Final answer:
The equation relating the quotient of the vertical and horizontal side lengths to the slope of a line is m = (Y2 - Y1) / (X2 - X1). To justify whether a point is on the line, you compare the quotient of the differences in y and x of the point to the line's slope; if they match, the point is on the line.
Step-by-step explanation:
To create an equation relating the quotient of the vertical and horizontal side lengths of a slope triangle to the slope of a line, we use the concept of 'rise over run'. The slope (m) is the ratio of the change in the y-value (Δy or Y₂-Y₁) over the change in the x-value (Δx or X₂-X₁). Therefore, the equation that represents the slope of a line is:
m = (Y₂ - Y₁) / (X₂ - X₁)
For example, using Figure A1, with a slope of 3, and a y-intercept of 9, we can create the equation:
y = 3x + 9
This equation shows that for every one unit increase in x (horizontal), there is a three-unit increase in y (vertical). To justify whether a point is on a line, we find the quotient of the vertical distance and horizontal distance between the given point and another known point on the line. If this quotient equals the slope of the line, then the point lies on the line.