Equation of a Line (y = mx + c)

Two numbers describe every straight line

m is the slope — how much y changes for each unit of x. c is the intercept — the value of y when x is zero.

Change m and the line pivots. Change c and it slides up or down without tilting. Those are the only two things a straight line can do.

Finding the line through two points

Given (1, 5) and (3, 11):

m = (11 − 5) / (3 − 1) = 6 / 2 = 3

Substitute either point back in to get c. Using (1, 5): 5 = 3(1) + c, so c = 2.

The line is y = 3x + 2. Check with the other point: 3(3) + 2 = 11. Correct.

Where it shows up in machine learning

Linear regression is the search for the m and c that put the line closest to a cloud of points.

A neuron computes z = wx + b. That is the same equation with different letters: the weight is the slope, the bias is the intercept.

Higher dimensions keep the idea and add slopes. With two inputs, z = m₁x + m₂y + c is a plane.

What usually goes wrong

Confusing sign with steepness. A slope of −5 is steeper than a slope of 2. The minus sign only says which way it leans.

Expecting every line to fit this form. A vertical line has an undefined slope — x never changes, so the run is zero. Vertical lines are written x = k instead.

In one line

m tilts, c slides — and a neuron's weight and bias are the same two numbers under different names.