Last week, in Number Sense 029 we added and subtracted nothing from our equations, and found that to be a useful technique.
Up until now, we have been dealing with one unknown, one goat, and we have come up with methods for finding the value of that goat. But what do we do when we have more than one unknown?
Real problems are considerably messier than the simple problems we set up to teach basic ideas, simple problems are rarely encountered in the real world, or, when they are, can be solved with the most simple mechanical methods.
Have you ever played cards? When you dealt the cards, did you think your were dividing a certain number of cards by the number of players? Probably not. You just dealt the cards, one at a time, to each player in turn. A simple real world mathematical problem, solved so simply that few people even think they are doing math when they solve that problem.
This week we will begin to look at problems where there are two unknown quantities, somehow related to each other.
When we first took a look at numbers, we placed them on a number line.
Here the red arrow represents some unknown number. We know it is a positive number, because it is pointing to the right, and our convention says pointing right means positive. But we won't know it's actual value until we find out what the unit length is, put the tick marks in place, and find the numerical length of the arrow.
Now, it is true that we can put a second unknown on the same number line
But these two unknowns are not related in any way, other than they both are numbers of some kind.
Suppose the numbers are related, though. Perhaps one number is how many goats there are, and the other number is how much pasture you need for the goats to get enough to eat. Your local Agricultural Extension Agent tells you a goat needs 1/2 acre of grassland to forage for food (y is the number of goats, x is the acres of grassland)
and, you'll need to harvest hay from an additional 2 acres to give the goats a treat when they come home at the end of the day.
So, this is how these two unknowns are related. If we decide to keep a certain number of goats, we can figure out how much land we will need. If we already have a certain amount of land, we can figure out how many goats it will support. The number of goats depends on the amount of land, and the amount of land depends on the number of goats.
Two goats require three acres
Four goats require four acres
Six goats require five acres
There is no way to usefully represent these numbers on a number line. Sure, we could put them all on a number line, but it would not show the relationship.
We cannot tell which green arrows (grassland acres) are connected to which red arrows (size of goat herd) from looking at this number line. And this only shows three possible herds. What if we had three goats, or seven, or ten? The number line is already a jumbled mess, adding more arrows would only make it worse.
This problem has been around for a long time, of course, and it was solved many years ago, by French philosopher and mathematician Rene Descartes. Back in his day, works were published in Latin, and authors used “Latin” pen names, Descartes' pen name was Renatus Cartesius. His philosophical ideas and mathematical innovations are known, therefore, as “Cartesian.”
The Cartesian solution to this problem of two related unknown numbers is simple: use two number lines. Descartes brilliant innovation was to place the second number line at right angles to the first
Each pair of numbers, 2 and 3, 4 and 4, 6 and 5, identifies a unique spot. Descartes also noticed that connecting these dots formed a line
and showed that all solutions to the problem were on that line.
Some caution is needed, though. Just because all solutions are on that line, does not mean that everything on that line is a solution. One of the spots on the line corresponds to zero goats and two acres of grassland. Obviously we do not need two acres of grassland if we are keeping no goats.
There are places on the line corresponding to 2 1/2 goats, or 1 3/4 goats. These likewise are not possible solutions: half a goat eats no grass at all, because half a goat would be in the meat locker waiting to become goat stew.
Have fun in the comments.