Solve x^2 + ( y - 3 ) 2 + z^2 = 16

Question

Solve the equation

Solve for x

Solve for y

Solve for z

x = 22 - 2 y - z^{2} x = - 22 - 2 y - z^{2}

Evaluate

x^{2} + (y - 3) \times 2 + z^{2} = 16

Multiply the terms

x^{2} + 2 (y - 3) + z^{2} = 16

Rewrite the expression

x^{2} + 2 y - 6 + z^{2} = 16

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

x^{2} = 16 - (2 y - 6 + z^{2})

Subtract the terms

More Steps

Evaluate

16 - (2 y - 6 + z^{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

16 - 2 y + 6 - z^{2}

Add the numbers

22 - 2 y - z^{2}

x^{2} = 22 - 2 y - z^{2}

Take the root of both sides of the equation and remember to use both positive and negative roots

x = \pm 22 - 2 y - z^{2}

Solution

x = 22 - 2 y - z^{2} x = - 22 - 2 y - z^{2}

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Find the partial derivative

Find \frac{\partial z}{\partial x} by differentiating the equation directly

Find \frac{\partial z}{\partial y} by differentiating the equation directly

\frac{\partial z}{\partial x} = - \frac{x}{z}

Evaluate

x^{2} + (y - 3) \times 2 + z^{2} = 16

Multiply the terms

x^{2} + 2 (y - 3) + z^{2} = 16

Find \frac{\partial z}{\partial x} by taking the derivative of both sides with respect to x

\frac{\partial}{\partial x} (x^{2} + 2 (y - 3) + z^{2}) = \frac{\partial}{\partial x} (16)

Use differentiation rule \frac{\partial}{\partial x} (f (x) \pm g (x)) = \frac{\partial}{\partial x} (f (x)) \pm \frac{\partial}{\partial x} (g (x))

\frac{\partial}{\partial x} (x^{2}) + \frac{\partial}{\partial x} (2 (y - 3)) + \frac{\partial}{\partial x} (z^{2}) = \frac{\partial}{\partial x} (16)

Use \frac{\partial}{\partial x} x^{n} = n x^{n - 1} to find derivative

2 x + \frac{\partial}{\partial x} (2 (y - 3)) + \frac{\partial}{\partial x} (z^{2}) = \frac{\partial}{\partial x} (16)

Use \frac{\partial}{\partial x} (c) = 0 to find derivative

2 x + 0 + \frac{\partial}{\partial x} (z^{2}) = \frac{\partial}{\partial x} (16)

Evaluate

More Steps

Evaluate

\frac{\partial}{\partial x} (z^{2})

Use the chain rule \frac{\partial}{\partial x} (f (g)) = \frac{\partial}{\partial g} (f (g)) \times \frac{\partial}{\partial x} (g) where the g = z, to find the derivative

\frac{\partial}{\partial z} (z^{2}) \frac{\partial z}{\partial x}

Find the derivative

2 z \frac{\partial z}{\partial x}

2 x + 0 + 2 z \frac{\partial z}{\partial x} = \frac{\partial}{\partial x} (16)

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

2 x + 2 z \frac{\partial z}{\partial x} = \frac{\partial}{\partial x} (16)

Find the partial derivative

2 x + 2 z \frac{\partial z}{\partial x} = 0

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

2 z \frac{\partial z}{\partial x} = 0 - 2 x

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

2 z \frac{\partial z}{\partial x} = - 2 x

Divide both sides

\frac{2 z \frac{\partial z}{\partial x}}{2 z} = \frac{- 2 x}{2 z}

Divide the numbers

\frac{\partial z}{\partial x} = \frac{- 2 x}{2 z}

Solution

More Steps

Evaluate

\frac{- 2 x}{2 z}

Cancel out the common factor 2

\frac{- x}{z}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

- \frac{x}{z}

\frac{\partial z}{\partial x} = - \frac{x}{z}

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