1. **State the problem:** Solve the linear equation $12x + 72 = 10x + 78$ for $x$. 2. **Write down the equation:** $$12x + 72 = 10x + 78$$ 3. **Isolate the variable terms on one side:** Subtract $10x$ from both sides: $$12x + 72 - 10x = 10x + 78 - 10x$$ $$12x - \cancel{10x} + 72 = \cancel{10x} + 78$$ $$2x + 72 = 78$$ 4. **Isolate the constant term on the other side:** Subtract $72$ from both sides: $$2x + 72 - 72 = 78 - 72$$ $$2x + \cancel{72} = 6$$ $$2x = 6$$ 5. **Solve for $x$ by dividing both sides by 2:** $$\frac{2x}{\cancel{2}} = \frac{6}{\cancel{2}}$$ $$x = 3$$ 6. **Final answer:** $$x = 3$$