y ) \times x - y \times \frac{d}{d x} ( x )}{x ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d y}{d x} \times x - y \times \frac{d}{d x} ( x )}{x ^{2}}

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d y}{d x} \times x - y \times 1}{x ^{2}}

Use the commutative property to reorder the terms

\frac{d ^{2} y}{d x ^{2}} = \frac{x \frac{d y}{d x} - y \times 1}{x ^{2}}

Any expression multiplied by 1 remains the same

\frac{d ^{2} y}{d x ^{2}} = \frac{x \frac{d y}{d x} - y}{x ^{2}}

Use equation \frac{d y}{d x} = \frac{y}{x} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{x \times \frac{y}{x} - y}{x ^{2}}

Solution

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\frac{d ^{2} y}{d x ^{2}} = 0

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