Solve 4x^5y-3z^24=0

Question

Solve the equation

Solve for x

Solve for y

Solve for z

x = \frac{5 3 z ^{2} y ^{4}}{y}

Evaluate

4 x^{5} y - 3 z^{2} \times 4 = 0

Multiply the terms

4 x^{5} y - 12 z^{2} = 0

Rewrite the expression

4 y x^{5} - 12 z^{2} = 0

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

4 y x^{5} = 0 + 12 z^{2}

Add the terms

4 y x^{5} = 12 z^{2}

Divide both sides

\frac{4 y x ^{5}}{4 y} = \frac{12 z ^{2}}{4 y}

Divide the numbers

x^{5} = \frac{12 z ^{2}}{4 y}

Cancel out the common factor 4

x^{5} = \frac{3 z ^{2}}{y}

Take the 5 -th root on both sides of the equation

5 x^{5} = 5 \frac{3 z ^{2}}{y}

Calculate

x = 5 \frac{3 z ^{2}}{y}

Solution

More Steps

Evaluate

5 \frac{3 z ^{2}}{y}

To take a root of a fraction,take the root of the numerator and denominator separately

\frac{5 3 z ^{2}}{5 y}

Multiply by the Conjugate

\frac{5 3 z ^{2} \times 5 y ^{4}}{5 y \times 5 y ^{4}}

Calculate

\frac{5 3 z ^{2} \times 5 y ^{4}}{y}

The product of roots with the same index is equal to the root of the product

\frac{5 3 z ^{2} y ^{4}}{y}

x = \frac{5 3 z ^{2} y ^{4}}{y}

Show Solution

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{5 x ^{4} y}{6 z}

Evaluate

4 x^{5} y - 3 z^{2} \times 4 = 0

Multiply the terms

4 x^{5} y - 12 z^{2} = 0

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

\frac{\partial}{\partial x} (4 x^{5} y - 12 z^{2}) = \frac{\partial}{\partial x} (0)

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} (4 x^{5} y) - \frac{\partial}{\partial x} (12 z^{2}) = \frac{\partial}{\partial x} (0)

Evaluate

More Steps

Evaluate

\frac{\partial}{\partial x} (4 x^{5} y)

Use differentiation rule \frac{\partial}{\partial x} (c f (x)) = c \times \frac{\partial}{\partial x} (f (x))

4 y \times \frac{\partial}{\partial x} (x^{5})

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

4 y \times 5 x^{4}

Multiply the terms

20 x^{4} y

20 x^{4} y - \frac{\partial}{\partial x} (12 z^{2}) = \frac{\partial}{\partial x} (0)

Evaluate

More Steps

Evaluate

\frac{\partial}{\partial x} (12 z^{2})

Use differentiation rule \frac{\partial}{\partial x} (c f (x)) = c \times \frac{\partial}{\partial x} (f (x))

12 \times \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

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

Find the derivative

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

Multiply the terms

24 z \frac{\partial z}{\partial x}

20 x^{4} y - 24 z \frac{\partial z}{\partial x} = \frac{\partial}{\partial x} (0)

Find the partial derivative

20 x^{4} y - 24 z \frac{\partial z}{\partial x} = 0

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

- 24 z \frac{\partial z}{\partial x} = 0 - 20 x^{4} y

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

- 24 z \frac{\partial z}{\partial x} = - 20 x^{4} y

Divide both sides

\frac{- 24 z \frac{\partial z}{\partial x}}{- 24 z} = \frac{- 20 x ^{4} y}{- 24 z}

Divide the numbers

\frac{\partial z}{\partial x} = \frac{- 20 x ^{4} y}{- 24 z}

Solution

\frac{\partial z}{\partial x} = \frac{5 x ^{4} y}{6 z}

Show Solution