First, make a score
Suppose each point has coordinates x₁ and x₂. A linear classifier computes a score: s = w₁x₁ + w₂x₂ + b. In this example, a nonnegative score means class A; a negative score means class B. The score of zero marks the boundary.
Move a point across the line
The experiment fixes the score to x₁ + x₂ − 3. Try moving x₁ or x₂ until the point crosses the line. Watch the score change sign. The line is a geometric picture of the same decision rule, not a second model. Try a point on each side of the line, then move it onto the boundary. Which side matches the sign of the score?
INTERACTIVE EXPLORATION
Move a point across the boundary
The model scores x₁ + x₂ − 3. Move the point and observe when its predicted class changes.
What if a line cannot separate the data?
Put matching labels at opposite corners of a square: no single straight line can perfectly separate those two classes. Changing weights moves or rotates the line, but it cannot change that limitation. This motivates the next question: can nonlinear operations make a more flexible boundary?
Check your understanding
If all weights and the bias are multiplied by 2, does the boundary change?
No. The score doubles but its sign stays the same; the zero set is unchanged.