* Creation of Linear Functions from Two Given Points *

Euclidean Geometry states that Two Points determine a Line

therefore given two points in a Rectangular Coordinate System

a Linear Function can be created with one specific equation.

The Standard Form for a Linear Function is: Ax + By = C

The Point Slope Form for a Linear Function is: y - y₁ = m(x - x₁)

The Slope, which is defined as the Rise / Run, can be determined by

the difference of the Y values over the difference of the X values.

Given two points (ordered pairs) in a Rectangular Coordinate System.

Therefore Slope is defined as follows: m = ( Y₂ – Y₁ ) / (X₂ – X₁)

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Given: P₂ = ( -3, +1 ) & P₁ = ( +2, -3 ) Determine equation of a Linear Function

Slope as defined as Rise / Run: m = ( +1 - -3) / (-3 - +2) = (+4) / (-5) = -4/5

Using Slope Relation again: ( Y₂ – Y₁ ) = m (X₂ – X₁) and either of the points.

(Y₂ - -3) = (-4/5) (X₂ - +2) Y +3 = -4/5 (X- 2) +5Y +15 = -4X +8

Therefore the Linear Function for the given two points: +4X +5Y = -7

Check by determing the X and Y intercepts.

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Given: P₂ = ( +1,+2 ) and P₁ = ( -3, -2 ) Determine the equation of a Linear Function

Slope as defined as Rise / Run: m = ( +2 - -2) / (+1 - -3) = (+4) / (+4) = +1

Using Slope Relation again: ( Y₂ – Y₁ ) = m (X₂ – X₁) and either of the points.

(Y₂ - -2) = (+1) (X₂ - -3) Y + 2 = +1 (X+ 3) +Y + 2 = +X + 3

Therefore the Linear Function for the given two points: -X +Y = +1

Check by determing the X and Y intercepts.

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