1. **State the problem:** Solve the equation $$45 + 2x = 5x - 9$$ and prove that $$x = 18$$. 2. **Formula and rules:** To solve linear equations, we use properties of equality such as the Subtraction Property, Addition Property, and Division Property. 3. **Step-by-step solution:** 1) Given equation: $$45 + 2x = 5x - 9$$ 2) Apply the Subtraction Property of Equality by subtracting $$2x$$ from both sides: $$45 + \cancel{2x} - \cancel{2x} = 5x - 9 - 2x$$ Simplifies to: $$45 = 3x - 9$$ 3) Apply the Addition Property of Equality by adding $$9$$ to both sides: $$45 + 9 = 3x - 9 + 9$$ Simplifies to: $$54 = 3x$$ 4) Apply the Division Property of Equality by dividing both sides by $$3$$: $$\frac{54}{\cancel{3}} = \frac{3x}{\cancel{3}}$$ Simplifies to: $$18 = x$$ 5) By the Symmetric Property of Equality, we write: $$x = 18$$ 4. **Answer:** The solution to the equation is $$x = 18$$. 5. **Algebraic reason for transition from statement 1 to 2:** The Subtraction Property of Equality justifies subtracting $$2x$$ from both sides to isolate terms.