Algebra Know-It-ALL

As before, we have

mΔx=Δy

Observe that in Fig. 15-7,

y=y 0 +Δy

Let’s substitute mΔx for Δy here. That gives us

y=y 0 +mΔx

Now we can see from Fig. 15-7 that

x=x 0 +Δx

Subtracting x 0 from each side, we obtain

x − x 0 =Δx

Substituting (x − x 0 ) for Δx in the equation for y in terms of y 0 and mΔx, we get

y=y 0 +m(x − x 0 )

Subtracting y 0 from each side gets us to the PS form

y − y 0 =m(x − x 0 )

Are you confused?

The preceding two examples show lines with positive slope. There’s good reason to wonder, “What
happens when the slope of the line is negative?” The answer is, “Nothing special, as long as we’re
careful.”
Think of what happens when we move to the right in the Cartesian plane. If the slope of a line is posi-
tive, then Δy is always positive as we move to the right. If the slope is negative, then Δy is always negative
as we move to the right. Either way, we add Δy when we move to the right. If we keep adding a positive Δy,
we go higher and higher. If we keep adding a negative Δy, we go lower and lower. In this context, “higher”
means “in the positive y direction,” and “lower” means “in the negative y direction.”
Similar sign-related confusion can occur when we work with the SI form of a linear equation. The
standard form, once again, is

y=mx+b

This equation has a plus sign whether b is positive or negative. If m= 3 and b=−2, for example,

y= 3 x+ (−2)

Equations from Graphs 247