1. **State the problem:** Solve the simultaneous equations: $$y = 3x + 2$$ $$y = 4x + 1$$ 2. **Set the equations equal to each other:** Since both expressions equal $y$, we can set them equal: $$3x + 2 = 4x + 1$$ 3. **Isolate $x$:** Subtract $3x$ from both sides: $$\cancel{3x} + 2 = 4x + 1 - \cancel{3x}$$ $$2 = x + 1$$ 4. **Solve for $x$:** Subtract 1 from both sides: $$2 - 1 = x + \cancel{1} - \cancel{1}$$ $$1 = x$$ 5. **Find $y$:** Substitute $x=1$ into one of the original equations, for example $y = 3x + 2$: $$y = 3(1) + 2 = 3 + 2 = 5$$ 6. **Final answer:** The solution to the system is: $$x = 1, \quad y = 5$$