Solve x y-x=7 - ScanMath

Question

Solve the equation

Solve for x

Solve for y

x = \frac{7}{y - 1}

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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{cos ( θ ) + cos ^{2} ( θ ) + 14 sin ( 2 θ )}{sin ( 2 θ )} r = \frac{cos ( θ ) - cos ^{2} ( θ ) + 14 sin ( 2 θ )}{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 + 1}{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 - 2}{x ^{2}}

Calculate

x y - x = 7

Take the derivative of both sides

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

Calculate the derivative

More Steps

y + x \frac{d y}{d x} - 1 = \frac{d}{d x} (7)

Calculate the derivative

y + x \frac{d y}{d x} - 1 = 0

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

x \frac{d y}{d x} = 0 - (y - 1)

Subtract the terms

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

Divide both sides

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

Divide the numbers

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

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

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

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

Solution

More Steps

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

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Conic

\frac{( x ^{'} - \frac{2}{2} ) ^{2}}{14} - \frac{( y ^{'} - \frac{2}{2} ) ^{2}}{14} = 1

Evaluate

x y - x = 7

Move the expression to the left side

x y - x - 7 = 0

The coefficients A,B and C of the general equation are A= 0,B= 1 and C= 0

A = 0 B = 1 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}{1}

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 x y - x - 7 = 0

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

Calculate

More Steps

\frac{1}{2} (x^{'})^{2} - \frac{1}{2} (y^{'})^{2} - \frac{2}{2} x^{'} + \frac{2}{2} y^{'} - 7 = 0

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

\frac{1}{2} (x^{'})^{2} - \frac{1}{2} (y^{'})^{2} - \frac{2}{2} x^{'} + \frac{2}{2} y^{'} = 0 - (- 7)

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{1}{2} (x^{'})^{2} - \frac{1}{2} (y^{'})^{2} - \frac{2}{2} x^{'} + \frac{2}{2} y^{'} = 0 + 7

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

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

Use the commutative property to reorder the terms

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

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

\frac{1}{2} (x^{'})^{2} - \frac{2}{2} x^{'} + \frac{1}{4} - \frac{1}{2} (y^{'})^{2} + \frac{2}{2} y^{'} = 7 + \frac{1}{4}

Factor out \frac{1}{2} from the expression

\frac{1}{2} ((x^{'})^{2} - 2 \times x^{'} + \frac{1}{2}) - \frac{1}{2} (y^{'})^{2} + \frac{2}{2} y^{'} = 7 + \frac{1}{4}

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

\frac{1}{2} (x^{'} - \frac{2}{2})^{2} - \frac{1}{2} (y^{'})^{2} + \frac{2}{2} y^{'} = 7 + \frac{1}{4}

Add the numbers

More Steps

\frac{1}{2} (x^{'} - \frac{2}{2})^{2} - \frac{1}{2} (y^{'})^{2} + \frac{2}{2} y^{'} = \frac{29}{4}

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

\frac{1}{2} (x^{'} - \frac{2}{2})^{2} - \frac{1}{2} (y^{'})^{2} + \frac{2}{2} y^{'} - \frac{1}{4} = \frac{29}{4} - \frac{1}{4}

Factor out - \frac{1}{2} from the expression

\frac{1}{2} (x^{'} - \frac{2}{2})^{2} - \frac{1}{2} ((y^{'})^{2} - 2 \times y^{'} + \frac{1}{2}) = \frac{29}{4} - \frac{1}{4}

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

\frac{1}{2} (x^{'} - \frac{2}{2})^{2} - \frac{1}{2} (y^{'} - \frac{2}{2})^{2} = \frac{29}{4} - \frac{1}{4}

Subtract the numbers

More Steps

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

Multiply both sides of the equation by \frac{1}{7}

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

Multiply the terms

More Steps

\frac{1}{14} (x^{'} - \frac{2}{2})^{2} - \frac{1}{14} (y^{'} - \frac{2}{2})^{2} = 7 \times \frac{1}{7}

Multiply the terms

More Steps

\frac{1}{14} (x^{'} - \frac{2}{2})^{2} - \frac{1}{14} (y^{'} - \frac{2}{2})^{2} = 1

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

\frac{( x ^{'} - \frac{2}{2} ) ^{2}}{14} - \frac{1}{14} (y^{'} - \frac{2}{2})^{2} = 1

Solution

\frac{( x ^{'} - \frac{2}{2} ) ^{2}}{14} - \frac{( y ^{'} - \frac{2}{2} ) ^{2}}{14} = 1

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