Solve 2x^3y-z=4 - ScanMath

Question

Solve the equation

Solve for x

Solve for y

Solve for z

x = \frac{3 16 y ^{2} + 4 z y ^{2}}{2 y}

Evaluate

2 x^{3} y - z = 4

Rewrite the expression

2 y x^{3} - z = 4

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

2 y x^{3} = 4 + z

Divide both sides

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

Divide the numbers

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

Take the 3 -th root on both sides of the equation

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

Calculate

x = 3 \frac{4 + z}{2 y}

Simplify the root

More Steps

Evaluate

3 \frac{4 + z}{2 y}

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

\frac{3 4 + z}{3 2 y}

Multiply by the Conjugate

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

Calculate

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

Calculate

More Steps

Evaluate

3 4 + z \times 3 2^{2} y^{2}

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

3 (4 + z) \times 2^{2} y^{2}

Calculate the product

3 2^{4} y^{2} + 2^{2} z y^{2}

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

Calculate

\frac{3 16 y ^{2} + 2 ^{2} z y ^{2}}{2 y}

x = \frac{3 16 y ^{2} + 2 ^{2} z y ^{2}}{2 y}

Solution

x = \frac{3 16 y ^{2} + 4 z y ^{2}}{2 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} = 6 x^{2} y

Evaluate

2 x^{3} y - z = 4

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

\frac{\partial}{\partial x} (2 x^{3} y - z) = \frac{\partial}{\partial x} (4)

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} (2 x^{3} y) - \frac{\partial}{\partial x} (z) = \frac{\partial}{\partial x} (4)

Evaluate

More Steps

Evaluate

\frac{\partial}{\partial x} (2 x^{3} y)

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

2 y \times \frac{\partial}{\partial x} (x^{3})

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

2 y \times 3 x^{2}

Multiply the terms

6 x^{2} y

6 x^{2} y - \frac{\partial}{\partial x} (z) = \frac{\partial}{\partial x} (4)

Evaluate

6 x^{2} y - \frac{\partial z}{\partial x} = \frac{\partial}{\partial x} (4)

Find the partial derivative

6 x^{2} y - \frac{\partial z}{\partial x} = 0

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

- \frac{\partial z}{\partial x} = 0 - 6 x^{2} y

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

- \frac{\partial z}{\partial x} = - 6 x^{2} y

Solution

\frac{\partial z}{\partial x} = 6 x^{2} y

Show Solution