Ray's New Elementary Algebra · Arts. 173–177 · Unit 5: Chapter V: Supplement to simple equations
36. The couriers; indeterminate and impossible problems
Goal: Your student follows the Problem of the Couriers through its cases and can tell when a set of equations has no answer, one answer, or many.
You'll need
- The book, open to the page
- Pencil and plenty of paper (or a whiteboard)
- A card to cover printed answers
What's in the book
The Problem of the Couriers in two cases and several suppositions; independent and indeterminate equations; redundant and incompatible conditions; an impossible problem; and the proof that a simple equation has one root. A general review follows.
The lesson, step by step
Warm-up
Ask: Two people 10 miles apart walk toward each other at 2 and 3 miles an hour. Where do they meet? (4 miles from the first.)
The couriers, toward each other
Read Case I. The meeting point is x = am/(m + n). Check it with the warm-up: a = 10, m = 2, n = 3 gives 4.
Same direction
Read Case II: x = am/(m − n). Discuss the suppositions: if m = n the answer is infinite, since the one never catches the other; if a = 0 they are already together.
Indeterminate
Read Arts. 174 and 175.
When the number of unknown quantities exceeds the number of independent equations, the problem is indeterminate.
Example: x − y = 5 has endless answers (6 and 1, 7 and 2, ...).
Redundant and impossible
Read Art. 176. Solve x + y = 9, x + 2y = 13, 2x + 3y = 21 two at a time and see the answers disagree. Then the impossible question on p. 153 gives 0 = 72.
One root only
Read Art. 177. Say: Any simple equation can be written mx = n, and n divided by m has only one answer.
Wrap-up
Use the General Review on p. 176 for the first half of its questions (through the Couriers).
Tip
Act out the couriers with two toy cars on a ruler. It makes the infinite and zero cases easy to see.