What is the answer to solvefor x,xy=e^{4x-y}-1 ?

The answer to solvefor x,xy=e^{4x-y}-1 is x=-(\W(-4/(e^{\frac{y^2+4){y}}y})y+4)/(4y)

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solvefor x,xy=e^{4x-y}-1

solve for

x, x y = e^{4 x - y} - 1

x = - \frac{W ( - \frac{4}{e ^{\frac{y ^{2} + 4}{y}} y} ) y + 4}{4 y}

Solution steps

x y = e^{4 x - y} - 1

Prepare

x y = e^{4 x - y} - 1

for Lambert form:

1 = (x y + 1) e^{- 4 x + y}

Rewrite the equation with

- 4 x - \frac{4}{y} = u

and

x = - \frac{y u + 4}{4 y}

1 = ((- \frac{y u + 4}{4 y}) y + 1) e^{- 4 (- \frac{y u + 4}{4 y}) + y}

Simplify

1 = e^{\frac{y u + 4}{y} + y} (- \frac{y u + 4}{4} + 1)

Rewrite

1 = e^{\frac{y u + 4}{y} + y} (- \frac{y u + 4}{4} + 1)

in Lambert form:

e^{u} u = - \frac{4}{e ^{\frac{y ^{2} + 4}{y}} y}

Solve

e^{u} u = - \frac{4}{e ^{\frac{y ^{2} + 4}{y}} y} : u = W (- \frac{4}{e ^{\frac{y ^{2} + 4}{y}} y})

u = W (- \frac{4}{e ^{\frac{y ^{2} + 4}{y}} y})

Substitute back

u = - 4 x - \frac{4}{y},

solve for

x

Solve

- 4 x - \frac{4}{y} = W (- \frac{4}{e ^{\frac{y ^{2} + 4}{y}} y}) : x = - \frac{W ( - \frac{4}{e ^{\frac{y ^{2} + 4}{y}} y} ) y + 4}{4 y}

x = - \frac{W ( - \frac{4}{e ^{\frac{y ^{2} + 4}{y}} y} ) y + 4}{4 y}

(\frac{2}{3})^{x} = \frac{4}{9}

3^{x - 2} = 7

(- 2)^{x} = - 32

4.7 = 3000000 (1 + 0.23)^{n}

1 - e^{2 x} = 0

What is the answer to solvefor x,xy=e^{4x-y}-1 ?
The answer to solvefor x,xy=e^{4x-y}-1 is x=-(\W(-4/(e^{\frac{y^2+4){y}}y})y+4)/(4y)