Algebra Know-It-ALL

We Renamed, We Replaced, We Can Graph

In Chap. 16, we solved the following two-by-two linear system for v and w using the rename-

and-replace scheme, also called the substitution method:

− 7 v+w+ 10 = 0

and

4 v+ 8 w=− 40

We found that v= 2/3 and w=−16/3. Let’s graph this system, calling v the independent vari-

able and w the dependent variable.

Find two points for each line

Our first step is to get both of the original equations into SI form with w as the dependent

variable. For the first equation:

− 7 v+w+ 10 = 0

w+ 10 = 7 v

w= 7 v− 10

This tells us that the w-intercept for one of the lines is −10, so we can plot the point (0, −10)

on the Cartesian plane. For the second equation:

4 v+ 8 w=− 40

8 w=− 4 v− 40

w= (−4/8)v− 40/8

w= (−1/2)v− 5

Thew-intercept for the other line is −5, so we can plot (0, −5) on the coordinate plane. We

know that the lines intersect at the solution point where v= 2/3 and w=−16/3, so we can

plot (2/3, −16/3). These three known points are plotted in Fig. 17-5.

Connect the points

One line passes through (0, −10) and (2/3, −16/3). The other line passes through (0, −5) and

(2/3,−16/3). The points here are badly “bunched up.” Let’s find new points on both lines so

we can obtain accurate graphs. Look again at the SI equation

w= 7 v− 10

The slope of the line is 7. It ramps steeply upward as we move to the right. If we go 1 unit to

the right, for example, we’ll go 7 units upward. If we go 2 units to the right, we’ll go 14 units

We Renamed, We Replaced, We Can Graph 271