The point-slope form of a linear equation takes the form:

(y - y₁) = m(x - x₁)

where m = slope and (x₁, y₁) is a coordinate point on the line.

This form is derived from the formula for slope: 

m = delta y/delta x = change in y/change in x = (y - y₁)/(x - x₁)

If m = (y - y₁)/(x - x₁), then multiply both sides by (x - x₁).  We get our point slope form:

(y - y₁) = m(x - x₁).

To put an equation into slope-intercept form, perform algebra manipulations as follows:

(y - y₁) = m(x - x₁)         Our equation in point-slope form

y - y₁ = mx - mx₁          Distribute m over x - x1

y = mx - mx₁ + y₁        Add y1 to both sides.  The quantity -mx₁ + y₁ is the y-intercept b.

The point-slope form of a line is helpful in that we can find an equation of a line given either:

• a point and a slope, or
• two points

If given a point and a slope, we replace m with the slope value and (x₁, y₁) with the value of the given point.  We can then put the equation into slope-intercept form (as described above) or any other form wanted.

When given two points, we use the two points to find the slope.   Then we pick either point to use as the point in the point-slope formula.  We now have a point and a slope and can find the equation of the line as described in the previous paragraph.

The class notes provide further explanation and examples.