THE COMPOUND // FIELD NOTES // WEEK 23
How lines interact, nonlinear systems, and the slope formula
Field Book
Lessons 84 to 87
Due
Test 20 due ยท Supplement 23
Method
Paper first, every step
Two lines can only do three things: cross once, never meet, or be the same line. Name which before you solve.
What you have to be able to do
- Classify a linear system by how its lines interact
- Solve a nonlinear system by substitution
- Use the axioms of inequality and the slope formula
The procedures
Three interactionsField Book Lesson 84
- Crossing lines give one solution. Parallel lines give none. The same line twice gives infinitely many.
- Compare slopes and intercepts to classify before solving.
Nonlinear systemsField Book Lesson 85
- Solve the linear equation for one variable and substitute into the curve.
- Expect up to two intersection points: a line can cut a circle twice.
Order axiomsField Book Lesson 86
- Trichotomy: for any two numbers, exactly one of less, equal, greater holds.
- Transitivity chains comparisons. Irrational roots come in conjugate pairs.
The slope formulaField Book Lesson 87
- m = y2 - y1x2 - x1: rise over run from two points.
- Keep the subtraction order matched top and bottom.
Worked: slope from two points
- Find the slope through (1, 2) and (4, 8).
- m = 8 - 24 - 1 = 63.
- m = 2.
Terms on the record
| Term | Meaning |
|---|
| trichotomy axiom | exactly one of less than, equal, greater than holds for any pair |
| nonlinear system | a system where at least one equation graphs as a curve |
| slope formula | the change in y over the change in x between two points |