Algebra Know-It-ALL

Chapter 15

1. We have these two ordered pairs defining the points P and Q, respectively:

P= (u 1 ,v 1 )= (−1,−6)

and

Q= (u 2 ,v 2 )= (2, 2)

The slope m is equal to the difference in the dependent-variable coordinates divided by

the difference in the independent-variable coordinates, or Δv/Δu. If we move along the

line from P to Q, we find the slope on the basis of the ratio between the differences in the

point values with the “destination” values listed first:

m= (v 2 −v 1 ) / (u 2 −u 1 )

Plugging in the values v 2 = 2, v 1 =−6,u 2 = 2, and u 1 =−1, we get

m= [2 − (−6)] / [2 − (−1)]

= (2 + 6) / (2 + 1)

= 8/3

2. We still have the same two points, defined by the same two ordered pairs. Points P and Q,
   respectively, are still defined by

P= (u 1 ,v 1 )= (−1,−6)

and

Q= (u 2 ,v 2 )= (2, 2)

If we want to go from Q to P rather than from P to Q, we must reverse the order of v 1 and

v 2 in the numerator of the slope equation, and we must also reverse the order of u 1 and u 2

in the denominator. When we do that, we get

m= (v 1 −v 2 ) / (u 1 −u 2 )

= (− 6 − 2) / (− 1 − 2)

=−8 / (−3)

= 8/3

The slope in either direction is equal to the difference in the v values divided by the differ-

ence in the u values, or Δv/Δu. Reversing the direction in which we move along the line

simply multiplies both Δv and Δu by −1. The ratio turns out the same either way.

Chapter 15 633