1. **State the problem:** We need to find the missing angle $y$ in a circle intersected by secants and chords with given angles $34^\circ$, $18^\circ$, and $x^\circ$.\n\n2. **Recall the relevant theorem:** When two secants intersect outside a circle, the measure of the angle formed is half the difference of the intercepted arcs. The formula is:\n$$\text{Angle} = \frac{1}{2} |\text{arc}_1 - \text{arc}_2|$$\n\n3. **Identify the arcs and angles:** Given angles $34^\circ$ and $18^\circ$ correspond to arcs intercepted by the secants. The angle $x$ is related to these arcs, and $y$ is the angle we want to find.\n\n4. **Express $x$ in terms of arcs:** Using the formula for angle formed by two secants,\n$$x = \frac{1}{2} |34^\circ - 18^\circ| = \frac{1}{2} \times 16^\circ = 8^\circ$$\n\n5. **Find $y$ using the same principle:** The angle $y$ is formed by the other pair of arcs, which are $68^\circ$ and $x^\circ$. So,\n$$y = \frac{1}{2} |68^\circ - x| = \frac{1}{2} |68^\circ - 8^\circ| = \frac{1}{2} \times 60^\circ = 30^\circ$$\n\n6. **Final answer:**\n$$\boxed{y = 30^\circ}$$