- Let’s state the two equations again for reference, and then try to solve them using
double elimination:
2 x+y= 3
and
6 x+ 3 y= 12
Let’s eliminate x. We can multiply the first equation through by −3 to get
−6x− 3y =− 9
Here’s what happens when we add this to the second original equation:
− 6 x− 3 y=− 9
6 x+ 3 y= 12
⎯⎯⎯⎯⎯
0 = 3
That’s nonsense! No matter what other method we use in an attempt to solve this system,
we’ll arrive at some sort of contradiction. When this happens with a two-by-two linear
system, the system is said to be inconsistent. (Most two-by-two linear systems are consis-
tent, meaning that they have a single solution that can be expressed as an ordered pair.)
Nothing is technically wrong with either equation here. They simply don’t get along
together. Inconsistent linear systems have no solutions.
- Let’s put the two equations from Prob. 5 into SI form, and see if that tells us anything
about what their graphs look like. First, this:
2 x+y= 3
When we subtract 2x from each side, we get
y=− 2 x+ 3
This indicates that the slope of the graph, which is a straight line, is −2. The y-intercept
is 3. Now for the second equation:
6 x+ 3 y= 12
When we subtract 6x from each side, we get
3 y=− 6 x+ 12
We can divide through by 3 to obtain
y=− 2 x+ 4
Chapter 16 641
