Algebra Know-It-ALL

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.