Algebra Know-It-ALL

Are you confused?

Suppose the graphs of a two-by-two linear system show up as parallel, vertical lines, both with undefined
slope. If you call x the independent variable and y the dependent variable, then neither of the equations
is a function of x, although they are both relations between x and y. This sort of system can always be
reduced to the form

x=a

and

x=b

where a and b are constants, and a≠b.
If the graphs of a two-by-two linear system show up as parallel, horizontal lines in the same coordinate
system, then they both have slopes of 0, and they both represent functions. Such a system can always be
reduced to the form

y=c

and

y=d

where c and d are constants, and c≠d.

Here’s a challenge!

In exercise 10 at the end of Chap. 16, we tried to solve the following pair of equations as a linear system
by substitution, but failed when we got the meaningless result 0 = 0 :

s= 2 r− 3

and

− 10 r+ 5 s+ 15 = 0

We showed that these equations are equivalent to a single function of r. State and graph that function,
lettingr be the independent variable and s be the dependent variable. Label a few of the infinitely many
ordered pairs that satisfy the system.

Solution

The function can be stated as the first equation above. It tells us that the s-intercept is −3, so we can plot
(0,−3) on the Cartesian plane. Additional points can be found by moving to the right or left from (0, −3)
and moving upward or downward by twice that distance. The slope is 2, so if we start at any point on
the line and move to the right by 1 unit (add 1 to r), we must move upward by 2 units (add 2 to s) to
stay on the line. If we move to the left by 1 unit (subtract 1 from r), we must move downward by 2 units

We Couldn’t Solve, but We Can Graph 277