You mean that I really put an example on this page where the "method" will not work? Well, I didn't want to, but out of fairness (and to keep you, the teacher, from getting hung out to dry in class) I felt I should warn you of a major pitfall.
On one of the previous worksheets, I mentioned a type of equation in standard form that went beyond "hard." The term I used was "impossible." I probably should've used the term "impossible to avoid fractions."
Maybe if you've tinkered with this method enough, you've already found the snag. But, in case you haven't, here we go.
In the hard equations, like 4x + 6y = 14, we never had a problem finding a solution that had integer terms for the x and y coordinates. If you were going to ever get clued in on the problem we are about to face, it was about right here.
What would happen if we had an equation like 4x + 6y = 15? It looks about the same as the last equation we saw, except for that "14" became a "15." If we go through the same method we used in the "hard" equations, we crank out the multiples of 4 and 6 and look for a pair that would somehow wind up at 15. The multiples are shown below:
When we try to pair up the multiples (both positives and negatives) and add or subtract, we see a common theme developing: You can't add or subtract even integers and expect to get an odd integer.
What are we adding? Evens.
What do we need to get? An odd.....Problem.
This problem can be done, but you are now locked into finding fractional
points on the line. The easiest route to take now that you have found this
equation to be "impossible," may be to make the equation like
an "easy." Set one of the variables (say x) equal to zero, suck
it up, and get a fractional y-coordinate. In this case if x = 0, one of
the points on the line will be
(0, 5/2). Repeating the process for setting y = 0 leaves us with a coordinate
of (15/4, 0).