# How do you solve 1 + (2+x-y)/(x+y) = 2/y?

Mar 10, 2017

$x = 0$ or $y = 1$

#### Explanation:

First rearrange the equation to put the fractions together:

$1 = \frac{2}{y} - \frac{2 + x - y}{x + y}$

Multiply out the denominators. This can be done in one step, but I've done it in two to show exactly what's happening:

Multiply out $x + y$:

$1 \left(x + y\right) = \frac{2 \left(x + y\right)}{y} - \left(2 + x - y\right)$

Multiply out the $y$:

$1 \left(x + y\right) y = 2 \left(x + y\right) - y \left(2 + x - y\right)$

Simplify:

$x y + {y}^{2} = 2 x + 2 y - 2 y - x y + {y}^{2}$

$x y + x y + \cancel{{y}^{2}} = 2 x + \cancel{2 y} - \cancel{2 y} + \cancel{{y}^{2}}$

$2 x y = 2 x$

$2 x y - 2 x = 0$

$2 x \left(y - 1\right) = 0$

and dividing by $2$ we get $x \left(y - 1\right) = 0$

as product of $x$ and $\left(y - 1\right)$ is $0$. we have

$x = 0$ or $y = 1$