Getting equation from differential equations I have:
$\dfrac {dx} {dt}$=$-x+y$
$\dfrac {dy}{dt}$=$-x-y$
and I am trying to find $x(t)$ and $y(t)$ given that $x(0)=0$ and $y(0)=1$
I know to do this I need to decouple the equations so that I only have to deal with one variable but the decoupling is what I am having trouble with
Do I set them equal to each other and then just move like terms to separate sides getting two different equations and then integrate?

Suppose you change your variables from $(x,y)$ to $(u,v)$ according to the rule
$$\begin{cases}x=a u+bv \\y=cu+dv \end{cases}. $$
This can be written in a matrix form $\mathbf x=P \mathbf u$ where $P=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$.
The ODE transforms accordingly, from
$$\frac{ \mathrm d \mathbf x}{\mathrm d t}=A \mathbf x $$ to
$$P\frac{ \mathrm d \mathbf u}{\mathrm d t}=AP \mathbf u,   $$ or
$$\frac{ \mathrm d \mathbf u}{\mathrm d t}=P^{-1}AP \mathbf u.  $$
If you choose the "right" matrix $P$ the equations for $u,v$ become uncoupled and you can solve them. Then it takes a simple multiplication by $P^{-1}$ to get back to $x,y$. 
