Solution: Given $ x + y = 4.5 $, multiplying by 5 yields $ 5x + 5y = 5(x + y) = 5 \cdot 4.5 = oxed{22.5} $.
["Mastering Simple Algebra: How Multiplying Equations Simplifies Problem-Solving", "Effective problem-solving in algebra often begins with a single, elegant step—one that transforms a straightforward equation into a powerful tool. Consider the simple relationship:\n$$ x + y = 4.5 $$\nAt first glance, this equation defines a linear relationship between two variables. But what happens when we apply a fundamental algebraic principle—multiplication by a constant?", "---", "### The Power of Multiplication in Algebra", "When we multiply both sides of an equation by the same non-zero value, the equality remains unchanged. This property preserves balance and allows us to simplify expressions efficiently. In this case, multiplying both sides of $ x + y = 4.5 $ by 5 enables us to rewrite the expression more conveniently.", "$$\n5(x + y) = 5 \cdot 4.5\n$$", "Thanks to the distributive property of multiplication over addition:", "$$\n5(x + y) = 5x + 5y\n$$", "So:", "$$\n5x + 5y = 5 \cdot 4.5 = \boxed{22.5}\n$$", "This transformation does more than just compute a number—it strengthens our understanding of variable relationships and paves the way for solving more complex equations.", "---", "### Why This Technique Matters", "1. Efficiency in Calculation\n Multiplying through simplifies expressions, making subsequent algebra easier and reducing the chance of error.", "2. Foundation for Higher Math\n This principle extends beyond basic addition—core to solving systems of equations, linear programming, and calculus.", "3. Real-World Applications\n Multiplying linear equations by constants appears in physics, economics, and engineering when scaling measurements or adjusting rates.", "---", "### Step-by-Step Breakdown", "1. Start with the equation:\n $$ x + y = 4.5 $$", "2. Multiply both sides by 5:\n $$ 5(x + y) = 5 \cdot 4.5 $$", "3. Apply the distributive law:\n $$ 5x + 5y = 22.5 $$", "This final expression reveals the weighted sum of $ x $ and $ y $ scaled uniformly—file for immediate use in further computations.", "---", "### Conclusion", "In algebra, transformation is key. By recognizing $ x + y = 4.5 $ as a multiplier-friendly form, multiplying by 5 turns a sum into a scaled expression with clarity and precision. Whether you're a student building foundational skills or a professional applying mathematical models, mastering this concept accelerates learning and enhances problem-solving agility.", "Start small. Multiply smartly — because every equation holds the potential for a powerful simplified truth."]
