Lines can be drawn in three dimensions, but most analytical geometry courses stick to lines in two dimensions, generally using the Cartesian or XY coordinate system. The generic equation for a line may follow the form:

y = mx + b

where m is the *slope* (measure of tilt or steep-ness) of the line, while b is its *intercept* or intersection with the y-axis.

## Equation for a Line from Two Points

A line can be determined and an equation derived from two points. In the Cartesian system, for instance, take two points, ( 2 , 3 ) and ( – 1 , 5 ). The first number in each pair represents the x-value of a point and the second number in each pair represents the y-value.

Writing these points into the general equation y = mx + b, we have for Point 1,

3 = m ( 2 ) + b

For point 2, we have,

5 = m ( – 1 ) + b

Solving the two-equation system for both in terms of b, we get, b = 3 – 2 m and b = 5 + m So 3 – 2 m = 5 + m and so, 3 m = – 2 and so,

m = – 2/3

Picking either of the two points and putting in this m value into the generic equation, 3 = – 2/3 ( 2 ) + b or, b = 9/3 + 4/3 = 13/3

b = 13/3

The equation for the line derived from the two points of our example is, therefore

y = – 2/3 x + 13/3

Verification of this equation is achieved by inserting the values of the second point into the equation,

## Verification?

5 = – 2/3 ( – 1 ) + 13/3 (Yes!)

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Yes, I think I remember this from a long time ago.