1. **State the problem:** Solve the linear equation $7 + 3x = 9x + 1$ for $x$. 2. **Write down the equation:** $$7 + 3x = 9x + 1$$ 3. **Isolate variable terms on one side:** Subtract $3x$ from both sides: $$7 + \cancel{3x} - \cancel{3x} = 9x - 3x + 1$$ which simplifies to $$7 = 6x + 1$$ 4. **Isolate constant terms on the other side:** Subtract $1$ from both sides: $$7 - 1 = 6x + \cancel{1} - \cancel{1}$$ which simplifies to $$6 = 6x$$ 5. **Solve for $x$ by dividing both sides by 6:** $$\frac{6}{\cancel{6}} = \frac{6x}{\cancel{6}}$$ which simplifies to $$1 = x$$ 6. **Final answer:** $$x = 1$$