Solve y 9x-5y=-4 - ScanMath

Question

Solve the equation

Solve for x

Solve for y

x = \frac{- 4 + 5 y}{9 y}

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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

Not symmetry with respect to the origin

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Rewrite the equation

r = \frac{5 sin ( θ ) + 25 sin ^{2} ( θ ) - 72 sin ( 2 θ )}{9 sin ( 2 θ )} r = \frac{5 sin ( θ ) - 25 sin ^{2} ( θ ) - 72 sin ( 2 θ )}{9 sin ( 2 θ )}

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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{9 y}{9 x - 5}

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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}} = \frac{162 y}{81 x ^{2} - 90 x + 25}

Calculate

y 9 x - 5 y = - 4

Simplify the expression

9 y x - 5 y = - 4

Take the derivative of both sides

\frac{d}{d x} (9 y x - 5 y) = \frac{d}{d x} (- 4)

Calculate the derivative

More Steps

9 y + 9 x \frac{d y}{d x} - 5 \frac{d y}{d x} = \frac{d}{d x} (- 4)

Calculate the derivative

9 y + 9 x \frac{d y}{d x} - 5 \frac{d y}{d x} = 0

Collect like terms by calculating the sum or difference of their coefficients

9 y + (9 x - 5) \frac{d y}{d x} = 0

Move the constant to the right side

(9 x - 5) \frac{d y}{d x} = 0 - 9 y

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

(9 x - 5) \frac{d y}{d x} = - 9 y

Divide both sides

\frac{( 9 x - 5 ) \frac{d y}{d x}}{9 x - 5} = \frac{- 9 y}{9 x - 5}

Divide the numbers

\frac{d y}{d x} = \frac{- 9 y}{9 x - 5}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

\frac{d y}{d x} = - \frac{9 y}{9 x - 5}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (- \frac{9 y}{9 x - 5})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (- \frac{9 y}{9 x - 5})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d}{d x} ( 9 y ) \times ( 9 x - 5 ) - 9 y \times \frac{d}{d x} ( 9 x - 5 )}{( 9 x - 5 ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{9 \frac{d y}{d x} \times ( 9 x - 5 ) - 9 y \times \frac{d}{d x} ( 9 x - 5 )}{( 9 x - 5 ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = - \frac{9 \frac{d y}{d x} \times ( 9 x - 5 ) - 9 y \times 9}{( 9 x - 5 ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = - \frac{81 x \frac{d y}{d x} - 45 \frac{d y}{d x} - 9 y \times 9}{( 9 x - 5 ) ^{2}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = - \frac{81 x \frac{d y}{d x} - 45 \frac{d y}{d x} - 81 y}{( 9 x - 5 ) ^{2}}

Use equation \frac{d y}{d x} = - \frac{9 y}{9 x - 5} to substitute

\frac{d ^{2} y}{d x ^{2}} = - \frac{81 x ( - \frac{9 y}{9 x - 5} ) - 45 ( - \frac{9 y}{9 x - 5} ) - 81 y}{( 9 x - 5 ) ^{2}}

Solution

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\frac{d ^{2} y}{d x ^{2}} = \frac{162 y}{81 x ^{2} - 90 x + 25}

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Conic

\frac{( y ^{'} + \frac{5 2}{18} ) ^{2}}{\frac{8}{9}} - \frac{( x ^{'} - \frac{5 2}{18} ) ^{2}}{\frac{8}{9}} = 1

Evaluate

y \times 9 x - 5 y = - 4

Move the expression to the left side

y \times 9 x - 5 y - (- 4) = 0

Calculate

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9 y x - 5 y + 4 = 0

The coefficients A,B and C of the general equation are A= 0,B= 9 and C= 0

A = 0 B = 9 C = 0

To find the angle of rotation θ,substitute the values of A,B and C into the formula cot (2 θ) = \frac{A - C}{B}

cot (2 θ) = \frac{0 - 0}{9}

Calculate

cot (2 θ) = 0

Using the unit circle,find the smallest positive angle for which the cotangent is 0

2 θ = \frac{π}{2}

Calculate

θ = \frac{π}{4}

To rotate the axes,use the equation of rotation and substitute \frac{π}{4} for θ

x = x^{'} cos (\frac{π}{4}) - y^{'} sin (\frac{π}{4}) y = x^{'} sin (\frac{π}{4}) + y^{'} cos (\frac{π}{4})

Calculate

x = x^{'} \times \frac{2}{2} - y^{'} sin (\frac{π}{4}) y = x^{'} sin (\frac{π}{4}) + y^{'} cos (\frac{π}{4})

Calculate

x = x^{'} \times \frac{2}{2} - y^{'} \times \frac{2}{2} y = x^{'} sin (\frac{π}{4}) + y^{'} cos (\frac{π}{4})

Calculate

x = x^{'} \times \frac{2}{2} - y^{'} \times \frac{2}{2} y = x^{'} \times \frac{2}{2} + y^{'} cos (\frac{π}{4})

Calculate

x = x^{'} \times \frac{2}{2} - y^{'} \times \frac{2}{2} y = x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}

Substitute x and y into the original equation 9 y x - 5 y + 4 = 0

9 (x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}) (x^{'} \times \frac{2}{2} - y^{'} \times \frac{2}{2}) - 5 (x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}) + 4 = 0

Calculate

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Calculate

9 (x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}) (x^{'} \times \frac{2}{2} - y^{'} \times \frac{2}{2}) - 5 (x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}) + 4

Use the commutative property to reorder the terms

9 (\frac{2}{2} x^{'} + y^{'} \times \frac{2}{2}) (x^{'} \times \frac{2}{2} - y^{'} \times \frac{2}{2}) - 5 (x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}) + 4

Use the commutative property to reorder the terms

9 (\frac{2}{2} x^{'} + \frac{2}{2} y^{'}) (x^{'} \times \frac{2}{2} - y^{'} \times \frac{2}{2}) - 5 (x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}) + 4

Use the commutative property to reorder the terms

9 (\frac{2}{2} x^{'} + \frac{2}{2} y^{'}) (\frac{2}{2} x^{'} - y^{'} \times \frac{2}{2}) - 5 (x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}) + 4

Use the commutative property to reorder the terms

9 (\frac{2}{2} x^{'} + \frac{2}{2} y^{'}) (\frac{2}{2} x^{'} - \frac{2}{2} y^{'}) - 5 (x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}) + 4

Use the commutative property to reorder the terms

9 (\frac{2}{2} x^{'} + \frac{2}{2} y^{'}) (\frac{2}{2} x^{'} - \frac{2}{2} y^{'}) - 5 (\frac{2}{2} x^{'} + y^{'} \times \frac{2}{2}) + 4

Use the commutative property to reorder the terms

9 (\frac{2}{2} x^{'} + \frac{2}{2} y^{'}) (\frac{2}{2} x^{'} - \frac{2}{2} y^{'}) - 5 (\frac{2}{2} x^{'} + \frac{2}{2} y^{'}) + 4

Expand the expression

More Steps

\frac{9}{2} (x^{'})^{2} - \frac{9}{2} (y^{'})^{2} - 5 (\frac{2}{2} x^{'} + \frac{2}{2} y^{'}) + 4

Expand the expression

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\frac{9}{2} (x^{'})^{2} - \frac{9}{2} (y^{'})^{2} - \frac{5 2}{2} x^{'} - \frac{5 2}{2} y^{'} + 4

\frac{9}{2} (x^{'})^{2} - \frac{9}{2} (y^{'})^{2} - \frac{5 2}{2} x^{'} - \frac{5 2}{2} y^{'} + 4 = 0

Move the constant to the right-hand side and change its sign

\frac{9}{2} (x^{'})^{2} - \frac{9}{2} (y^{'})^{2} - \frac{5 2}{2} x^{'} - \frac{5 2}{2} y^{'} = 0 - 4

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

\frac{9}{2} (x^{'})^{2} - \frac{9}{2} (y^{'})^{2} - \frac{5 2}{2} x^{'} - \frac{5 2}{2} y^{'} = - 4

Use the commutative property to reorder the terms

\frac{9}{2} (x^{'})^{2} - \frac{5 2}{2} x^{'} - \frac{9}{2} (y^{'})^{2} - \frac{5 2}{2} y^{'} = - 4

To complete the square, the same value needs to be added to both sides

\frac{9}{2} (x^{'})^{2} - \frac{5 2}{2} x^{'} + \frac{25}{36} - \frac{9}{2} (y^{'})^{2} - \frac{5 2}{2} y^{'} = - 4 + \frac{25}{36}

Factor out \frac{9}{2} from the expression

\frac{9}{2} ((x^{'})^{2} - \frac{5 2}{9} x^{'} + \frac{25}{162}) - \frac{9}{2} (y^{'})^{2} - \frac{5 2}{2} y^{'} = - 4 + \frac{25}{36}

Use a^{2} - 2 ab + b^{2} = (a - b)^{2} to factor the expression

\frac{9}{2} (x^{'} - \frac{5 2}{18})^{2} - \frac{9}{2} (y^{'})^{2} - \frac{5 2}{2} y^{'} = - 4 + \frac{25}{36}

Add the numbers

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\frac{9}{2} (x^{'} - \frac{5 2}{18})^{2} - \frac{9}{2} (y^{'})^{2} - \frac{5 2}{2} y^{'} = - \frac{119}{36}

To complete the square, the same value needs to be subtract from both sides

\frac{9}{2} (x^{'} - \frac{5 2}{18})^{2} - \frac{9}{2} (y^{'})^{2} - \frac{5 2}{2} y^{'} - \frac{25}{36} = - \frac{119}{36} - \frac{25}{36}

Factor out - \frac{9}{2} from the expression

\frac{9}{2} (x^{'} - \frac{5 2}{18})^{2} - \frac{9}{2} ((y^{'})^{2} + \frac{5 2}{9} y^{'} + \frac{25}{162}) = - \frac{119}{36} - \frac{25}{36}

Use a^{2} + 2 ab + b^{2} = (a + b)^{2} to factor the expression

\frac{9}{2} (x^{'} - \frac{5 2}{18})^{2} - \frac{9}{2} (y^{'} + \frac{5 2}{18})^{2} = - \frac{119}{36} - \frac{25}{36}

Subtract the numbers

More Steps

\frac{9}{2} (x^{'} - \frac{5 2}{18})^{2} - \frac{9}{2} (y^{'} + \frac{5 2}{18})^{2} = - 4

Multiply both sides of the equation by - \frac{1}{4}

\frac{9}{2} (x^{'} - \frac{5 2}{18})^{2} - \frac{9}{2} (y^{'} + \frac{5 2}{18})^{2} (- \frac{1}{4}) = - 4 (- \frac{1}{4})

Multiply the terms

More Steps

- \frac{9}{8} (x^{'} - \frac{5 2}{18})^{2} + \frac{9}{8} (y^{'} + \frac{5 2}{18})^{2} = - 4 (- \frac{1}{4})

Multiply the terms

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- \frac{9}{8} (x^{'} - \frac{5 2}{18})^{2} + \frac{9}{8} (y^{'} + \frac{5 2}{18})^{2} = 1

Use a = \frac{1}{\frac{1}{a}} to transform the expression

- \frac{( x ^{'} - \frac{5 2}{18} ) ^{2}}{\frac{8}{9}} + \frac{9}{8} (y^{'} + \frac{5 2}{18})^{2} = 1

Use a = \frac{1}{\frac{1}{a}} to transform the expression

- \frac{( x ^{'} - \frac{5 2}{18} ) ^{2}}{\frac{8}{9}} + \frac{( y ^{'} + \frac{5 2}{18} ) ^{2}}{\frac{8}{9}} = 1

Solution

\frac{( y ^{'} + \frac{5 2}{18} ) ^{2}}{\frac{8}{9}} - \frac{( x ^{'} - \frac{5 2}{18} ) ^{2}}{\frac{8}{9}} = 1

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