Step 1: Remove the decimals by multiplying both equations by 10.
$$x+3y=10 \quad \cdots (i)$$
$$2x-y=-1 \quad \cdots (ii)$$
Step 2: From (ii), express $y$ in terms of $x$.
$$y=2x+1$$
Step 3: Substitute this in (i).
$$x+3(2x+1)=10$$
$$x+6x+3=10 \;\Rightarrow\; 7x=7 \;\Rightarrow\; x=1$$
Step 4: Put $x=1$ back in $y=2x+1$.
$$y=2(1)+1=3$$
Step 5: Verification: $0.1(1)+0.3(3)=0.1+0.9=1$ ✓ and $0.2(1)-0.1(3)=0.2-0.3=-0.1$ ✓
Hence $x=1$ and $y=3$.