The differential equation $\frac{dy}{dx} = \frac{x+y-1}{x+y+1}$ reduces to variable separable form by making the substitution

$x+y = v$

The correct answer is option (1) : $x+y = v$

Let $x+y=v$. Then, $1+\frac{dy}{dx} =\frac{dv}{dx}.$

Substituting these values in the given differential equation,

we get

$\frac{dv}{dx}-1=\frac{v-1}{v+1}$

$⇒\frac{dv}{dx}=\frac{2v}{v+1}$

$⇒\frac{v+1}{2v}dv=dx$

Clearly, it is in variable separable form.