decoupling and integrating differential equations I am having trouble with the process of decoupling. If I have
$$\frac{dx}{dt}=-x+y$$
$$\frac{dy}{dt}=-x-y$$
I am trying to figure out how to solve for $x(t)$ and $y(t)$ by decoupling the system so that I only have one variable but I can't seem to get anywhere

Compute $d^2 x/dt^2$, giving
$$\frac{d^2 x}{dt^2} = - \frac{dx}{dt} + \frac{dy}{dt}$$
Use the second equation to replace $\frac{dy}{dt}$ with an expression in x and y, and use the first equation to replace $y$ with an expression in $dx/dt$ and $x$. The result is a second order equation in x.
Edit: For clarity, this gives
$$x'' = -x' + y' = -x' + (-x - y) = -x' -x - y = -x' - x - (x' + x)$$
$$x'' = -2x' - 2x$$
