Question: A high school student participating in a STEM competition is asked to find the point on the line $ y = 3x - 2 $ that is closest to the point $ (4, 1) $. Determine the coordinates of this point.

Mon, 04 May 2026 18:10:52 +0000

**Discover the Surmath of Coordinates: A Gardener’s Path by Math**
When a high school student steps into a STEM competition, one of the common challenges isn’t just solving equations—it’s making sense of geometric relationships in real time. Take this focused question: *What point on the line* $ y = 3x - 2 $ *is closest to the point* $ (4, 1) $? It’s a problem that blends precision, analysis, and practical insight—skills the next generation of innovators rely on. More than a routine calculus problem, it reveals how data and spatial reasoning intersect across science, technology, and everyday curiosity. Solving it step by step builds confidence and understanding. **Why This Question Matters in Today’s STEM Landscape**
As US education emphasizes STEM as a gateway to innovation, problems like this connect classroom math to real-world applications—from robotics navigation to architectural design. Discovered frequently in student forums and competition prep resources, this question reflects a growing trend: young learners building foundational analytical habits through real-world geometry. It’s not flashy, but it trains critical thinking in a way that matters, mirroring the structured, evidence-based reasoning colleges and employers value. **How to Find the Closest Point on the Line**
The goal is simple: find the point on the line $ y = 3x - 2 $ where the distance to $ (4, 1) $ is minimized. This occurs at the perpendicular projection—where the line from $ (4, 1) $ meets the target line at 90 degrees. Begin by noting the slope of the given line is $ 3 $. The perpendicular line carries slope $ -\frac{1}{3} $. Using the point-slope form, the equation of this perpendicular line through $ (4, 1) $ becomes:
$ y - 1 = -\frac{1}{3}(x - 4) $
Simplify to:
$ y = -\frac{1}{3}x + \frac{7}{3} $ Now solve the system of equations:
$ y = 3x - 2 $
$ y = -\frac{1}{3}x + \frac{7}{3} $