Solve 2(8x) 4(7y)=12

Question

Solve the equation

Solve for x

Solve for y

x = \frac{3}{112 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

Symmetry with respect to the origin

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Rewrite the equation

r = \frac{42 sin ( 2 θ )}{28 ∣ sin ( 2 θ ) ∣} r = - \frac{42 sin ( 2 θ )}{28 ∣ 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{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}} = \frac{2 y}{x ^{2}}

Calculate

2 (8 x) 4 (7 y) = 12

Simplify the expression

448 x y = 12

Take the derivative of both sides

\frac{d}{d x} (448 x y) = \frac{d}{d x} (12)

Calculate the derivative

More Steps

448 y + 448 x \frac{d y}{d x} = \frac{d}{d x} (12)

Calculate the derivative

448 y + 448 x \frac{d y}{d x} = 0

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

448 x \frac{d y}{d x} = 0 - 448 y

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

448 x \frac{d y}{d x} = - 448 y

Divide both sides

\frac{448 x \frac{d y}{d x}}{448 x} = \frac{- 448 y}{448 x}

Divide the numbers

\frac{d y}{d x} = \frac{- 448 y}{448 x}

Divide the numbers

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\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

More Steps

\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 ( - \frac{y}{x} ) - y}{x ^{2}}

Solution

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

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Conic

\frac{( x ^{'} ) ^{2}}{\frac{3}{56}} - \frac{( y ^{'} ) ^{2}}{\frac{3}{56}} = 1

Evaluate

2 (8 x) \times 4 (7 y) = 12

Move the expression to the left side

2 (8 x) \times 4 (7 y) - 12 = 0

Calculate

More Steps

448 x y - 12 = 0

The coefficients A,B and C of the general equation are A= 0,B= 448 and C= 0

A = 0 B = 448 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}{448}

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 448 x y - 12 = 0

448 (x^{'} \times \frac{2}{2} - y^{'} \times \frac{2}{2}) (x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}) - 12 = 0

Calculate

More Steps

224 (x^{'})^{2} - 224 (y^{'})^{2} - 12 = 0

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

224 (x^{'})^{2} - 224 (y^{'})^{2} = 0 - (- 12)

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

224 (x^{'})^{2} - 224 (y^{'})^{2} = 0 + 12

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

224 (x^{'})^{2} - 224 (y^{'})^{2} = 12

Multiply both sides of the equation by \frac{1}{12}

(224 (x^{'})^{2} - 224 (y^{'})^{2}) \times \frac{1}{12} = 12 \times \frac{1}{12}

Multiply the terms

More Steps

\frac{56}{3} (x^{'})^{2} - \frac{56}{3} (y^{'})^{2} = 12 \times \frac{1}{12}

Multiply the terms

More Steps

\frac{56}{3} (x^{'})^{2} - \frac{56}{3} (y^{'})^{2} = 1

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

\frac{( x ^{'} ) ^{2}}{\frac{3}{56}} - \frac{56}{3} (y^{'})^{2} = 1

Solution

\frac{( x ^{'} ) ^{2}}{\frac{3}{56}} - \frac{( y ^{'} ) ^{2}}{\frac{3}{56}} = 1

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