Algebra Know-It-ALL

Suppose we move from (0, b) to some point (x,y) on the line by going Δx units to the
right and Δy units upward. The x coordinate of the point (x,y) will be 0 +Δx, because we have
moved Δx units horizontally from a point where x= 0. The y coordinate of the point (x,y) will
beb+Δy, because we have moved Δy units vertically from a point where y=b.
If we can get an equation that allows us to calculate y in terms of x for the arbitrary point
(x,y), then we will have demonstrated how y is a function of x. As things turn out, we’ll also
get the SI form of the equation for the line.
We can express Δy in terms of the slope m and the increment Δx by morphing the formula
that defines slope. That formula, once again, is

m=Δy/Δx

Multiplying through by Δx, we get

mΔ x=Δy

Now remember that

y=b+Δy

We can substitute mΔx for Δy in this equation, getting

y=b+mΔ x

+x

+y

• x
  
  
  • y
  
  
  
  

y

x

(0,b)

x= 0 + x (x,y)

y=b+ y

m= y x

Figure 15-6 The SI form of a linear equation can be derived

from this generic graph.

Equations from Graphs 245