1. **State the problem:** Solve the linear equation $2x + 3 = 5x - 6$ for $x$. 2. **Formula and rules:** To solve for $x$, we want to isolate $x$ on one side of the equation. We can do this by moving all terms involving $x$ to one side and constants to the other. 3. **Step 1: Move $x$ terms to one side.** Subtract $2x$ from both sides: $$2x + 3 - 2x = 5x - 6 - 2x$$ which simplifies to $$3 = 3x - 6$$ 4. **Step 2: Move constants to the other side.** Add $6$ to both sides: $$3 + 6 = 3x - 6 + 6$$ which simplifies to $$9 = 3x$$ 5. **Step 3: Solve for $x$.** Divide both sides by $3$: $$\frac{9}{\cancel{3}} = \frac{3x}{\cancel{3}}$$ which simplifies to $$3 = x$$ 6. **Final answer:** $$x = 3$$