Solution: The $ y $-intercept of a linear equation $ E(x) = mx + b $ is the value of $ E(x) $ when $ x = 0 $. Substituting $ x = 0 $ into $ E(x) = 3.5x + 12 $, we get $ E(0) = 3.5(0) + 12 = 12 $.

Sat, 18 Apr 2026 07:47:17 +0000

**The Strong Significance of the $ y $-Intercept in Linear Equations — Why It Still Matters Today** Curious about how math shapes the way we understand change, patterns, and growth? A simple concept lies at the heart of linear relationships: the $ y $-intercept. Known formally as the value of $ E(x) $ when $ x = 0 $, it reveals exactly where a line crosses the vertical axis. When working with the equation $ E(x) = 3.5x + 12 $, substituting $ x = 0 $ delivers a clear result: $ E(0) = 12 $. This foundational step forms the basis for interpreting trends in economics, science, and daily data analysis — sparking curiosity where numbers meet real life. **How This Linear Insight Shapes Real-World Problem Solving** The $ y $-intercept represents the starting point — the baseline before any change occurs. In practical terms, whether evaluating cost models with a flat fee or tracking growth from day one, that initial value anchors all predictions. For instance, a delivery service’s base rate plus per-mile charges follows a linear equation where $ E(0) $ reflects the flat fee before distance is added. This simplicity allows professionals and learners alike to decode complex systems quickly, fostering clearer decision-making across industries across the U.S.