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

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

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solvefor x,x+y=e^{x-y}

solve for

x, x + y = e^{x - y}

x = - W (- \frac{1}{e ^{2 y}}) - y

Solution steps

x + y = e^{x - y}

Prepare

x + y = e^{x - y}

for Lambert form:

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

Rewrite the equation with

- x - y = u

and

x = - u - y

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

Simplify

- e^{u + 2 y} u = 1

Rewrite

- e^{u + 2 y} u = 1

in Lambert form:

e^{u} u = - \frac{1}{e ^{2 y}}

Solve

e^{u} u = - \frac{1}{e ^{2 y}} : u = W (- \frac{1}{e ^{2 y}})

u = W (- \frac{1}{e ^{2 y}})

Substitute back

u = - x - y,

solve for

x

Solve

- x - y = W (- \frac{1}{e ^{2 y}}) : x = - W (- \frac{1}{e ^{2 y}}) - y

x = - W (- \frac{1}{e ^{2 y}}) - y

2^{3 x} = 64

e^{x} x + 2 e^{x} = 0

- 1^{2} \cdot (- 1^{n}) = - 1^{2 n}

(\frac{1}{2})^{x} = 0

x, 2 7^{2 x} = 9^{x - 3}

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