Solve y: 7x - 3y = 21

Question

Solve the equation

Solve for x

Solve for y

x = \frac{y}{147 + 21 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 = 0 r = \frac{sec ( θ ) - 147 csc ( θ )}{21}

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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 - 21 x ^{2}}

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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{42 y}{x - 42 x ^{2} + 441 x ^{3}}

Calculate

y : (7 x) - 3 y = 21

Simplify the expression

\frac{y}{7 x} - 3 y = 21

Take the derivative of both sides

\frac{d}{d x} (\frac{y}{7 x} - 3 y) = \frac{d}{d x} (21)

Calculate the derivative

More Steps

\frac{x \frac{d y}{d x} - y - 21 x ^{2} \frac{d y}{d x}}{7 x ^{2}} = \frac{d}{d x} (21)

Calculate the derivative

\frac{x \frac{d y}{d x} - y - 21 x ^{2} \frac{d y}{d x}}{7 x ^{2}} = 0

Simplify

x \frac{d y}{d x} - y - 21 x^{2} \frac{d y}{d x} = 0

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

(x - 21 x^{2}) \frac{d y}{d x} - y = 0

Move the constant to the right side

(x - 21 x^{2}) \frac{d y}{d x} = 0 + y

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

(x - 21 x^{2}) \frac{d y}{d x} = y

Divide both sides

\frac{( x - 21 x ^{2} ) \frac{d y}{d x}}{x - 21 x ^{2}} = \frac{y}{x - 21 x ^{2}}

Divide the numbers

\frac{d y}{d x} = \frac{y}{x - 21 x ^{2}}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{y}{x - 21 x ^{2}})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{y}{x - 21 x ^{2}})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( y ) \times ( x - 21 x ^{2} ) - y \times \frac{d}{d x} ( x - 21 x ^{2} )}{( x - 21 x ^{2} ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d y}{d x} \times ( x - 21 x ^{2} ) - y \times \frac{d}{d x} ( x - 21 x ^{2} )}{( x - 21 x ^{2} ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d y}{d x} \times ( x - 21 x ^{2} ) - y ( 1 - 42 x )}{( x - 21 x ^{2} ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{x \frac{d y}{d x} - 21 x ^{2} \frac{d y}{d x} - y ( 1 - 42 x )}{( x - 21 x ^{2} ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{x \frac{d y}{d x} - 21 x ^{2} \frac{d y}{d x} - ( y - 42 y x )}{( x - 21 x ^{2} ) ^{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} - 21 x ^{2} \frac{d y}{d x} - y + 42 y x}{( x - 21 x ^{2} ) ^{2}}

Use equation \frac{d y}{d x} = \frac{y}{x - 21 x ^{2}} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{x \times \frac{y}{x - 21 x ^{2}} - 21 x ^{2} \times \frac{y}{x - 21 x ^{2}} - y + 42 y x}{( x - 21 x ^{2} ) ^{2}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{42 y}{x - 42 x ^{2} + 441 x ^{3}}

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