Solve the system of differential equations by systematic eli

Solve the system of differential equations by systematic elimination. (dx/dt) + 3x + (dy/dt) = 1 (dx/dt) - x + (dy/dt) - y = e^t

I have no idea how to even start this problem. Please explain and show all steps in your solution. Thank you!

SUbtracting the two equations gives

4x+y=1-e^t

y=1-4x-e^t

Differentiating gives

dy/dt=-4dx/dt-e^t

Substituting this in first equations gives

-4dx/dt-e^t+3x+dx/dt=1

-3dx/dt+3x=1+e^t

dx/dt-x=-(1+e^t)/3

Integrating factor is: e^{-t}

Multiplying gives

(dx/dt-x)e^{-t}=-(e^{-t}+1)/3

(xe^{-t})\'=-(e^{-t}+1)/3

Integrating gives

xe^{-t}=e^{-t}/3-t/3+C

y=1-4x-e^t=1-4(e^{-t}/3-t/3+C)-e^t

x=1/3-te^{t}/3+Ce^t