Question: A financial model predicts revenue from two products with $ x + y = 4.5 $ and $ x^2 + y^2 = 10.25 $. What is $ 5x + 5y $?
["Title: Solve for Revenue Prediction: Using $ x + y = 4.5 $ and $ x^2 + y^2 = 10.25 $ to Find $ 5x + 5y $", "Understanding financial models is essential for accurate revenue forecasting, especially when multiple product lines contribute to overall revenue. In one such advanced scenario, financial analysts often work with equations involving two variables—representing sales or revenue streams—to determine optimal outcomes.", "### Given Equations:\nIn a key forecasting model, we are provided with:\n- $ x + y = 4.5 $\n- $ x^2 + y^2 = 10.25 $", "Where $ x $ and $ y $ represent revenue from two distinct products or market segments. The goal is to calculate $ 5x + 5y $ using these constraints.", "### Step-by-Step Solution", "Step 1: Recognize the identity for $ (x + y)^2 $\nWe know:\n$$\n(x + y)^2 = x^2 + 2xy + y^2\n$$\nSubstitute the given values:\n$$\n(4.5)^2 = 10.25 + 2xy\n\Rightarrow 20.25 = 10.25 + 2xy\n\Rightarrow 2xy = 10\n\Rightarrow xy = 5\n$$", "Step 2: Use $ 5x + 5y $ for simplification\nFactor out the 5:\n$$\n5x + 5y = 5(x + y)\n$$\nSubstitute $ x + y = 4.5 $:\n$$\n5(x + y) = 5 \ imes 4.5 = 22.5\n$$", "### Conclusion:\nThe predicted total revenue contribution from both products is $ 22.5. This demonstrates how mathematical relationships in financial modeling support precise forecasting, enabling better decision-making for budgeting, resource allocation, and strategic planning.", "For financial professionals and business analysts, solving such equations efficiently translates directly into actionable insights—proving that models like the one above are powerful tools in predicting and optimizing revenue streams.", "---", "Keywords: financial model, revenue prediction, $ x + y = 4.5 $, $ x^2 + y^2 = 10.25 $, 5x + 5y calculation, financial forecasting, algebra in business, solve for variables", "Meta Description: Learn how to use $ x + y = 4.5 $ and $ x^2 + y^2 = 10.25 $ to calculate $ 5x + 5y $ in financial modeling. Step-by-step solution with financial relevance."]
