Last week, in Number Sense 32, we took two linear equations and plotted them on a graph, and discovered that the intersection of the two lines gave us a solution that worked for both equations. This is a very powerful idea.
Neither of the linear equations had a single solution. We could pick many different values for the amount of liquid milk, and then use one of the linear equations to calculate how much we had left to make cheese, and still not know how much was actually used.
If we used the other equation, again, we could pick many different values for the amount of liquid milk, and calculate the amount of milk used for cheese, and still not know how much was actually used.
But! when we combined those two equations, there was only one combination of liquid milk and cheese milk that worked for both. There was AN ANSWER! A single, gotta be this, honest to god answer, just like the answers we got in arithmetic when we were asked to add 3 plus 6.
But not all situations are linear. Sometimes equations are based on the square of an unknown number:
If we made a t chart it would look something like this
If we examine this t-chart (an expanded version, with more than two columns) we notice that x squared is always a positive number. This means that it doesn't matter whether x is a positive or negative number, y will come out the same. We can see this work out for x = 1 and x = -1. When we look at x = -2, it is easy to see that if x were +2, the rest of the row would have identical numbers, and y would also be 0 when x = +2.
When plotted, this equation, using x squared, gives a curved line, rather than a straight one. It is called a parabola, and it is found quite frequently in the real world.
Parabolas describe the path of a bouncing ball. Mirrors shaped like parabolas focus sunlight to a point, they can also be used to focus radio waves, so satellite dishes also use this shape. Water squirted from a garden hose follows a parabolic path. Suspension bridge cables are close to parabolas.
For graphical problem solving, though, parabolas (and other curves) have a very interesting property. We can draw a line (from a linear equation, perhaps) that intersects a parabola in two places. We might find a problem that has two valid solutions! We can also imagine a line crossing the graph that doesn't touch the parabola at all. We could prove, by this method, that a particular problem has no solution at all.
Sometimes this makes people frustrated. Problems with no solutions. Arithmetic problems always had answers. But algebra, we can come up with problems that have no solutions, and, not only that, prove they have no solutions? Frustrating, but useful. If we know, for a mathematical fact, there is no solution, we can stop looking for answers to that particular problem, and focus our energy on other things.
The goats would approve. Have fun in the comments.