Step 1: Write the equations.
$$3x+5y=8 \quad \cdots (i)$$
$$5x-3y=2 \quad \cdots (ii)$$
Step 2: To eliminate $y$, multiply (i) by 3 and (ii) by 5.
$$9x+15y=24 \quad \cdots (iii)$$
$$25x-15y=10 \quad \cdots (iv)$$
Step 3: Add (iii) and (iv).
$$34x=34 \;\Rightarrow\; x=1$$
Step 4: Substitute $x=1$ in (i).
$$3(1)+5y=8 \;\Rightarrow\; 5y=5 \;\Rightarrow\; y=1$$
Step 5: Verification in (ii): $5(1)-3(1)=2$ ✓
Hence $x=1$ and $y=1$.