1. **State the problem:** We need to find the value of $x$ such that $f(x) = g(x)$, where $f(x) = 2 + 3x$ and $g(x) = 7x + 1$. 2. **Set the functions equal:** Since $f(x) = g(x)$, we write the equation: $$2 + 3x = 7x + 1$$ 3. **Rearrange the equation:** Move all terms involving $x$ to one side and constants to the other: $$2 + 3x - 7x = 1$$ $$2 - 4x = 1$$ 4. **Isolate the variable term:** Subtract 2 from both sides: $$2 - 4x - 2 = 1 - 2$$ $$\cancel{2} - 4x - \cancel{2} = -1$$ $$-4x = -1$$ 5. **Solve for $x$:** Divide both sides by $-4$: $$x = \frac{-1}{-4}$$ $$x = \cancel{\frac{-1}{-4}} = 0.25$$ 6. **Final answer:** The value of $x$ is $0.25$ as a decimal.