Solve 9x(3 y/3)-3y=9

Question

Solve the equation

Solve for x

Solve for y

x = \frac{3 + y}{3 y}

Show Solution

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

Show Solution

Rewrite the equation

r = \frac{sin ( θ ) + sin ^{2} ( θ ) + 18 sin ( 2 θ )}{3 sin ( 2 θ )} r = \frac{sin ( θ ) - sin ^{2} ( θ ) + 18 sin ( 2 θ )}{3 sin ( 2 θ )}

Evaluate

9 x (3 \times \frac{y}{3}) - 3 y = 9

Evaluate

More Steps

9 x y - 3 y = 9

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

9 cos (θ) \times r sin (θ) \times r - 3 sin (θ) \times r = 9

Factor the expression

9 cos (θ) sin (θ) \times r^{2} - 3 sin (θ) \times r = 9

Simplify the expression

\frac{9}{2} sin (2 θ) \times r^{2} - 3 sin (θ) \times r = 9

Subtract the terms

\frac{9}{2} sin (2 θ) \times r^{2} - 3 sin (θ) \times r - 9 = 9 - 9

Evaluate

\frac{9}{2} sin (2 θ) \times r^{2} - 3 sin (θ) \times r - 9 = 0

Solve using the quadratic formula

r = \frac{3 sin ( θ ) \pm ( - 3 sin ( θ ) ) ^{2} - 4 \times \frac{9}{2} sin ( 2 θ ) ( - 9 )}{9 sin ( 2 θ )}

Simplify

r = \frac{3 sin ( θ ) \pm 9 sin ^{2} ( θ ) + 162 sin ( 2 θ )}{9 sin ( 2 θ )}

Separate the equation into 2 possible cases

r = \frac{3 sin ( θ ) + 9 sin ^{2} ( θ ) + 162 sin ( 2 θ )}{9 sin ( 2 θ )} r = \frac{3 sin ( θ ) - 9 sin ^{2} ( θ ) + 162 sin ( 2 θ )}{9 sin ( 2 θ )}

Evaluate

More Steps

r = \frac{sin ( θ ) + sin ^{2} ( θ ) + 18 sin ( 2 θ )}{3 sin ( 2 θ )} r = \frac{3 sin ( θ ) - 9 sin ^{2} ( θ ) + 162 sin ( 2 θ )}{9 sin ( 2 θ )}

Solution

More Steps

r = \frac{sin ( θ ) + sin ^{2} ( θ ) + 18 sin ( 2 θ )}{3 sin ( 2 θ )} r = \frac{sin ( θ ) - sin ^{2} ( θ ) + 18 sin ( 2 θ )}{3 sin ( 2 θ )}

Show Solution

Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = - \frac{3 y}{3 x - 1}

Show Solution

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{18 y}{9 x ^{2} - 6 x + 1}

Calculate

9 x (3 \times \frac{y}{3}) - 3 y = 9

Simplify the expression

9 x y - 3 y = 9

Take the derivative of both sides

\frac{d}{d x} (9 x y - 3 y) = \frac{d}{d x} (9)

Calculate the derivative

More Steps

9 y + 9 x \frac{d y}{d x} - 3 \frac{d y}{d x} = \frac{d}{d x} (9)

Calculate the derivative

9 y + 9 x \frac{d y}{d x} - 3 \frac{d y}{d x} = 0

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

9 y + (9 x - 3) \frac{d y}{d x} = 0

Move the constant to the right side

(9 x - 3) \frac{d y}{d x} = 0 - 9 y

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

(9 x - 3) \frac{d y}{d x} = - 9 y

Divide both sides

\frac{( 9 x - 3 ) \frac{d y}{d x}}{9 x - 3} = \frac{- 9 y}{9 x - 3}

Divide the numbers

\frac{d y}{d x} = \frac{- 9 y}{9 x - 3}

Divide the numbers

More Steps

\frac{d y}{d x} = - \frac{3 y}{3 x - 1}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (- \frac{3 y}{3 x - 1})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (- \frac{3 y}{3 x - 1})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d}{d x} ( 3 y ) \times ( 3 x - 1 ) - 3 y \times \frac{d}{d x} ( 3 x - 1 )}{( 3 x - 1 ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{3 \frac{d y}{d x} \times ( 3 x - 1 ) - 3 y \times \frac{d}{d x} ( 3 x - 1 )}{( 3 x - 1 ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{3 \frac{d y}{d x} \times ( 3 x - 1 ) - 3 y \times 3}{( 3 x - 1 ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{9 x \frac{d y}{d x} - 3 \frac{d y}{d x} - 3 y \times 3}{( 3 x - 1 ) ^{2}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = - \frac{9 x \frac{d y}{d x} - 3 \frac{d y}{d x} - 9 y}{( 3 x - 1 ) ^{2}}

Use equation \frac{d y}{d x} = - \frac{3 y}{3 x - 1} to substitute

\frac{d ^{2} y}{d x ^{2}} = - \frac{9 x ( - \frac{3 y}{3 x - 1} ) - 3 ( - \frac{3 y}{3 x - 1} ) - 9 y}{( 3 x - 1 ) ^{2}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{18 y}{9 x ^{2} - 6 x + 1}

Show Solution

Conic

\frac{( x ^{'} - \frac{2}{6} ) ^{2}}{2} - \frac{( y ^{'} + \frac{2}{6} ) ^{2}}{2} = 1

Evaluate

9 x (3 \times \frac{y}{3}) - 3 y = 9

Move the expression to the left side

9 x (3 \times \frac{y}{3}) - 3 y - 9 = 0

Calculate

More Steps

9 x y - 3 y - 9 = 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 x y - 3 y - 9 = 0

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

Calculate

More Steps

Calculate

9 (x^{'} \times \frac{2}{2} - y^{'} \times \frac{2}{2}) (x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}) - 3 (x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}) - 9

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

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

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

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^{'}) - 3 (x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}) - 9

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^{'}) - 3 (\frac{2}{2} x^{'} + y^{'} \times \frac{2}{2}) - 9

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^{'}) - 3 (\frac{2}{2} x^{'} + \frac{2}{2} y^{'}) - 9

Expand the expression

More Steps

\frac{9}{2} (x^{'})^{2} - \frac{9}{2} (y^{'})^{2} - 3 (\frac{2}{2} x^{'} + \frac{2}{2} y^{'}) - 9

Expand the expression

More Steps

\frac{9}{2} (x^{'})^{2} - \frac{9}{2} (y^{'})^{2} - \frac{3 2}{2} x^{'} - \frac{3 2}{2} y^{'} - 9

\frac{9}{2} (x^{'})^{2} - \frac{9}{2} (y^{'})^{2} - \frac{3 2}{2} x^{'} - \frac{3 2}{2} y^{'} - 9 = 0

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

\frac{9}{2} (x^{'})^{2} - \frac{9}{2} (y^{'})^{2} - \frac{3 2}{2} x^{'} - \frac{3 2}{2} y^{'} = 0 - (- 9)

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

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

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

\frac{9}{2} (x^{'})^{2} - \frac{9}{2} (y^{'})^{2} - \frac{3 2}{2} x^{'} - \frac{3 2}{2} y^{'} = 9

Use the commutative property to reorder the terms

\frac{9}{2} (x^{'})^{2} - \frac{3 2}{2} x^{'} - \frac{9}{2} (y^{'})^{2} - \frac{3 2}{2} y^{'} = 9

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

\frac{9}{2} (x^{'})^{2} - \frac{3 2}{2} x^{'} + \frac{1}{4} - \frac{9}{2} (y^{'})^{2} - \frac{3 2}{2} y^{'} = 9 + \frac{1}{4}

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

\frac{9}{2} ((x^{'})^{2} - \frac{2}{3} x^{'} + \frac{1}{18}) - \frac{9}{2} (y^{'})^{2} - \frac{3 2}{2} y^{'} = 9 + \frac{1}{4}

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

\frac{9}{2} (x^{'} - \frac{2}{6})^{2} - \frac{9}{2} (y^{'})^{2} - \frac{3 2}{2} y^{'} = 9 + \frac{1}{4}

Add the numbers

More Steps

\frac{9}{2} (x^{'} - \frac{2}{6})^{2} - \frac{9}{2} (y^{'})^{2} - \frac{3 2}{2} y^{'} = \frac{37}{4}

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

\frac{9}{2} (x^{'} - \frac{2}{6})^{2} - \frac{9}{2} (y^{'})^{2} - \frac{3 2}{2} y^{'} - \frac{1}{4} = \frac{37}{4} - \frac{1}{4}

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

\frac{9}{2} (x^{'} - \frac{2}{6})^{2} - \frac{9}{2} ((y^{'})^{2} + \frac{2}{3} y^{'} + \frac{1}{18}) = \frac{37}{4} - \frac{1}{4}

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

\frac{9}{2} (x^{'} - \frac{2}{6})^{2} - \frac{9}{2} (y^{'} + \frac{2}{6})^{2} = \frac{37}{4} - \frac{1}{4}

Subtract the numbers

More Steps

\frac{9}{2} (x^{'} - \frac{2}{6})^{2} - \frac{9}{2} (y^{'} + \frac{2}{6})^{2} = 9

Multiply both sides of the equation by \frac{1}{9}

\frac{9}{2} (x^{'} - \frac{2}{6})^{2} - \frac{9}{2} (y^{'} + \frac{2}{6})^{2} \times \frac{1}{9} = 9 \times \frac{1}{9}

Multiply the terms

More Steps

\frac{1}{2} (x^{'} - \frac{2}{6})^{2} - \frac{1}{2} (y^{'} + \frac{2}{6})^{2} = 9 \times \frac{1}{9}

Multiply the terms

More Steps

\frac{1}{2} (x^{'} - \frac{2}{6})^{2} - \frac{1}{2} (y^{'} + \frac{2}{6})^{2} = 1

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

\frac{( x ^{'} - \frac{2}{6} ) ^{2}}{2} - \frac{1}{2} (y^{'} + \frac{2}{6})^{2} = 1

Solution

\frac{( x ^{'} - \frac{2}{6} ) ^{2}}{2} - \frac{( y ^{'} + \frac{2}{6} ) ^{2}}{2} = 1

Show Solution