1. **State the problem:** We are given that $y = 4x$ and that angles $x$ and $y$ are adjacent angles on a straight line. 2. **Understand the relationship:** Adjacent angles on a straight line add up to 180 degrees. So, we have the equation: $$x + y = 180$$ 3. **Substitute the given expression:** Since $y = 4x$, substitute this into the equation: $$x + 4x = 180$$ 4. **Simplify the equation:** $$5x = 180$$ 5. **Solve for $x$:** $$x = \frac{180}{5}$$ 6. **Simplify the fraction:** $$x = 36$$ 7. **Conclusion:** The size of angle $x$ is 36 degrees.