Solve y 14x-24y = 50

Question

Solve the equation

Solve for x

Solve for y

x = \frac{25 + 12 y}{7 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{12 sin ( θ ) + 144 sin ^{2} ( θ ) + 350 sin ( 2 θ )}{7 sin ( 2 θ )} r = \frac{12 sin ( θ ) - 144 sin ^{2} ( θ ) + 350 sin ( 2 θ )}{7 sin ( 2 θ )}

Evaluate

y \times 14 x - 24 y = 50

Use the commutative property to reorder the terms

14 y x - 24 y = 50

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

14 sin (θ) \times r cos (θ) \times r - 24 sin (θ) \times r = 50

Factor the expression

14 sin (θ) cos (θ) \times r^{2} - 24 sin (θ) \times r = 50

Simplify the expression

7 sin (2 θ) \times r^{2} - 24 sin (θ) \times r = 50

Subtract the terms

7 sin (2 θ) \times r^{2} - 24 sin (θ) \times r - 50 = 50 - 50

Evaluate

7 sin (2 θ) \times r^{2} - 24 sin (θ) \times r - 50 = 0

Solve using the quadratic formula

r = \frac{24 sin ( θ ) \pm ( - 24 sin ( θ ) ) ^{2} - 4 \times 7 sin ( 2 θ ) ( - 50 )}{14 sin ( 2 θ )}

Simplify

r = \frac{24 sin ( θ ) \pm 576 sin ^{2} ( θ ) + 1400 sin ( 2 θ )}{14 sin ( 2 θ )}

Separate the equation into 2 possible cases

r = \frac{24 sin ( θ ) + 576 sin ^{2} ( θ ) + 1400 sin ( 2 θ )}{14 sin ( 2 θ )} r = \frac{24 sin ( θ ) - 576 sin ^{2} ( θ ) + 1400 sin ( 2 θ )}{14 sin ( 2 θ )}

Evaluate

More Steps

r = \frac{12 sin ( θ ) + 144 sin ^{2} ( θ ) + 350 sin ( 2 θ )}{7 sin ( 2 θ )} r = \frac{24 sin ( θ ) - 576 sin ^{2} ( θ ) + 1400 sin ( 2 θ )}{14 sin ( 2 θ )}

Solution

More Steps

r = \frac{12 sin ( θ ) + 144 sin ^{2} ( θ ) + 350 sin ( 2 θ )}{7 sin ( 2 θ )} r = \frac{12 sin ( θ ) - 144 sin ^{2} ( θ ) + 350 sin ( 2 θ )}{7 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{7 y}{7 x - 12}

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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{98 y}{49 x ^{2} - 168 x + 144}

Calculate

y 14 x - 24 y = 50

Simplify the expression

14 y x - 24 y = 50

Take the derivative of both sides

\frac{d}{d x} (14 y x - 24 y) = \frac{d}{d x} (50)

Calculate the derivative

More Steps

14 y + 14 x \frac{d y}{d x} - 24 \frac{d y}{d x} = \frac{d}{d x} (50)

Calculate the derivative

14 y + 14 x \frac{d y}{d x} - 24 \frac{d y}{d x} = 0

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

14 y + (14 x - 24) \frac{d y}{d x} = 0

Move the constant to the right side

(14 x - 24) \frac{d y}{d x} = 0 - 14 y

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

(14 x - 24) \frac{d y}{d x} = - 14 y

Divide both sides

\frac{( 14 x - 24 ) \frac{d y}{d x}}{14 x - 24} = \frac{- 14 y}{14 x - 24}

Divide the numbers

\frac{d y}{d x} = \frac{- 14 y}{14 x - 24}

Divide the numbers

More Steps

\frac{d y}{d x} = - \frac{7 y}{7 x - 12}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (- \frac{7 y}{7 x - 12})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (- \frac{7 y}{7 x - 12})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d}{d x} ( 7 y ) \times ( 7 x - 12 ) - 7 y \times \frac{d}{d x} ( 7 x - 12 )}{( 7 x - 12 ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{7 \frac{d y}{d x} \times ( 7 x - 12 ) - 7 y \times \frac{d}{d x} ( 7 x - 12 )}{( 7 x - 12 ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = - \frac{7 \frac{d y}{d x} \times ( 7 x - 12 ) - 7 y \times 7}{( 7 x - 12 ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = - \frac{49 x \frac{d y}{d x} - 84 \frac{d y}{d x} - 7 y \times 7}{( 7 x - 12 ) ^{2}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = - \frac{49 x \frac{d y}{d x} - 84 \frac{d y}{d x} - 49 y}{( 7 x - 12 ) ^{2}}

Use equation \frac{d y}{d x} = - \frac{7 y}{7 x - 12} to substitute

\frac{d ^{2} y}{d x ^{2}} = - \frac{49 x ( - \frac{7 y}{7 x - 12} ) - 84 ( - \frac{7 y}{7 x - 12} ) - 49 y}{( 7 x - 12 ) ^{2}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{98 y}{49 x ^{2} - 168 x + 144}

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Conic

\frac{( x ^{'} - \frac{6 2}{7} ) ^{2}}{\frac{50}{7}} - \frac{( y ^{'} + \frac{6 2}{7} ) ^{2}}{\frac{50}{7}} = 1

Evaluate

y \times 14 x - 24 y = 50

Move the expression to the left side

y \times 14 x - 24 y - 50 = 0

Use the commutative property to reorder the terms

14 y x - 24 y - 50 = 0

The coefficients A,B and C of the general equation are A= 0,B= 14 and C= 0

A = 0 B = 14 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}{14}

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 14 y x - 24 y - 50 = 0

14 (x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}) (x^{'} \times \frac{2}{2} - y^{'} \times \frac{2}{2}) - 24 (x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}) - 50 = 0

Calculate

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Calculate

14 (x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}) (x^{'} \times \frac{2}{2} - y^{'} \times \frac{2}{2}) - 24 (x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}) - 50

Use the commutative property to reorder the terms

14 (\frac{2}{2} x^{'} + y^{'} \times \frac{2}{2}) (x^{'} \times \frac{2}{2} - y^{'} \times \frac{2}{2}) - 24 (x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}) - 50

Use the commutative property to reorder the terms

14 (\frac{2}{2} x^{'} + \frac{2}{2} y^{'}) (x^{'} \times \frac{2}{2} - y^{'} \times \frac{2}{2}) - 24 (x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}) - 50

Use the commutative property to reorder the terms

14 (\frac{2}{2} x^{'} + \frac{2}{2} y^{'}) (\frac{2}{2} x^{'} - y^{'} \times \frac{2}{2}) - 24 (x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}) - 50

Use the commutative property to reorder the terms

14 (\frac{2}{2} x^{'} + \frac{2}{2} y^{'}) (\frac{2}{2} x^{'} - \frac{2}{2} y^{'}) - 24 (x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}) - 50

Use the commutative property to reorder the terms

14 (\frac{2}{2} x^{'} + \frac{2}{2} y^{'}) (\frac{2}{2} x^{'} - \frac{2}{2} y^{'}) - 24 (\frac{2}{2} x^{'} + y^{'} \times \frac{2}{2}) - 50

Use the commutative property to reorder the terms

14 (\frac{2}{2} x^{'} + \frac{2}{2} y^{'}) (\frac{2}{2} x^{'} - \frac{2}{2} y^{'}) - 24 (\frac{2}{2} x^{'} + \frac{2}{2} y^{'}) - 50

Expand the expression

More Steps

7 (x^{'})^{2} - 7 (y^{'})^{2} - 24 (\frac{2}{2} x^{'} + \frac{2}{2} y^{'}) - 50

Expand the expression

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7 (x^{'})^{2} - 7 (y^{'})^{2} - 122 \times x^{'} - 122 \times y^{'} - 50

7 (x^{'})^{2} - 7 (y^{'})^{2} - 122 \times x^{'} - 122 \times y^{'} - 50 = 0

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

7 (x^{'})^{2} - 7 (y^{'})^{2} - 122 \times x^{'} - 122 \times y^{'} = 0 - (- 50)

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

7 (x^{'})^{2} - 7 (y^{'})^{2} - 122 \times x^{'} - 122 \times y^{'} = 0 + 50

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

7 (x^{'})^{2} - 7 (y^{'})^{2} - 122 \times x^{'} - 122 \times y^{'} = 50

Use the commutative property to reorder the terms

7 (x^{'})^{2} - 122 \times x^{'} - 7 (y^{'})^{2} - 122 \times y^{'} = 50

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

7 (x^{'})^{2} - 122 \times x^{'} + \frac{72}{7} - 7 (y^{'})^{2} - 122 \times y^{'} = 50 + \frac{72}{7}

Factor out 7 from the expression

7 ((x^{'})^{2} - \frac{12 2}{7} x^{'} + \frac{72}{49}) - 7 (y^{'})^{2} - 122 \times y^{'} = 50 + \frac{72}{7}

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

7 (x^{'} - \frac{6 2}{7})^{2} - 7 (y^{'})^{2} - 122 \times y^{'} = 50 + \frac{72}{7}

Add the numbers

More Steps

7 (x^{'} - \frac{6 2}{7})^{2} - 7 (y^{'})^{2} - 122 \times y^{'} = \frac{422}{7}

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

7 (x^{'} - \frac{6 2}{7})^{2} - 7 (y^{'})^{2} - 122 \times y^{'} - \frac{72}{7} = \frac{422}{7} - \frac{72}{7}

Factor out - 7 from the expression

7 (x^{'} - \frac{6 2}{7})^{2} - 7 ((y^{'})^{2} + \frac{12 2}{7} y^{'} + \frac{72}{49}) = \frac{422}{7} - \frac{72}{7}

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

7 (x^{'} - \frac{6 2}{7})^{2} - 7 (y^{'} + \frac{6 2}{7})^{2} = \frac{422}{7} - \frac{72}{7}

Subtract the numbers

More Steps

7 (x^{'} - \frac{6 2}{7})^{2} - 7 (y^{'} + \frac{6 2}{7})^{2} = 50

Multiply both sides of the equation by \frac{1}{50}

7 (x^{'} - \frac{6 2}{7})^{2} - 7 (y^{'} + \frac{6 2}{7})^{2} \times \frac{1}{50} = 50 \times \frac{1}{50}

Multiply the terms

More Steps

\frac{7}{50} (x^{'} - \frac{6 2}{7})^{2} - \frac{7}{50} (y^{'} + \frac{6 2}{7})^{2} = 50 \times \frac{1}{50}

Multiply the terms

More Steps

\frac{7}{50} (x^{'} - \frac{6 2}{7})^{2} - \frac{7}{50} (y^{'} + \frac{6 2}{7})^{2} = 1

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

\frac{( x ^{'} - \frac{6 2}{7} ) ^{2}}{\frac{50}{7}} - \frac{7}{50} (y^{'} + \frac{6 2}{7})^{2} = 1

Solution

\frac{( x ^{'} - \frac{6 2}{7} ) ^{2}}{\frac{50}{7}} - \frac{( y ^{'} + \frac{6 2}{7} ) ^{2}}{\frac{50}{7}} = 1

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