2Question: Find the point on the line $ y = -2x + 5 $ that is closest to the point $ (3, -1) $.

Fri, 17 Apr 2026 03:48:47 +0000

**2Question: Find the point on the line $ y = -2x + 5 $ that is closest to the point $ (3, -1) $** — this query reflects a quiet but growing interest among US users navigating geometry-based decision-making, particularly in personal finance, urban planning, or app development contexts. Whether optimizing delivery routes, analyzing heatmaps, or exploring pure math curiosity, this question reveals how everyday people engage with precise spatial relationships. Now, discovering *how* to pinpoint that closest point unlocks not just a formula, but a practical mindset rooted in clarity and logic—ideal for moving browsers beyond gut feeling and into confident understanding. **Why This Geometry Problem Is Rising in Infinite Interest**
The query reflects a broader pattern: Americans increasingly seek accurate, actionable answers in technical fields, where precision matters. The line $ y = -2x + 5 $ isn’t just abstract math—it’s a cognitive shortcut for mapping relationships in life, from route planning to data modeling. Platforms like 2Question thrive here, serving mobile-first users who value clarity over clickbait, turning abstract equations into usable knowledge. As digital literacy blooms and spatial reasoning becomes foundational in education and tech, questions about proximity and optimization naturally grow—especially when presented with simple, intuitive methods. **How to Actually Find the Closest Point — Step by Step**
To locate the closest point on the line to $ (3, -1) $, we use a proven geometric principle: the shortest distance from a point to a line is along the perpendicular. Start with your line equation: $ y = -2x + 5 $. Its slope is $ -2 $, so the perpendicular line carries a slope that’s the negative reciprocal—$ \frac{1}{2} $. Using point-slope form, the perpendicular line through $ (3, -1) $ becomes $ y + 1 = \frac{1}{2}(x - 3) $. Now solve the system: $ y = -2x + 5 $ and $ y = \frac{1}{2}x - \frac{5}{2} $. Set them equal:
$ -2x + 5 = \frac{1}{2}x - \frac{5}{2} $
Multiply through by 2 to eliminate fractions:
$ -4x + 10 = x - 5 $
$ -5x = -15 $
$ x = 3 $