246 Graphs of Linear Relations
But in this situation, Δx is exactly equal to x! That’s because, by traversing the increment Δx,
we have moved from the y axis (where x= 0) horizontally by x units. Because of this lucky
coincidence, we can substitute x for Δx in the above equation, getting
y=b+mx
If we want to be picayune, we can reverse the order of the addends to state it as
y=mx+b
Known point and slope
Imagine a line in the Cartesian plane that passes through a point whose coordinates are (x 0 y 0 ),
where x 0 and y 0 are known constants. Suppose the line has slope m as shown in Fig. 15-7.
If we move away from (x 0 , y 0 ) along the line, the slope is always equal to Δy/Δx.
Let’s go from (x 0 ,y 0 ) to some arbitrary point (x,y) on the line, just as we did when we
derived the SI equation. The x coordinate of (x,y) will be x 0 +Δx, because we have moved
Δx units horizontally from a point where x=x 0. The y coordinate of the point (x,y) will be
y 0 +Δy, because we have moved Δy units vertically from a point where y=y 0. Now remember,
once again, how slope is defined:
m=Δy/Δx
+x
+y
- x
- y
y
x
(x,y)
m= y x
(x 0 ,y 0 )
= + xx
y= + y
x 0
y 0
Figure 15-7 The PS form of a linear equation can be derived
from this generic graph.