Chapter 3
Systems of Linear Equations
Someone is thinking of two numbers and tells you they add to twenty. That is not much help: one and nineteen fit, seven and thirteen fit, and so do endlessly many other pairs. Then comes a second clue about those very same numbers, and the field of candidates suddenly narrows. Equations that must hold at the same time like this are called a system, and they turn up whenever a situation states more than one fact at once. Neither equation settles anything alone, so what can two of them do together that one cannot?
What You'll Explore
5 lessons.
- Equations with Two Variables
One equation, two unknown letters. Which pairs of numbers make it true, how many of them are there, and is there any order to them? You will hunt for solutions to a single equation in two variables, then see what all of those pairs look like once they are drawn on a grid.
- Solving Systems by Substitution
Estimating where two equations agree is easy on a graph, but a drawing will never tell you whether the answer is two thirds or seven tenths. Is there a way to reach the exact pair using only algebra, and what should you conclude when the hunt produces no pair at all?
- Solving Systems by Elimination
Adding two equations together sounds like nonsense, and yet both of them are true at the same moment. Does the sum inherit that truth, and can the addition be set up beforehand so that one of the unknowns simply does not appear in it?
- Word Problems with Systems
Nobody hands you a pair of equations in real life. Coins in a jar, two cars setting off at different times, a stall working out when it stops losing money: each arrives as a paragraph of English. The interesting question is how a paragraph becomes two equations, and how you know which two facts it is really giving you.
- Systems with More Variables
Three unknowns, three facts. Does a bigger system need a bigger idea, or can it be shrunk down to the kind of problem already behind you? The picture changes as well, and it is worth asking where a solution in three letters can live, once a flat page is no longer big enough to hold it.