Step 1: Two angles are complementary when their sum is $90^{\circ}$. So the first equation is
$$x+y=90 \quad \cdots (i)$$
Step 2: The ratio is given as $x:y=1:2$, i.e. $\dfrac{x}{y}=\dfrac{1}{2}$. Cross-multiplying,
$$2x=y \;\Rightarrow\; 2x-y=0 \quad \cdots (ii)$$
Step 3: So the required system of linear equations is $x+y=90$ and $2x-y=0$.
Step 4: Add (i) and (ii) to eliminate $y$.
$$3x=90 \;\Rightarrow\; x=30$$
Step 5: Substitute $x=30$ in (i).
$$30+y=90 \;\Rightarrow\; y=60$$
Step 6: Verification: $30+60=90$ ✓ and $30:60=1:2$ ✓
Hence $x=30^{\circ}$ and $y=60^{\circ}$.